Cedar Crest College Statistics 110
Lecture Notes
Author: E-Mail Address: James Hammer [email protected]
June 23, 2015 Preface
This is meant as a teaching aid. It can be freely distributed and edited in any way. For a copy of the LATEX document, please email the author.
ii Contents
1 Nature of Probability and Statistics1 1.1 Descriptive and Inferential Statistics...... 1 1.2 Variables and Types of Data...... 3 1.3 Data Collection and Sampling Techniques...... 5 1.4 Experimental Design...... 6
2 Frequency Distribution and Graphs7 2.1 Organizing Data...... 7 2.2 Histograms, Frequency Polygons, and Ogives...... 10 2.3 Other Types of Graphs...... 12
3 Data Description 19 3.1 Measures of Central Tendencies...... 19 3.2 Measures of Variation...... 26 3.3 Measures of Position...... 33 3.4 Exploratory Data Analysis...... 38
4 Probability and Counting Rules 41 4.1 Sample Spaces and Probability...... 41 4.2 The Addition Rules for Probability...... 47 4.3 The Multiplication Rules & Conditional Probability...... 49 4.4 Counting Rules...... 56 4.5 Probability and Counting Rules...... 62
5 Discrete Probability Distributions 65 5.1 Probability Distributions...... 65 5.2 Mean, Variance, Standard Deviation, and Expectation...... 67 5.3 The Binomial Distribution...... 72
6 The Normal Distribution 75 6.1 Normal Distributions...... 75 6.2 Applications of the Normal Distribution...... 77 6.3 The Central Limit Theorem...... 80
iii 7 Confidence Intervals and Sample Size 83 7.1 Confidence Intervals for the Mean Shen σ is Known...... 83 7.2 Confidence Intervals for the Mean When σ is Unknown...... 87 7.3 Confidence Intervals and Sample Size for Populations...... 89
8 Hypothesis Testing 93 8.1 Traditional Method...... 93 8.2 z Test for a Mean...... 95 8.3 t Test for a Mean...... 98 8.4 z Test for a Proportion...... 100
10 Correlation and Regression 103 10.1 Scatter Plots and Correlation...... 103
iv Chapter 1
Nature of Probability and Statistics
1.1 Descriptive and Inferential Statistics
Definition 1 (Statistics). Statistics is the science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data.
Definition 2 (Variable). A variable is a characteristic of attribute that can assume different values.
Definition 3 (Data). The data are the values (measurements or observations) that the variables can assume. Variables whose values are determined by chance are called random variables.
Definition 4 (Population). A population consists of all subjects (human or otherwise) that are being studied.
Definition 5 (Sample). A sample is a group of subjects selected from a population.
Definition 6 (Descriptive Statistics). Descriptive statistics consists of the collection, organization, summarization, and presentation of data.
Definition 7 (Inferential Statistics). Inferential statistics consists of generalizing from sample to populations, performing estimations and hypothesis tests, determining relationship among variables and making predictions.
Definition 8 (Probability). Probability is the measure of the likeliness that an event will occur. Probability is quantified as a number between 0 and 1. The higher the probability of an event, the more likely the event will occur.
Definition 9 (Hypothesis Testing). The area of inferential statistics called hypothesis testing is a decision-making process for evaluating claims about a population based on information obtained from samples.
1 Hammer 2 Statistics 110 Lecture Notes
Problem 10. Determine whether descriptive statistics or inferential statistics were used. Explain why.
a. The average jackpot for the top five lottery winners was $367.6 million.
b. A study done by the American Academy of Neurology suggested that older people who had a high caloric diet more than doubled their risk of memory loss.
c. Based on a survey of 9317 consumers done by the National Retail Federation, the average amount that consumers spent on Valentines Day in 2011 was $116.
d. Scientists at the University of Oxford in England found that a good laugh significantly raises a person’s pain level tolerance.
Suggested Problems: 1-19.
Cedar Crest College June 23, 2015 1.2. Variables and Types of Data Hammer 3
1.2 Variables and Types of Data
Definition 11 (Qualitative Variables). Qualitative variables are variables that have dis- tinct categories according to some characteristic or attribute.
Definition 12 (Quantitative Variables). Quantitative variables are variables that can be counted or measured.
Definition 13 (Discrete Variables). Discrete variables assume values that can be counted.
Definition 14 (Continuous Variables). Continuous variables can assume an infinite num- ber of values between any two specific values. They are obtained by measuring. They often include fractions and decimals.
Problem 15. Classify each of the variables as a discrete variable or as a continuous variable.
a. The highest wind speed of a hurricane.
b. The weight of baggage on an airplane.
c. The number of pages in these lecture notes.
d. The amount of money a person spends per year for online purchases.
June 23, 2015 Cedar Crest College Hammer 4 Statistics 110 Lecture Notes
Definition 16 (Boundary). The boundary of a number is defined as a class in which a data value would be placed before the data values are rounded.
Problem 17. Find the boundaries of each variable.
a.8 .4 quarts
b. 138 mmHg
c. 137.63 mg/dL
Definition 18 (Nominal Level of Measurement). The nominal level of measurement classifies data into mutually exclusive (nonoverlapping) categories in which no order or rank- ing can be imposed on the data.
Definition 19 (Ordinal Level of Measurement). The ordinal level of measurement clas- sifies data into categories that can be ranked; however, precise differences between the ranks do not exist.
Definition 20 (Interval Level of Measurement). The interval level of measurement ranks data, and precise differences between units of measure do exist; however, there is no meaningful zero.
Definition 21 (Ratio Level of Measurement). The ratio level of measurement possesses all of the characteristics of interval measurement, and there exists a true zero. In addi- tion, true ratios exist when the same variable is measured on two different members of the population.
Problem 22. What level of measurement would be used to measure each variable?
a. The age of patients in a local hospital.
b. The rating of movies released this month.
c. Colors of athletic shirts sold by the Iron Pigs.
d. Temperatures of hot tubs in local health clubs.
Suggested Problems: 5 − 22
Cedar Crest College June 23, 2015 1.3. Data Collection and Sampling Techniques Hammer 5
1.3 Data Collection and Sampling Techniques
Definition 23 (Random Sample). A random Sample is a sample in which all members of the population have an equal chance of being selected.
Definition 24 (Systematic Sample). A systematic sample is a sample obtained by select- ing every kth member of the population.
Definition 25 (Stratified Sample). A stratified sample is a sample obtained by dividing the population into subgroups or strata according to some characteristic relevant to the study. The subjects are selected from each subgroup.
Definition 26 (Cluster Sample). A cluster sample is obtained by dividing the population into sections or clusters and then selecting one or more clusters and using all members in the cluster(s) as the members of the sample.
Definition 27 (Sampling Error). Sampling error is the difference between the results obtained from a sample and the results obtained from the population from which the sample was selected.
Definition 28 (Nonsampling Error). A nonsampling error occurs when the data are obtained erroneously or the sample is biased (non-representative).
Problem 29. State which sampling method was used in the following:
a. Out of 10 hospitals in the municipality, a researcher selects one and collects records for a 24-hour period on the types of emergencies that were treated here.
b. A researcher divides a group of students according to gender; major field; and low, average, and high grade point average. Then she randomly selects six students from each group to answer questions in the survey.
c. The subscribers to a magazine are numbered. Then a sample of these people is selected using random numbers.
d. Every 10th bottle of Super-Duper Cola is selected, and the amount of liquid in the bottle is measured. The purpose is to see if the machines that fill the bottles are working properly.
Suggested Problems: 11 − 16
June 23, 2015 Cedar Crest College Hammer 6 Statistics 110 Lecture Notes
1.4 Experimental Design
Definition 30 (Observational Study). In an observational study, the researcher merely observes what is happening or what has happened in the past and tries to draw conclusions based on these observations.
Definition 31 (Experimental Study). In an experimental study, the researcher manip- ulates one of the variables and tries to determine how the manipulation influences other variables.
Definition 32 (Variable Types). The independent variable in an experimental study is the one that is being manipulated by the researcher. The independent variable is also called the explanatory variable. The resultant variable is called the dependent variable or the outcome variable.
Definition 33 (Confounding Variables). A confounding variable or sometimes called a nuisance variable is one that influences the dependent or outcome variables but was not separated from the independent variable.
Problem 34. Researchers randomly assign 10 people to each of three different groups. Group 1 was instructed to write an essay about the hassles in their lives. Group 2 was instructed to write an essay about circumstances that made them feel thankful. Group 3 was asked to write an essay about events that they felt neutral about. After the exercise, they were given a questionnaire on their outlook on life. The researchers found that those who wrote about circumstances that made them feel thankful had a more optimistic outlook on life. The conclusion is that focusing on the positive makes you more optimistic about life in general. Based on this study, answer the following questions:
a. Was this an observational or an experimental study?
b. What is the independent variable?
c. What is the dependent variable?
d. What may be a confounding variable in this study?
e. What can you say about the sample size?
f. Do you agree with the conclusion? Please explain your answer.
Suggested Problems: 19 − 31, 37 − 42
Cedar Crest College June 23, 2015 Chapter 2
Frequency Distribution and Graphs
2.1 Organizing Data
Definition 35 (Raw Data). When the data are in original form, they are called raw data.
