NAME:______BLOCK:______DATE:______
7.1-2 Sampling Distributions The sampling distribution of a statistic is the distribution of values taken on by the statistic. It is based on all possible samples of the same size from the population. When we sample, we sample with replacement. This means that the same value can be used over again. A sampling distribution is a sample space. It describes everything that can happen when we sample.
Exercise 1: The population values {1, 3, 5, 7} are written on slips of paper and put in a box. Two slips of paper are randomly selected, with replacement.
a) Use your calculator to find the mean, variance, and standard deviation of the population
=
b) Graph the probability histogram for the population values {1,3,5,7}
c) List all the possible samples of size and calculate the mean of each sample.
x1 x2 x1 x2
d) Construct the probability distribution of the sample means
1 NAME:______BLOCK:______DATE:______
f Probability
Total
e) Use your calculator to find the mean, variance, and standard deviation of the sampling distribution of the sample means.
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f) Graph the probability histogram for the sampling distribution of the sample means. TITLE:
Exercise 2: Each group of three should repeat the following experiment five times. Generate random number pairs using the command RandInt(1,7,2). Reject any pair that contains an even number. Complete the following table and report your five mean values to the class recorder.
2 NAME:______BLOCK:______DATE:______
Trial # 1 2 3 4 5
Central Limit Theorem:
Theorem Case Conclusion 7.1 Population is normally The sampling distribution of sample means is distributed. normally distributed for any sample size n. 7.2 Population distribution is The sampling distribution of sample means any shape, and . approximates a normal distribution. The greater the sample size, the better the approximation. 7.1,7.2 Population Mean 7.1,7.2 Population Standard Deviation (Standard Error of the Mean)
Exercise 3: Cellular phone bills for residents of a city have a mean of $63 and a standard deviation of $11. Random samples of 100 cellular phone bills are drawn from this population and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution of sample means.
Solution:
Sketch: Distribution of Sample Means = (n=100)
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Exercise 4: Suppose the training heart rates of all 20-year-old athletes are normally distributed, with a mean of 135 beats per minute and standard deviation of 18 beats per minute. Random samples of size 4 are drawn from this population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution of sample means
Sketch of Sampling Distribution
Exercise 5: The graph shows the length of time people spend driving each day. You randomly select 50 drivers ages 15 to 19. What is the probability that the mean time they spend driving each day is between 24.7 and 25.5 minutes? Assume that σ = 1.5 minutes.
4 NAME:______BLOCK:______DATE:______
Exercise 6: Suppose a team of biologists has been studying the Pinedale Children’s fishing pond. Let x represent the length of a single trout taken at random from the pond. Assume x has a normal distribution with and standard deviation
a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches?
b) What is the probability that the mean length of 5 trout taken at random is between 8 and 12 inches?
= =
c) Explain the difference between parts a) and b).
.
Exercise 7: An education finance corporation claims that the average credit card debts carried by undergraduates are normally distributed, with a mean of $3173 and a standard deviation of $1120. a) What is the probability that a randomly selected undergraduate, who is a credit card holder, has a credit card balance less than $2700?
b) You randomly select 25 undergraduates who are credit card holders. What is the probability that their mean credit card balance is less than $2700? = = 3173 =
c) Write interpretive statements for the two calculations above
7.3 Sampling Distributions for Proportions
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Exercise 1: The annual crime rate in the Capital Hill neighborhood of Denver is 111 victims per 1000 residents. This means that 111 out of 1000 residents have been the victim of at least one crime. The Arms is an apartment building in Capital Hill. It has 50 year round residents. Suppose we view each of the n residents as a binomial trial. The random variable r (which takes on values 0, 1, 2, 3 … 50) represents the number of victims of at least one crime in the next year
a) What is the population probability p that a resident in the Capital Hill neighborhood will be the victim of a crime next year? What is the probability q that a resident will not be a victim?
p = and q =
b) Can we approximate the distribution with a normal distribution? Explain.
50 ( ) = 50 ( ) =
Since both np and nq are greater than 5, we can approximate the distribution with a normal distribution.
c) What are the mean and standard deviation for the distribution?
= =
d) What is the probability that between 10% and 20% of the Arms residents will be victims of a crime next year? Interpret the results.
Continuity Correction:
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Interpretive Statement:
Exercise 2: Consider tossing a fair coin 5 times. Calculate the proportion of the 5 tosses that result in heads. Calculate the sampling distribution of.
a) Compute the possible values of
6 NAME:______BLOCK:______DATE:______
r 0 1 2 3 4 5
b) Compute the possible values of
r ) 0 1 2 3 4 5
c) G
Exercise 3: According to a study by the Department of Transportation, 44% of college students drive while distracted. A professor surveyed 244 students at her college and 36% of them admitted to driving while distracted in the past week. Compute the probability that in a sample of 244 students