Definition 36 (Frequency Distribution). A frequency distribution is the organization of raw data in table form, using classes and frequencies.
Problem 37. Twenty-five army inductees were given a blood test to determine their blood type. The data set is: A B B AB O O O B AB B BBOAO A O O O AB AB A O B A. Construct a frequency distribution for the data.
7 Hammer 8 Statistics 110 Lecture Notes
Definition 38 (Lower Class Limit). The lower class limit is the smallest data value that can be included in the class.
Definition 39 (Upper Class Limit). The upper class limit is the largest data value that can be included in the class.
Definition 40 (Class Boundaries). The class boundaries are ranges used to separate the classes so that there are no gaps in the frequency distribution.
Definition 41 (Class Width). The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class limit of the next class.
Definition 42 (Class Midpoint). The class midpoint Xm is obtained by adding the lower and upper boundaries and dividing by two or adding the lower and upper limits and dividing my two.
Definition 43 (Open-Ended Distribution). An open-ended distribution is a frequency distribution where either the first class has no lower bound or the last class has no upper bound.
Problem 44. These data represent the record high temperatures in degrees Fahrenheit for each of the 50 states. Construct a grouped frequency distribution for the data, using 7 classes. 112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 105 109 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114
Cedar Crest College June 23, 2015 2.1. Organizing Data Hammer 9
Definition 45 (Cumulative Frequency Distribution). A cumulative frequency distribu- tion is a distribution that shows the number of data values less than or equal to a specific value (usually an upper boundary).
Definition 46 (Ungrouped Frequency Distribution). An ungrouped frequency distribu- tion is a distribution where the range of data values is relatively small; therefore, a frequency distribution can be constructed using single data values for each class.
Problem 47. The data shown here represent the number of miles per gallon (mpg) that 30 selected four-wheel-drive sports utility vehicles obtained in city driving. Construct a frequency distribution, and analyze the distribution.
12 17 12 14 16 18 16 18 12 16 17 15 15 16 12 15 16 16 12 14 15 12 15 15 19 13 16 18 16 14
Suggested Problems: 5 − 8, 16 − 26
June 23, 2015 Cedar Crest College Hammer 10 Statistics 110 Lecture Notes
2.2 Histograms, Frequency Polygons, and Ogives
Definition 48 (Histogram). A histogram is a graph that displays the data by using con- tiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.
Definition 49 (Frequency Polygon). The frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the points.
Definition 50 (Ogive). The ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution
Definition 51 (Lower Class Limit). The lower class limit represents the smallest data value that can be included in the class.
Definition 52 (Upper Class Limit). The upper class limit represents the largest data value that can be included in the class.
Definition 53 (Class Midpoint). The class midpoint, Xm, is obtained by adding the lower and the upper boundaries and dividing by 2.
Problem 54. Construct a histogram, frequency polygon, and an ogive to represent the data shown for the record high temperatures for each of the 50 states.
Class Bondaries Frequency 99.5 − 104.5 2 104.5 − 109.5 8 109.5 − 114.5 18 114.5 − 119.5 13 119.5 − 124.5 7 124.5 − 129.5 1 129.5 − 134.5 1
Cedar Crest College June 23, 2015 2.2. Histograms, Frequency Polygons, and Ogives Hammer 11
Definition 55 (Relative Frequency Graphs). Histograms, frequency polygons, and ogives can be constructed from raw data as done above or by using proportions. When proportions are used instead of raw data, the graphs are called relative frequency graphs.
Problem 56. Construct a histogram, frequency polygon, and ogive using relative frequencies for the distribution of miles that 20 randomly selected runners ran during a given week.
Class Bondaries Frequency 5.5 − 10.5 1 10.5 − 15.5 2 15.5 − 20.5 3 20.5 − 25.5 5 25.5 − 30.5 4 30.5 − 35.5 3 35.5 − 40.5 2
Suggested Problems: 1 − 11
June 23, 2015 Cedar Crest College Hammer 12 Statistics 110 Lecture Notes
2.3 Other Types of Graphs
Definition 57 (Bar Graph). A bar graph represents the data by using vertical or horizontal bars whose heights or lengths represent the frequencies of the data.
Problem 58. The table shows the average money spent by first-year college students. Draw a horizontal and vertical bar graph for the data.
Electronics $728 Dorm Decor $344 Clothing $141 Shoes $72
Cedar Crest College June 23, 2015 2.3. Other Types of Graphs Hammer 13
Definition 59 (Pareto Chart). A pareto chart is used to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest.
Problem 60. The data shown here consists of the number of police calls for specific cate- gories that a local municipality received in 2011. Draw and analyze a pareto chart for the data. Category Number Juvenile complaint 92 Loud noise/music/party 27 Drug offenses 79 Driving under the influence 38 Disabled vehicle 65
June 23, 2015 Cedar Crest College Hammer 14 Statistics 110 Lecture Notes
Definition 61 (Time Series Graph). A time series graph represents data that occur over a specific period of time.
Definition 62 (Compound Time Series Graph). When two or more data sets are compared on the same graph, the graph is called a compound time series graph.
Problem 63. The data show in the percentage of U.S. adults who smoke. Draw and analyze a time series graph for the data.
Year 1970 1980 1990 2000 2010 Percent 37 33 25 23 19
Cedar Crest College June 23, 2015 2.3. Other Types of Graphs Hammer 15
Definition 64 (Pie Graph). A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.
Problem 65. This frequency distribution shows the number of pounds of each snack food eaten during the Super Bowl. Construct a pie graph for the data.
Snack Pounds (frequency) Potato Chips 11.2million Tortilla Chips 8.2million Pretzels 4.3million Popcorn 3.8million Snack Nuts 2.5million
June 23, 2015 Cedar Crest College Hammer 16 Statistics 110 Lecture Notes
Definition 66 (Dotplot). A dotplot is a statistical graph in which each data value is plotted as a point (dot) above the horizontal axis.
Problem 67. The data show the number of named storms each year for the last 40 years. Construct and analyze a dotplot for the data.
19 15 14 7 6 11 11 9 16 8 8 11 9 8 16 12 13 14 13 12 7 15 15 19 11 4 6 13 10 15 7 12 6 10 28 12 8 7 12 9
Cedar Crest College June 23, 2015 2.3. Other Types of Graphs Hammer 17
Definition 68 (Stem and Leaf Plot). A stem and leaf plot is a data plot that uses part of the data values as the stem and part of the data values as the leaf to form groups or classes.
Problem 69. At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data.
25 31 20 32 13 14 43 02 57 23 36 32 33 32 44 32 52 44 51 45
Problem 70. The number of stories in two selected samples of tall buildings in Atlanta and Philadelphia is shown. Construct a back to back stem and leaf plot and compare the distributions. Atlanta Philadelphia 55 70 44 36 40 61 40 38 32 30 63 40 44 34 38 58 40 40 25 30 60 47 52 32 32 54 40 36 30 30 50 53 32 28 31 53 39 36 34 33 52 32 34 32 50 50 38 36 39 32 26 29
Suggested Problems: 1 − 20
June 23, 2015 Cedar Crest College Hammer 18 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 3
Data Description
3.1 Measures of Central Tendencies
Definition 71 (Statistic). A Statistic is a characteristic or measure obtained by using the data values from a sample
Definition 72 (Parameter). A parameter is a characteristic or measure obtained by using all the data values from a specific population.
Definition 73 (Mean). The mean is the sum of the values, divided by the total number of values. This is also called the arithmetic average.
Definition 74 (Sample Mean). Let n denote the number of data points. The sample mean, X, is calculated by using the sample data. The sample mean is a statistic.
Pn X X = i=1 i . n In a frequency distribution, the sample mean can be found by adding up the product of the frequencies of each group, fi, with the midpoint of each group, Xm, and then dividing by the number of participants. That is to say
#groups P f · X X = i=1 i m . n Definition 75 (Population Mean). let N denote the total number of values in the popula- tion. The population mean, µ, is calculated by using all of the values in the population. The population mean is a parameter.
PN X µ = i=1 i . N
19 Hammer 20 Statistics 110 Lecture Notes
Problem 76. The number of calls that a local police department responded to for a sample of 9 months is shown. Find the mean.
475, 447, 440, 761, 993, 1052, 783, 671, 621.
Problem 77. The data show the number of patients in a sample of six hospitals who acquired an infection while hospitalized. Find the mean.
110, 76, 29, 38, 105, 31.
Problem 78. Using the following frequency distribution, find the mean. The data represent the number of miles run during one week for a sample of 20 runners.
Class Bondaries Frequency 5.5 − 10.5 1 10.5 − 15.5 2 15.5 − 20.5 3 20.5 − 25.5 5 25.5 − 30.5 4 30.5 − 35.5 3 35.5 − 40.5 2
Cedar Crest College June 23, 2015 3.1. Measures of Central Tendencies Hammer 21
Definition 79 (Median). The median, MD, is the midpoint of the data array. Let n be the number of data points. If n is odd, select the middle data value as the median. If n is even, find the mean of the two middle values.
Problem 80. The number of police officers killed in the line of duty over the last 11 years is shown. Find the median.
177, 153, 122, 141, 189, 155, 162, 165, 149, 157, 240.
Problem 81. The number of tornadoes that have occurred in the U.S. over an 8-year period follows. Find the median.
684, 764, 656, 702, 856, 1133, 1132, 1303.
June 23, 2015 Cedar Crest College Hammer 22 Statistics 110 Lecture Notes
Definition 82 (Mode). The value that occurs most often in a data set is called the mode.A data set that only has one value that occurs with the greatest frequency is called unimodal, a data set that has two values that occur with the same greatest frequency is called bimodal, and a data set that has more than two values that occur with the same greatest frequency is called multimodal. A data set where every value occurs only once is said to have no mode.
Problem 83. Find the mode of the signing bonuses of eight NFL players for a specific year. The bonuses are in millions of dollars.
18.0, 14.0, 34.5, 10.0, 11.3, 10.0, 12.4, 10.0.
Problem 84. The data show the number of licensed nuclear reactors in the U.S. for a recent 15-year period. Find the mode.
104 104 104 104 104 107 109 109 109 110 . 109 111 112 111 109
Problem 85. The number of accidental deaths due to firearms for a six-year period is shown. Find the mode. 649, 789, 642, 613, 610, 600.
Cedar Crest College June 23, 2015 3.1. Measures of Central Tendencies Hammer 23
Definition 86 (Mode Class). The mode for grouped data is the modal class. The modal class is the class with the largest frequency.
Problem 87. Find the modal class for the frequency distribution of miles that 20 runners ran in one week. Class Bondaries Frequency 5.5 − 10.5 1 10.5 − 15.5 2 15.5 − 20.5 3 20.5 − 25.5 5 25.5 − 30.5 4 30.5 − 35.5 3 35.5 − 40.5 2
Problem 88. A small company consists of the owner, the manager, the salesperson, and two technicians, all of whose annual salaries are listed here. Find the mean, median, and mode. Staff Salary Owner $100, 000 Manager $40, 000 . Salesperson $24, 000 Technician $18, 000 Technician $18, 000
June 23, 2015 Cedar Crest College Hammer 24 Statistics 110 Lecture Notes
Definition 89 (Midrange). The midrange is defined as the sum of the lowest and highest values in the data set, divided by 2. The symbol MR is used for the midrange. lowest value + highest value MR = . 2 Problem 90. The number of bank failures for a recent five-year period is shown. Find the midrange. 3, 30, 148, 157, 71.
Problem 91. Find the midrange of data for NFL singning bonuses. The bonuses in millions of dollars are 18.0, 14.0, 34.5, 10.0, 11.3, 10.0, 12.4, 10.0. Find the midrange.
Cedar Crest College June 23, 2015 3.1. Measures of Central Tendencies Hammer 25
Definition 92 (Weighted Mean). Let w1, w2, . . . , wn be weights for the given values X1,X2, ...,Xn. The weighted mean of a variable X is found by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights. Pn i=1 wiXi X = Pn . i=1 wi Problem 93. A student received an A in English Composition I (3 credits), a C in Introduc- tion to Psychology (3 credits), a B in Biology I (4 credits), and a D in Physical Education (2 credits). Assuming that an A is 4 grade points, a B is 3 grade points, a C is 2 grade points, a D is 1 grade point, and an F is 0 grade points, find the student’s grade point average.
Suggested Problems: 1 − 11, 23 − 28
June 23, 2015 Cedar Crest College Hammer 26 Statistics 110 Lecture Notes
3.2 Measures of Variation
Problem 94. A testing lab wishes to test two experimental brands of outdoor paint to see how long each will last before fading. The testing lab makes 6 gallons of each paint to test. Since different chemical agents are added to each group and only six cans are involved, these two groups constitute two small populations. The results (in months) are shown. Find the mean of each group. Then analyze the data with a dot plot.
Brand A Brand B 10 35 60 45 50 30 30 35 40 40 20 25
Definition 95 (Range). The range is the highest value minus the lowest value. The symbol R is used for the range. R = highest value − lowest value.
Problem 96. Find the range for the paints in Problem 94.
Cedar Crest College June 23, 2015 3.2. Measures of Variation Hammer 27
Definition 97 (Population Variance). Let X denote the individual value, µ denote the population mean, and N denote the population size. The population variance is the average of the squares of the distance each value is from the mean. This statistic compares the distance each data point is from the mean. σ2.
PN (X − µ)2 σ2 = i=1 i . N Definition 98 (Population Standard Deviation). The population standard deviation is the square root of the variance. The standard deviation measures the amount of variation or dispersion of a set of data values. The symbol for the population standard deviation is σ. s √ PN (X − µ)2 σ = σ2 = i=1 i . N
Note 99. When the means are equal, the larger the variance or standard deviation is, the more variable the data are.
Problem 100. Find the variance and the standard deviation for the data set for brand A paint in Problem 94. The number of months brand A lasted before fading was
10, 60, 50, 30, 40, 20.
Problem 101. Find the variance and the standard deviation for the data set for brand B paint in Problem 94. The number of months brand B lasted before fading was
35, 45, 35, 30, 40, 25.
June 23, 2015 Cedar Crest College Hammer 28 Statistics 110 Lecture Notes
Definition 102 (Sample Variance). Let X denote the individual value, X denote the sample mean, and n denote the sample size. The formula for sample variance is
2 P X − X s2 = . n − 1
Definition 103 (Sample Standard Deviation). Let X denote the individual value, X de- note the sample mean, and n denote the sample size. The formula for sample standard deviationis s 2 P X − X s = . n − 1 Problem 104. The number of public school teacher strikes in Pennsylvania for a random sample of school years is shown. Find the sample variance and the sample standard deviation.
9, 10, 14, 7, 8, 3.
Cedar Crest College June 23, 2015 3.2. Measures of Variation Hammer 29
Note 105. The shortcut formulas for computing the variance and standard deviation for data obtained from samples are as follows. s n (P X2) − (P X)2 n (P X2) − (P X)2 s2 = s = . n (n − 1) n (n − 1)
Problem 106. The number of public school teacher strikes in Pennsylvania for a random sample of school years is shown. Use the shortcut formulas to find the sample variance and the sample standard deviation. 9, 10, 14, 7, 8, 3.
June 23, 2015 Cedar Crest College Hammer 30 Statistics 110 Lecture Notes
Definition 107 (Grouped Data Variance and Standard Deviation). Let Xm denote the class midpoint. Then the sample variance for grouped data is
n (P f · X2 ) − (P f · X )2 s2 = m m n (n − 1)
and the sample standard deviation for grouped data is s n (P f · X2 ) − (P f · X )2 s = m m . n (n − 1)
Problem 108. Find the sample variance and the sample standard deviation for the fre- quency distribution for the following data obtained from the number of miles that 20 runners ran during one week. Class Bondaries Frequency 5.5 − 10.5 1 10.5 − 15.5 2 15.5 − 20.5 3 20.5 − 25.5 5 25.5 − 30.5 4 30.5 − 35.5 3 35.5 − 40.5 2
Cedar Crest College June 23, 2015 3.2. Measures of Variation Hammer 31
Definition 109 (Coefficient of Variance or Covariance). The coefficient of variance or covariance is the standard deviation divided by the mean. The result is expressed as a percentage. This statistic allows you to compare standard deviations when the units are different. The covariance for sample data is s CVar = · 100 X and the covariance for population is σ CVar = · 100. µ Problem 110. The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5, 225, and the standard deviation is $773. Compare the variations of the two.
June 23, 2015 Cedar Crest College Hammer 32 Statistics 110 Lecture Notes
Theorem 111 (Chebyshev). The proportion of values from a data set that will fall within 1 k standard deviations of the mean will be at least 1 − k2 , where k is a number greater than 1 (not necessarily integer).
Problem 112. The mean price of houses in a certain neighborhood id $50, 000, and the standard deviation is $10, 000. Find the price range for which at least 75% of the houses will sell.
Problem 113. A survey of local companies found that the mean amount of travel allowance for couriers was $0.25 per mile. The standard deviation was $0.02. Find the minimum percentage of the data values that will fall within $0.20 and $0.30.
Suggested Problems: 1 − 20, 31 − 35
Cedar Crest College June 23, 2015 3.3. Measures of Position Hammer 33
3.3 Measures of Position
Definition 114 (standard score or z-score). A standard score or z-score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. value − mean z = . standard deviation For samples, the formula is X − X z = . s For population, the formula is X − µ z = . σ The z-score represents the number of standard deviations that a data value falls above or below the mean.
Problem 115. A student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests.
Problem 116. Find the z-score for each test, and state which is higher.
Test A X = 40 s = 5 Test B X = 100 s = 10
June 23, 2015 Cedar Crest College Hammer 34 Statistics 110 Lecture Notes
Definition 117 (Percentiles). Percentiles divide the data set into 100 equal groups.
Problem 118. The frequency distribution for the systolic blood pressure readings (in mil- limeters of mercury, mmHg) of 200 randomly selected college students is shown here. Con- struct a percentile graph.
Class Boundary Frequency Cumulative Frequency Cumulative Percentile 89.0 − 104.5 24 104.5 − 119.5 62 119.5 − 134.5 72 134.5 − 149.5 26 149.5 − 164.5 12 164.5 − 179.5 4
Cedar Crest College June 23, 2015 3.3. Measures of Position Hammer 35
Definition 119 (Percentile Rank Formula). The percentile corresponding to a given value X is computed by using the formula
(number of values below X) + 0.5 Percentile Rank = · 100. total number of values Problem 120. A teacher gives a 20-point test to 10 students. The scores are shown here. Find the percentile rank of a score of 12.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10.
Problem 121. Using the data in Problem 120, find the percentile rank for a score of 6.
June 23, 2015 Cedar Crest College Hammer 36 Statistics 110 Lecture Notes
Problem 122. Using the scores in Problem 120, find the value corresponding to the 25th percentile.
Problem 123. Using the data set in Problem 120, find the value that corresponds to the 60th percentile.
Definition 124 (Quartile). Quartiles divide the distribution into four equal groups denotes Q1,Q2, and Q3.
Problem 125. Find Q1,Q2, and Q3 for the data set
15, 13, 6, 5, 12, 50, 22, 18.
Cedar Crest College June 23, 2015 3.3. Measures of Position Hammer 37
Definition 126 (Interquartile Range). The interquartile range (IQR) is the difference between the third and the first quartiles.
IQR = Q3 − Q1.
Problem 127. Find the interquartile range for the data set in Problem 125.
Definition 128 (Outlier). An outlier is an extremely high or an extremely low data value when compared with the rest of the data values.
Problem 129. Check the following data set for outliers.
5, 6, 12, 13, 15, 18, 22, 50
Suggested Problems: 1 − 20.
June 23, 2015 Cedar Crest College Hammer 38 Statistics 110 Lecture Notes
3.4 Exploratory Data Analysis
Definition 130 (Boxplot). A boxplot is a graph of a data set obtained by drawing a horizontal line from the minimum data value to Q1, drawing a horizontal line from Q3 to the maximum data value, and drawing a box whose vertical sides pass through Q1 and Q3 with a vertical line inside the box passing through the median or Q2. Problem 131. The number of meteorites found in 10 states of the United States is
89, 47, 164, 296, 30, 215, 138, 78, 48, 39.
Construct a boxplot for the data.
Cedar Crest College June 23, 2015 3.4. Exploratory Data Analysis Hammer 39
Problem 132. A dietitian is interested in comparing the sodium content of real cheese with the sodium content of a cheese substitute. The data for two random samples are shown. Compare the distributions, using boxplots.
Real Cheese Cheese Substitute 310 420 45 40 270 180 250 290 220 240 180 90 130 260 340 310
Suggested Problems: 1 − 15
June 23, 2015 Cedar Crest College Hammer 40 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 4
Probability and Counting Rules
4.1 Sample Spaces and Probability
Definition 133 (Probability Experiment). A probability experiment is a chance process that leads to well-defined results called outcomes
Definition 134 (Outcome). An outcome is the result of a single trial of a probability experiment.
Definition 135 (Sample Space). A sample space is the set of all possible outcomes of a probability experiment.
Problem 136. Find the sample space for rolling two dice.
Problem 137. Find the sample space for drawing one card from an ordinary deck of cards.
41 Hammer 42 Statistics 110 Lecture Notes
Definition 138 (Tree Diagram). A tree diagram is a device consisting of line segments emanating from a starting point and also from an outcome point. It is used to determine all possible outcomes of a probability experiment.
Problem 139. Use a tree diagram to find the sample space for the gender of three children in a family.
Cedar Crest College June 23, 2015 4.1. Sample Spaces and Probability Hammer 43
Definition 140 (Event). An event consists of a set of outcomes of a probability experiment.
Definition 141 (Equally Likely Events). Equally likely events are events that have the same probability of occurring.
Definition 142 (Cardinality). The Cardinality of an event or a sample space is the size of that event or sample space. That is to say that it is the number of things in that event or sample space. Let E be an event. The cardinality of an event is denoted by |E| .
Formula 143 (Classical Probability). The probability of any event E happening in a sample space S is |E| P r(E) = P (E) = . |S| Problem 144. Find the probability of getting a red face card (Jack, Queen, or King) when randomly drawing a card from an ordinary deck.
Problem 145. If a family has three children, find the probability that exactly two of the three children are girls.
June 23, 2015 Cedar Crest College Hammer 44 Statistics 110 Lecture Notes
True Fact 146 (Rules of Probability). The rules of probability are as follows:
• Let E denote an event. Then 0 ≤ P r(E) ≤ 1.
• The sum of all of the outcomes in a sample space is 1.
• If an event E cannot occur, P r(E) = 0.
• If an event E is certain, then P r(E) = 1.
Problem 147. When a single die is rolled, find the probability of getting a 9.
Problem 148. When a single die is rolled, what is the probability of getting a number less than 7?
Cedar Crest College June 23, 2015 4.1. Sample Spaces and Probability Hammer 45
Definition 149 (Event Complement). The complement of an event E is the set of outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by E.
Formula 150 (Rule for Event Complement). P r E = 1 − P r(E).
Problem 151. In a study, it was found that 23% of the people surveyed said that vanilla was their favorite flavor of ice cream. If a person is selected randomly, find the probability that the person’s favorite ice cream is not vanilla.
June 23, 2015 Cedar Crest College Hammer 46 Statistics 110 Lecture Notes
Definition 152 (Empirical Probability). Empirical probability relies on actual experience to determine the likelihood of an outcome as opposed to classical probability which assumes that certain outcomes are equally likely.
Formula 153 (Rule for Empirical Probability). Given the frequency distribution, let f denote the frequency for a class and n denote the total frequencies in the distribution. The empirical probability (which is based on observation) for a given class is
f P r(E) = . n Problem 154. A researcher for the American Automobile Association (AAA) asked 50 peo- ple who plan to travel over the Thanksgiving holiday how they will get to their destination. Find the probability that a person will travel by airplane over the thanksgiving break.
Method Frequency Drive 41 Fly 6 Train or bus 3
Problem 155. In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.
a. A person has type O blood
b. A person has type A or B blood
c. A person has neither type A nor type O blood
d. A person does not have type AB blood.
Suggested Problems: 1 − 25
Cedar Crest College June 23, 2015 4.2. The Addition Rules for Probability Hammer 47
4.2 The Addition Rules for Probability
Definition 156 (Mutually Exclusive). Two events are mutually exclusive events or dis- joint events if if they cannot occur at the same time (i.e. they have no outcomes in common).
Formula 157 (Addition of Mutually Exclusive Events). When two events A and B are mutually exclusive, the probability that A or B will occur is
P r (A or B) = P r(A) + P r(B).
Problem 158. A city has 9 coffee shops: 3 Starbucks, 2 Caribou Coffees, and 4 Crazy Mocho Coffees. If a person selects one shop at random to buy a cup of coffee, find the probability that it is either a Starbucks or a Crazy Mocho Coffee.
Problem 159. The corporate research and development centers for three local companies have the following number of employees:
U.S. Steel 110 Alcoa 750 Bayer Material Science 250
If a research employee is selected at random, find the probability that the employee is employed by U.S. Steel or Alcoa.
June 23, 2015 Cedar Crest College Hammer 48 Statistics 110 Lecture Notes
Formula 160 (Addition Rule for Non-Mutually Exclusive Events). If A and B are not mutually exclusive, then
P r (A or B) = P r(A) + P r(B) − P r (A and B) .
Problem 161. A single card is drawn at random from an ordinary deck of cards. Find the probability that it is either an ace or a black card.
Problem 162. In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.
Problem 163. On New Years Eve, the probability of a person driving while intoxicated is 0.32, the probability of a person having a driving accident is 0.09, and the probability of a person having a driving accident while intoxicated is 0.06. What is the probability of a person driving while intoxicated or having a driving accident?
Cedar Crest College June 23, 2015 4.3. The Multiplication Rules & Conditional Probability Hammer 49
4.3 The Multiplication Rules & Conditional Probabil- ity
Definition 164 (Independent Events). Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring.
Formula 165 (Independent Multiplication Formula). When two events are independent, the probability of both occurring is
P r (A and B) = P r(A) · P r(B).
Problem 166. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.
Problem 167. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen and then an ace.
June 23, 2015 Cedar Crest College Hammer 50 Statistics 110 Lecture Notes
Problem 168. An urn contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and its color noted. Then it is replace. A second ball is selected and its color is noted. Find the probability of each of the following.
a. Selecting 2 blue balls
b. Selecting 1 blue ball and 1 white ball
c. Selecting 1 red ball and 1 blue ball.
Problem 169. A Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer great stress at least once a week?
Problem 170. Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If 3 men are selected at random, find the probability that all of them will have this type of red-green color blindness.
Cedar Crest College June 23, 2015 4.3. The Multiplication Rules & Conditional Probability Hammer 51
Definition 171 (Dependent Events). When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.
Definition 172 (Conditional Probability). The conditional probability of an event B in relationship to an event A is the probability that event B will occur after event A has already occurred. This is denoted as P r(B|A) and is sometimes read as, “The probability of B given A.”
Problem 173. In a recent survey, 33% of the respondents said that they feel that they are overqualified (O) for their present job. Of these, 24% said that they were looking for a new job (J). If a person is selected at random, find the probability that the person feels that he or she is overqualified and is also looking for a new job.
Problem 174. World Wide Insurance Company found that 53% of the residents of a city had homeowner’s insurance (H) with the company. Of these clients, 27% also had automobile insurance (A) with the company. If a resident is selected at random, find the probability that the resident has both homeowner’s insurance and automobile insurance with World Wide Insurance Company.
June 23, 2015 Cedar Crest College Hammer 52 Statistics 110 Lecture Notes
Problem 175. Three cards are drawn from an ordinary deck and not replaced. Find the probability of
a. Getting 3 Jacks.
b. Getting an Ace, a King, and a Queen in that order.
c. Getting a Club, a Spade, and a Heart in that order.
d. Getting 3 Clubs.
Problem 176. Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is tossed. If it falls heads up, Box 1 is selected and a ball is drawn. If it falls on tails, Box 2 is selected and a ball is drawn. Find the probability of selecting a red ball.
Cedar Crest College June 23, 2015 4.3. The Multiplication Rules & Conditional Probability Hammer 53
Formula 177 (Conditional Probability). The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. The formula is
P r (A and B) P r(B|A) = . P r(A)
Problem 178. A box contains black chips and white chips. A person selects two chips 15 without replacement. If the probability of selecting a black chip and a white chip is 56 and 3 the probability of selecting a black chip on the first draw is 8 , find the probability of selecting the white chip on the second draw given that the first chip selected was a black chip.
Problem 179. The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in a no-parking zone is 0.20. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.
June 23, 2015 Cedar Crest College Hammer 54 Statistics 110 Lecture Notes
Problem 180. A recent survey asked 100 people if they thought that women in the armed forces should be permitted to participate in combat. The results of the survey are shown.
Gender Yes No Total Male 32 18 50 Female 8 42 50 Total 40 60 100
Find the probabilities
a. The respondent answered yes given that the respondent was female.
b. The respondent was a male, given that the response answered no.
Cedar Crest College June 23, 2015 4.3. The Multiplication Rules & Conditional Probability Hammer 55
Problem 181. A person selects 3 cards from an ordinary deck and replaces each card after it is drawn. find the probability that the person will get at least one heart. Hint. Use the probability of the complement.
Problem 182. A coin is tossed 5 times. Find the probability of getting at least 1 tails.
Problem 183. The Neckware Association of America reported that 3% of ties sold in the United States are bow ties. If 4 customers who purchased a tie are randomly selected, find the probability that at least 1 purchased a bow tie?
Suggested Problems: 3 − 20, 32, 35 − 40
June 23, 2015 Cedar Crest College Hammer 56 Statistics 110 Lecture Notes
4.4 Counting Rules
Formula 184 (Fundamental Counting Rule). In a sequence of n events in which the first one has k1 possibilities, the second event has k2 possibilities, and so forth, the total number of possibilities of the sequence will be
k1 · k2 ····· kn.
Problem 185. A coin is tossed and a die is rolled. find the number of outcomes for the sequence.
Problem 186. A paint manufacturer wishes to manufacture several different paints. The categories include Color Red, Blue, White, Black, Green, Brown, Yellow Type Latex, Oil Texture Flat Semigloss, High Gloss Use Outdoor, Indoor How many different kinds of paint can be made if you can select one color, one type, one texture, and one use?
Cedar Crest College June 23, 2015 4.4. Counting Rules Hammer 57
Problem 187. There are four blood types, A, B, AB, and O. Blood can also be Rh+ or Rh-. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled?
Problem 188. The first year that the state of Pennsylvania issued railroad memorial license plates, the plate had a picture of a steam engine followed by four digits. Assuming that repetitions are allowed, how many railroad memorial plates could be issued?
June 23, 2015 Cedar Crest College Hammer 58 Statistics 110 Lecture Notes
Definition 189 (Factorial). For any counting number n, n! is read n factorial and is computed as n! = n(n − 1)(n − 2) ··· 1, and 0! = 1.
Definition 190 (Permutation). A Permutation is an arrangement of n objects in a specific order.
Problem 191. Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations?
Problem 192. Suppose the business owner in Problem 191 wishes to rank only the top 3 of the 5 locations. How many different ways can she rank them?
Cedar Crest College June 23, 2015 4.4. Counting Rules Hammer 59
Formula 193 (Permuation). The arrangement of n objects in a specific order using r objects at a time is written as nPr and is computed as n! P = . n r (n − r)!
Problem 194. A radio talk show host can select 3 of 6 special guests for her program. The order of appearance of the guests is important. How many different ways can this be done?
Problem 195. A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?
June 23, 2015 Cedar Crest College Hammer 60 Statistics 110 Lecture Notes
Formula 196 (Permutation with Repeats). The number of permutations of n objects when r1 objects are identical, r2 objects are identical, ..., rp objects are identical is n!
r1!r2! ··· rp!
where r1 + r2 + ··· + rp = n.
Problem 197. How many permutations of letters can be made from the word statistics?
Cedar Crest College June 23, 2015 4.4. Counting Rules Hammer 61
Definition 198 (Combination). A selection of distinct objects without regard to order is called a combination.
Formula 199 (Combination). The number of combinations of r objects selected from n n objects is denoted nCr or r . The formula is n n! C = = . n r r (n − r)!r!
Problem 200. How many combinations of 4 objects are there taken 2 at a time?
Problem 201. An advertising executive must select 3 different photographs for an adver- tising flier. If she has 10 different photographs that can be used, how many ways can she select 3 of them?
Problem 202. In a club there ate 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?
Suggested Problems: 1 − 40
June 23, 2015 Cedar Crest College Hammer 62 Statistics 110 Lecture Notes
4.5 Probability and Counting Rules
Problem 203. Find the probability of getting 4 aces when 5 cards are drawn from an ordinary deck of cards.
Problem 204. A box contains 24 transistors, 4 of which are defective. If 4 are sold at random, find the following probabilities.
a. Exactly 2 are defective
b. None are defective
c. All are defective
d. At least 1 is defective
Cedar Crest College June 23, 2015 4.5. Probability and Counting Rules Hammer 63
Problem 205. A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased.
Problem 206. In the Pennsylvania State Lottery, a person selects a three-digit number and repetitions are permitted. If a winning number is selected, find the probability that it will have all three digits the same.
Problem 207. There are 8 married couples in a tennis club. If 1 man and 1 woman are selected at random to plan the summer tournament, find the probability that they are married to each other.
Suggested Problems: 1 − 12
June 23, 2015 Cedar Crest College Hammer 64 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 5
Discrete Probability Distributions
5.1 Probability Distributions
Definition 208 (Random Variable). A random variable is a variable whose values are determined by chance.
Definition 209 (Discrete Probability Distribution). A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation.
Problem 210. Construct a probability distribution for rolling a single die.
Problem 211. Represent graphically the probability distribution for the sample space of tossing three coins and landing heads.
65 Hammer 66 Statistics 110 Lecture Notes
Problem 212. The baseball World Series is played by the winner of the National League and that of the American League. The first team to win four games wins the World Series. In other words, the series will consist of four to seven games, depending on the individual victories. The data shown consists of 40 World Series events. The number of games played in each series represented by the variable X. Find the probability P r(X) for each X, construct a probability distribution, and draw a graph for the data.
X Number of Series Played 4 8 5 7 6 9 7 16
Problem 213. Determine whether each distribution is a probability distribution X 5 8 11 14 a. P r(X) 0.2 0.6 0.1 0.3
X 1 2 3 4 5 b. 1 1 3 1 1 P r(X) 4 8 8 8 8
X 1 2 3 4 c. 1 1 1 1 P r(X) 4 4 4 4
X 4 8 12 d. P r(X) −0.5 0.6 0.4
Suggested Problems: 7 − 12, 19 − 25
Cedar Crest College June 23, 2015 5.2. Mean, Variance, Standard Deviation, and Expectation Hammer 67
5.2 Mean, Variance, Standard Deviation, and Expec- tation
Formula 214 (Mean of Probability Distribution). The mean of a random variable with a discrete probability distribution is
n X µ = (Xi · P r(Xi)) i=1 where X1,X2,...,Xn are the outcomes and P r(X1), P r(X2), . . . , P r(Xn) are the correspond- ing probabilities.
Problem 215. Find the mean of the number of spots (pips) that appear when a die is tossed.
Problem 216. In families with five children, find the mean number of children who will be girls.
June 23, 2015 Cedar Crest College Hammer 68 Statistics 110 Lecture Notes
Problem 217. If three coins are tossed, find the mean of the number of heads that occur.
Problem 218. The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more, etc.) Find the mean. Number of trips X 0 1 2 3 4 Probability P r(X) 0.06 0.70 0.20 0.03 0.01
Cedar Crest College June 23, 2015 5.2. Mean, Variance, Standard Deviation, and Expectation Hammer 69
Formula 219 (Variance of Probability Distribution). The variance of a probability dis- tribution is found as follows
n 2 X 2 2 σ = Xi · P r(Xi) − µ . i=1 The standard deviation is v √ u n 2 uX 2 2 σ = σ or σ = t (Xi · P r(Xi)) − µ i=1
Problem 220. Compute the variance and standard deviation for the probability distribution found for the number of spots (pips) that appear when a die is tossed.
Problem 221. A box contains 5 balls. Two are numbered 3, one is numbered 4, and two are numbered 5. The balls are mixed and one is selected at random. After a ball is selected, its number is recorded. Then it is replaced. If the experiment is repeated many times, find the variance and standard deviation of the numbers on the balls.
June 23, 2015 Cedar Crest College Hammer 70 Statistics 110 Lecture Notes
Problem 222. A talk radio station has four telephone lines. If the host is unable to talk or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2, 3, or 4 people will get through is shown in the probability distribution. Find the variance and standard deviation of the distribution. X 0 1 2 3 4 P r(X) 0.18 0.34 0.23 0.21 0.04
Definition 223 (Expected Value). The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable. The formula is
n X µ = E(x) = (Xi · P r(Xi)) . i=1 Problem 224. One thousand tickets are sold at $1 each for a color television valued at $350. What is the expected value of the gain if you purchase one ticket?
Cedar Crest College June 23, 2015 5.2. Mean, Variance, Standard Deviation, and Expectation Hammer 71
Problem 225. Six balls numbered 1, 2, 3, 5, 8 and 13 are placed in a box. A ball is selected at random, and its number is recorded and then it is replaced. Find the expected value of the numbers that will occur.
Problem 226. A financial adviser suggests that his client select one of two types of bonds in which to invest $5, 000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all of the investment.
Suggested Problems: 1 − 10.
June 23, 2015 Cedar Crest College Hammer 72 Statistics 110 Lecture Notes
5.3 The Binomial Distribution
Definition 227 (Binomial Experiment). A binomial distribution is a probability exper- iment that satisfies the following four requirements:
1. There must be a fixed number of trials.
2. Each trial can have only two outcomes. These outcomes can be considered as either success or failure.
3. The outcomes of each trial must be independent of one another.
4. The probability of a success must remain the same for each trial.
Problem 228. Decide whether each experiment is a binomial experiment. If not, state the reason why.
a. Selecting 20 university students and recording their class rank.
b. Selecting 20 students from a university and recording their gender.
c. Drawing five cards from a deck without replacement and recording whether they are red or black.
d. Selecting five students from a large school and asking them if they are on the deans list.
e. Recording the number of children in 50 randomly selected families.
Cedar Crest College June 23, 2015 5.3. The Binomial Distribution Hammer 73
Definition 229 (Binomial Distribution). The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.
Notation 230. The following notation is used for the binomial distribution:
P r(S) The probability of Success P r(F ) The probability of Failure p The numerical probability of success q The numerical probability of failure n The number of trials X The number of successes in n trials
Formula 231 (Binomial Probability). In a binomial experiment, the probability of exactly X successes in n trials is n P r(X) = pX qn−X . X Problem 232. A coin is tossed 3 times. Find the probability of getting exactly two heads.
’
Problem 233. A survey from Teenage Research Unlimited found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs.
June 23, 2015 Cedar Crest College Hammer 74 Statistics 110 Lecture Notes
Formula 234 (Binomial Distribution Mean, Variance, and Standard Deviation). The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas: √ √ Mean: µ = np Variance: σ2 = npq Standard Deviation: σ = σ2 = npq.
Problem 235. A coin is tossed 4 times. Find the mean, variance, and standard deviation of the number of heads that will be obtained.
Problem 236. An 8-sided die is rolled 560 times. Find the mean, variance, and standard deviation of the number of 7’s that will be rolled.
Suggested Problems: 1 − .10, 17 − 25
Cedar Crest College June 23, 2015 Chapter 6
The Normal Distribution
6.1 Normal Distributions
Definition 237 (Normal Distribution). If a random variable has a probability distribution whose graph is continuous, bell shaped, and symmetric, it is called a normal distribution. The graph is called a normal distribution curve.
Definition 238 (Standard Normal Distribution). The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Problem 239. Use the table on pages 788 and 789 of your textbook to find the area under the standard normal distribution curve to the left of z = 2.09.
Problem 240. Find the area under the standard normal distribution curve to the right of z = −1.14
Problem 241. Find the area under the standard normal distribution curve between z = 1.62 and z = −1.34.
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True Fact 242. The area under the standard normal distribution curve can also be thought of as a probability or as the proportion of the population with a given characteristic.
Problem 243. Find the probability for each assuming the standard normal distribution.
a. P r(0 < z < 2.53)
b. P r(z < 1.73)
c. P r(z > 1.98)
Problem 244. Find the z value such that the area under the standard normal distribution curve between 0 and z value is 0.2123.
Suggested Problems: 7 − 20, 27 − 35, 41 − 43.
Cedar Crest College June 23, 2015 6.2. Applications of the Normal Distribution Hammer 77
6.2 Applications of the Normal Distribution
Formula 245 (z-value). To solve problems by using the standard normal distribution, trans- form the original variable to a standard normal distribution variable by using the formula X − µ z = . σ Problem 246. An adult has on average 5.2 liters of blood. Assume the variable is normally distributed and has a standard deviation of 0.3. Find the percentage of people who have less than 5.4 liters of blood in their system.
Problem 247. Each month, an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the variable is approximately normally dis- tributed and the standard deviation is 2 pounds. If a household is selected at random, find the probability of its generating
a. Between 27 and 31 pounds per month.
b. More than 30.2 pounds per month.
June 23, 2015 Cedar Crest College Hammer 78 Statistics 110 Lecture Notes
Formula 248 (Normal Variable). The formula for finding the value of a normal variable X is X = zσ + µ.
Problem 249. To qualify for a police academy, candidates must score in the top 10% on a general abilities test. Assume the test scores are normally distributed and the tests has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify.
Problem 250. For a medical study, a researcher wishes to select people in the middle 60% of the population based on blood pressure. Assuming that blood pressure readings are normally distributed and the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify prople to participate in the study.
Cedar Crest College June 23, 2015 6.2. Applications of the Normal Distribution Hammer 79
Definition 251 (Pearson Coefficient). The skewness of a normal distribution can be checked with the Pearson Coefficient of skewness. The formula is 3 X − median PC = . s If the index is greater than or equal to +1 or less than or equal to −1, it can be concluded that the data are significantly skewed.
Problem 252. A survey of 18 high-tech firms showed the number of days’ inventory they had on hand. Determine if the data are approximately normally distributed.
5 29 34 44 45 63 68 74 74 81 88 91 97 98 113 118 151 158
Problem 253. The data shown consists of the number of games played each year in the career of Baseball Hall of Famer Bill Mazeroski. Determine if the data are approximately normally distributed. 81 148 152 135 151 152 159 142 34 162 130 162 163 143 67 112 70
Suggested Problems: 1 − 10, 18 − 25, 39 − 43
June 23, 2015 Cedar Crest College Hammer 80 Statistics 110 Lecture Notes
6.3 The Central Limit Theorem
Definition 254 (Sampling Distribution of Sample Means). A sampling distribution of sample means is a distribution using the means computed from all possible random samples of a specific size taken from a population.
Definition 255 (Sampling Error). Sampling error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.
Observation 256. Two observations can be made:
1. The mean of the sample means will be the same as the population mean.
2. The standard deviation of the sample means will be smaller than the standard deviation of the population, and it will be equal to the population standard deviation divided by the square root of the sample size.
Theorem 257 (Central Limits). As the sample size n increases without limit, the shape of the distribution of the sampling means taken with replacement from a population with mean µ and standard deviation σ will approach a normal distribution. As previously shown, this √σ distribution will have a mean µ and a standard deviation n . Formula 258 (z Values for Sample Mean). The central limit theorem can be used to answer questions about sample means in the same way that a normal distribution can be used to answer questions about individual values.
X − µ z = . √σ n Problem 259. A.C. Neilsen reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the variable is normally distributed and the standard deviation is 3 hours. If 20 children between the ages of 2 and 5 are randomly selected, find the probability that the mean of the number of hours they watch television will be greater than 26.3 hours.
Cedar Crest College June 23, 2015 6.3. The Central Limit Theorem Hammer 81
Problem 260. The average age of a vehicle registered in the United States is 8 year, or 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 90 and 100 months.
Problem 261. The average time spent by construction workers who work on weekends is 7.93 hours (over 2 days). Assume the distribution is approximately normal and has a standard deviation of 0.8 hours.
a. Find the probability that an individual who works at that trade works fewer than 8 hours on the weekend.
b. If a sample of 40 construction workers is randomly selected, find the probability that the mean of the sample will be less than 8 hours.
Suggested Homework: 7 − 15.
June 23, 2015 Cedar Crest College Hammer 82 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 7
Confidence Intervals and Sample Size
7.1 Confidence Intervals for the Mean Shen σ is Known
Definition 262 (Point Estimate). A point estimate is a specific numerical value estimate of a parameter. The best point estimate of the population mean µ is the sample mean X.
Properties 263. Three properties of a good estimator are
1. The estimator should be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.
2. The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.
3. The estimator should be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.
Definition 264 (Interval Estimate). An interval estimate of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.
Definition 265 (Confidence Level). The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated.
Definition 266 (Confidence Interval). A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.
Formula 267 (Confidence Interval of Mean where σ is known). Let α denote the total area α in both tails of the standard normal distribution curve, and 2 denote the area in each of the
83 Hammer 84 Statistics 110 Lecture Notes
tails. We call zα/2 a critical value. The interval formula for a confidence interval of the Mean for a specific α where σ is known is
σ σ X − z √ < µ < X + z √ . α/2 n α/2 n
For a 90% confidence interval, zα/2 = 1.65; for a 95% confidence interval, zα/2 = 1.96; and for a 99% confidence interval, zα/2 = 2.58. Definition 268 (Margin of Error). The margin of error, also called the maximum error of the estimate, is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter.
Problem 269. A researcher wishes to estimate the number of days it takes an automobile dealer to sell a Chevrolet Aveo. A random sample of 50 cars had a mean time on the dealer’s lot of 54 days. Assume the population standard deviation to be 6.0 days. Find the best point estimate of the population mean and the 95% confidence interval of the population mean.
Problem 270. A large department store found that it averages 362 customers per hour. Assume that the standard deviation is 29.6 and a random sample of 40 hours was used to determine the average. Find the 99% confidence interval of the population mean
Cedar Crest College June 23, 2015 7.1. Confidence Intervals for the Mean Shen σ is Known Hammer 85
Problem 271. The following data represent a random sample of the assets (in millions of dollars) of 30 credit unions in southwestern Pennsylvania. Assume that the ppopulation standard deviation is 14.405. Find the 90% confidence interval of the mean. 12.23 16.56 4.39 2.89 1.24 2.17 13.19 9.16 1.42 73.25 1.91 14.64 11.59 6.69 1.06 8.74 3.17 18.13 7.92 4.78 16.85 40.22 2.42 21.58 5.01 1.47 12.24 2.27 12.77 2.76
June 23, 2015 Cedar Crest College Hammer 86 Statistics 110 Lecture Notes
Formula 272 (Minimum Sample Size needed for Interval Estimate of the Population Mean). Let E be the margin of error and n denote the sample size. Then the minimum sample size n needed for an interval estimate of population mean is
z · σ 2 n = α/2 E Problem 273. A scientist wishes to estimate the average depth of a river. He wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.33 feet.
Suggested Problems: 7 − 15, 23 − 26
Cedar Crest College June 23, 2015 7.2. Confidence Intervals for the Mean When σ is Unknown Hammer 87
7.2 Confidence Intervals for the Mean When σ is Un- known
Definition 274 (t Distribution). The t distribution is siimilar to the standard normal dis- tribution in these ways:
1. It is bell-shaped.
2. It is symmetric about the mean.
3. The mean, median, and mode are equal to 0 and are located at the center of the distribution.
4. The curve approaches but never touches the x-axis.
The t distribution differs from the standard normal distribution in the following ways:
1. The variance is greater than 1.
2. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size.
3. As the sample size increases, the t distribution approaches the standard normal distri- bution.
Problem 275. Find the tα/2 value for a 95% confidence interval when the sample size is 22.
June 23, 2015 Cedar Crest College Hammer 88 Statistics 110 Lecture Notes
Formula 276 (Specific Confidence Interval for the Mean When σ is Unknown). The degrees of freedom are n − 1 and the confidence interval for the mean when σ is unknown is s s X − t √ < µ < X + t √ . α/2 n α/2 n
Problem 277. A random sample of 10 children found that their average growth for the first year was 9.8 inches. Assume the variable is normally distributed and the sample standard deviation is 0.96 inches. Find the 95% confidence interval of the population mean for growth during the first year.
Problem 278. The data represent a random sample of the number of home fires started by candles for the past several years (Data are from the National Fire Protection Association). Find the 99% confidence interval for the mean number of home fires started by candles each year. 5460, 5900, 6090, 6310, 7160, 8440, 9930
Suggested Problems: 5 − 10
Cedar Crest College June 23, 2015 7.3. Confidence Intervals and Sample Size for Populations Hammer 89
7.3 Confidence Intervals and Sample Size for Popula- tions
Notation 279. Let p denote the population proportion and pb denote the sample proportion. For a sample proportion, X n − X p = and q = or q = 1 − p b n b n b b Where X is the number of sample units that possess the characteristics of interest and n is the sample size.
Problem 280. A random sample of 200 workers found that 128 drove to work alone. Find pb and qb, where pb is the proportion of workers who drove to work alone.
June 23, 2015 Cedar Crest College Hammer 90 Statistics 110 Lecture Notes
Formula 281 (Specific Confidence Interval for a Proportion). Let npb and nqb be greater than or equal to 5. Then r r pq pq p − z bb < p < p + z bb. b α/2 n b α/2 n Problem 282. A survey conducted by Sallie Mae and Gallup of 1404 respondents found that 323 students paid for their education by student loans. Find the 90%
Problem 283. A survey of 1898 adults with lawns conducted by Harris Interactive Poll found that 45% of the adults said that dandelions were the toughest weeds to control in their yards. Find the 95% confidence interval of the true proportion who said that dandelions were the toughest weeds to control their yards.
Cedar Crest College June 23, 2015 7.3. Confidence Intervals and Sample Size for Populations Hammer 91
Formula 284 (Minimum Sample Size Needed for Interval Estimate of Proportion). n = 2 pq zα/2 bb E Problem 285. A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary.
Problem 286. In Problem 285, assume that no previous study was done. Find the minimum sample size necessary to be accurate within 2% of the true proportion.
June 23, 2015 Cedar Crest College Hammer 92 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 8
Hypothesis Testing
8.1 Traditional Method
Definition 287 (Statistical Hypothesis). A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true.
Definition 288 (Null Hypothesis). The null hypothesis, symbolized by H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.
Definition 289 (Alternative Hypothesis). The alternative hypothesis, symbolized by H1, is a statistical hypothesis that states the existence of a between a parameter and a specific value, or a state that there is a difference between two parameters.
Problem 290. State the null and alternative hypothesis for each conjecture.
a. A researcher thinks that if expectant mothers use vitamin pills, the birth weight of the babies will increase. The average birth weight of the population is 8.6 pounds.
b. An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of USB drives by using robots instead of humans for certain tasks. The mean number of defective drives per 1000 is 18.
c. A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores was 73.
93 Hammer 94 Statistics 110 Lecture Notes
Definition 291 (Statistical Test). A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.
Definition 292 (Test Value). The numerical value obtained from a statistical test is called the test value.
Definition 293 (Error Types). There are two types of error types with respect to the null hypothesis. H0 true H0 false Reject H0 Error Type I Correct Decision Do not reject H0 Correct Decision Error Type II Definition 294 (Level of Significance). The level of significance is the maximum probabil- ity of committing a type I error. The probability is symbolized by α. That is P r(type I error) = α.
Definition 295 (Rejection Region). The critical or rejection region is the range of test values that indicates that there is a significant difference and that the null hypothesis should be rejected.
Definition 296 (Nonrejection Region). The noncritical or nonrejection region is the range of test values that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.
Definition 297 (Critical Value). The critical value separates the critical region from the noncritical region. The symbol for critical value is C.V.
Definition 298 (Tail Test). A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. A one-tailed test is either a right-tailed test or a left-tailed test, depending on the direction of the inequality of the alternative hypothesis.
Definition 299 (Two-tailed Test). In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions.
Problem 300. Find the critical values for each situation and draw the appropriate figure, showing the critical regions.
a. A left-tailed test with α = 0.10
b. A two-tailed test with α = 0.02
c. A right-tailed test with α = 0.005
Suggested Problems: 7 − 14
Cedar Crest College June 23, 2015 8.2. z Test for a Mean Hammer 95
8.2 z Test for a Mean
Formula 301 (z Test). The z test is a statistical test for the mean of a population. It can be used either when n ≥ 30 or when the population is normally distributed and σ is known. Let X denote the sample mean, µ denote the hypothesized population mean, sigma denote the population standard deviation, and n denote the sample size. Then the formula is
X − µ z = √σ n Problem 302. In Pennsylvania, the average IQ score is 101.5. The variable is normally distributed, and the population standard deviation is 15. A school superintendent claims that the students in her school district have an IQ higher than the average. She selects a random sample of 30 students and finds the mean of the test scores is 106.4. Test the claim at α = 0.05.
June 23, 2015 Cedar Crest College Hammer 96 Statistics 110 Lecture Notes
Problem 303. For a specific year, the average score on the SAT Math test was 515. The variable is normally distributed, and the population standard deviation is 100. The same superintendent in Problem 302 wishes to see if her students scored significantly below the national average on the test. She randomly selected 36 student scores, as shown. At α = 0.10, is there enough evidence to support the claim?
496 506 507 505 438 499 505 522 531 762 513 493 522 668 543 519 349 506 519 516 714 517 511 551 287 523 576 516 515 500 243 509 523 503 414 504
Problem 304. The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24, 672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is $26, 343. The standard deviation of the population is $3, 251. At α = 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $24, 672?
Cedar Crest College June 23, 2015 8.2. z Test for a Mean Hammer 97
Definition 305 (P -value). The P -value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis it true.
Formula 306 (Rejecting or Not Rejecting Null Hypothesis with P -value). If the P -value is less than or equal to α, then reject the null hypothesis. If the P -value is greater than α, then do not reject the null hypothesis.
Problem 307. A researcher wishes to test the claim that the average cost of tuition and fees at a four-year public college is greater than $5, 700. She selects a random sample of 36 four-year public colleges and finds the mean to be $5, 950. The population standard deviation is $659. Is there evidence to support the claim that α = 0.05? Use the P -value method.
Problem 308. A researcher claims that the average wind speed in a certain city is 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the population is 0.6 miles per hour. At α = 0.05, is there enough evidence to reject the claim? Use the P -value method.
Suggested Problems: 1 − 5, 15 − 20
June 23, 2015 Cedar Crest College Hammer 98 Statistics 110 Lecture Notes
8.3 t Test for a Mean
Definition 309 (t Test). The t test is a statistical test for the mean of a population and is used when the population is normally or approximately normally distributed and σ is unknown. The test for the t test is X − µ t = √s n The degrees of freedom are d.f. = n − 1.
Problem 310. Find the critical t value for α = 0.05 with d.f. = 16 for a right tailed t test.
Problem 311. Find the critical t value for α = 0.01 with d.f. = 22 for a left tailed t test.
Problem 312. Find the critical values for α = 0.10 with d.f. = 18 for a two-tailed t test.
Problem 313. Find the critical value for α = 0.05 with d.f. = 28 for a right-tailed t test.
Cedar Crest College June 23, 2015 8.3. t Test for a Mean Hammer 99
Problem 314. A medical investigation claims that the average number of infections per week at a hospital in southwestern Pennsylvania is 16.3. A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is there enough evidence to reject the investigator’s claim at α = 0.05? Assume the variable is normally distributed.
Problem 315. According to payscale.com, the average starting salary for a nurse practi- tioner is $79, 500. A researcher wishes to test a claim that the starting salary is less than that. A random sample of 8 starting nurse practitioners is selected, and their starting salaries (in dollars) are shown. Is there enough evidence to support the researcher’s claim that α = 0.10? Assume the variable is normally distributed.
82, 000 68, 000 70, 200 75, 600 83, 500 64, 300 75, 600 79, 000
Suggested Problems: 7 − 15
June 23, 2015 Cedar Crest College Hammer 100 Statistics 110 Lecture Notes
8.4 z Test for a Proportion
Definition 316 (z Test for a Proportion). Let p = X denote the sample proportion, p b n denote the population proportion, and n denote the sample size. Then
pb− p z = p pq n Problem 317. A researcher claims that based on the information obtained from the Centers for Disease Control and Prevention, 17% of young people ages 2 − 19 are obese. To test the claim, she randomly selected 200 young people and found that 42 were obese. At α = 0.05, is there enough evidence to reject the claim?
Problem 318. The gallup Crime Survey stated that 23% of gun owners are women. A researcher believes that in the area where he lives, the percentage is less than 23%. He randomly selects a sample of 100 fun owners and finds that 11% of the gun owners are women. At α = 0.01, is the percentage of female gun owners in his area less than 23%?
Cedar Crest College June 23, 2015 8.4. z Test for a Proportion Hammer 101
Problem 319. A statistician reads that at least 77% of the population oppose replacing $1 bills with $1 coins. To see if the claim is valid, the statistician selected a random sample of 80 people and found that 55 were opposed to replacing the $1 bills. At α = 0.01, test the claim that at least 77% of the population are opposed to the chance.
Problem 320. An attorney claims that more than 25% of all lawyers advertise. A random sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α = 0.05, is there enough evidence to support the attorney’s claim? Use the P -method.
Suggested Problems: 5 − 10
June 23, 2015 Cedar Crest College Hammer 102 Statistics 110 Lecture Notes
Cedar Crest College June 23, 2015 Chapter 10
Correlation and Regression
10.1 Scatter Plots and Correlation
Definition 321 (Scatter Plot). A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y.
Definition 322 (Correlation). The correlation between two or more attributes or mea- surements is the degree to which those attributes or measurements show a tendency to vary together.
Definition 323 (Positive Correlation). A positive correlation is a relationship between two variables where if one variable increases, the other one also increases.
Definition 324 (Negative Correlation). A positive correlation is a relationship between two variables where if one variable increases, the other one decreases.
Problem 325. Construct a scatter plot for the data shown for car rental companies in the United States for a recent year. Describe whether the data set seems to have any correlation.
Company Cars (in ten thousands) Revenue (in billions of dollars) A 63.0 7.0 B 29.0 3.9 C 20.8 2.1 D 19.1 2.8 E 13.4 1.4 F 8.5 1.5
103 Hammer 104 Statistics 110 Lecture Notes
Problem 326. Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. Determine if there is any correlation. The data are shown here.
Student Number of absences x Final Grade y% A 6 82 B 2 86 C 15 43 D 9 74 E 12 58 F 5 90 G 8 78
Problem 327. A researcher wishes to see if there is a relationship between the ages of the wealthiest people in the world and their net worth. A random sample of 10 persons was selected from the Forbes list of 400 richest people for a recent year. The data are shown. Draw a scatter plot for the data and determine if there is any notable correlation.
Persons Age x Net Worth y (in billions of dollars) A 60 11 B 72 69 C 56 11.9 D 55 30 E 83 12.2 F 67 36 G 38 18.7 H 62 10.2 I 62 23.3 J 46 10.6
Cedar Crest College June 23, 2015 10.1. Scatter Plots and Correlation Hammer 105
Definition 328 (Population Correlation Coefficient). The population correlation coef- ficient, ρ, is the correlation computed by using all possible pairs of data values (x, y) taken from a population.
Definition 329 (Linear Correlation Coefficient). The Linear Correlation Coefficient computed from the sample data measures the strength and direction of a linear relationship between two quantitative variables. The symbol for the sample correlation coefficient is r.
June 23, 2015 Cedar Crest College Index
t Distributino, 87 Dependent Events, 51 z-score, 33 Dependent Variable,6 z-value, 77 Descriptive Statistics,1 Discrete Probability Distribution, 65 Alternative Hypothesis, 93 Discrete Variables,3 Arithmetic Average, 19 Disjoint Events, 47 Bar Graph, 12 Dotplot, 16 Bimodal, 22 Empirical Probability, 46 Binomial Distribution, 73 Equally Likely Events, 43 Binomial Distribution Mean, 74 Error Types, 94 Binomial Distribution Standard Deviation, 74 Event, 43 Binomial Distribution Variance, 74 Event Complement, 45 Binomial Experiment, 72 Expected Value, 70 Binomial Probability, 73 Experimental Study,6 Boundary,4 Explanatory Variable,6 Boxplot, 38
Cardinality, 43 Factorial, 58 Class Boundaries,8 Frequency Distribution,7 Class Midpoint, 10 Frequency Polygon, 10 Class Width,8 Histogram, 10 Cluster Sample,5 Hypothesis Testing,1 Coefficient of Variance, 31 Compound Time Series Graph, 14 Independent Events, 49 Conditional Probability, 51 Independent Variable,6 Confidence Interval, 83 Inferential Statistics,1 Confidence Interval σ Unknown, 88 Interquartile Range, 37 Confidence Interval for Proportion, 90 Interval Estimate, 83 Confidence Interval Known Mean, 83 Interval Level,4 Confidence Level, 83 Confounding Variable,6 Level of Significance, 94 Continuous Variables,3 Lower Class Limit,8, 10 Covariance, 31 Critical Region, 94 Margin of Error, 84 Critical Value, 94 Maximum Area of the Estimate, 84 Cumulative Frequency Distribution,9 Mean, 19 Mean of Distribution, 67 Data,1 Median, 21
106 Index Hammer 107
Midrange, 24 Random Variable, 65 Minimum Sample of Population Mean, 86 Random Variables,1 Minimum Sample Size Proportion, 91 Range, 26 Mode, 22 Ratio Level,4 Mode Class, 23 Raw Data,7 Multimodal, 22 Rejection Region, 94 Mutually Exclusive, 47 Relative Frequency Graphs, 11
Nominal Level,4 Sample,1 Noncritical Region, 94 Sample Mean, 19 Nonrejection Region, 94 Sample Mean z values, 80 Nonsampling Error,5 Sample Space, 41 Normal Distribution, 75 Sample Standard Deviation, 28 Normal Distribution Curve, 75 Sample Variance, 28 Normal Variable Formula, 78 Sampling Distribution, 80 Nuisance Variable,6 Sampling Error,5, 80 Null Hypothesis, 93 Standard Deviation of Grouped Data, 30 Standard Normal Distribution, 75 Observational Study,6 standard score, 33 Ogive, 10 Statistic, 19 Open-Ended Distribution,8 Statistical Hypothesis, 93 Ordinal Level,4 Statistical Test, 94 Outcome, 41 Statistics,1 Outcome Variable,6 Stem and Leaf Plot, 17 Outlier, 37 Stratified Sample,5 Systematic Sample,5 Parameter, 19 Pareto Chart, 13 Tail Test, 94 Pearson Coefficient, 79 Test Value, 94 Percentile Rank Formula, 35 Time Series Graph, 14 Percentiles, 34 Tree Diagram, 42 Permutation, 58 Two-tailed Test, 94 Pie Graph, 15 Point Estimate, 83 Ungrouped Frequency Distribution,9 population,1 Unimodal, 22 Population Mean, 19 Upper Class Limit,8, 10 Population Standard Deviation, 27 Variable,1 Population Variance, 27 Variance of Distribution, 69 Probability,1 Variance of Grouped Data, 30 Probability Experiment, 41 Weighted Mean, 25 Qualitative Variables,3 Quantitative Variables,3 Quartile, 36
Random Sample,5
June 23, 2015 Cedar Crest College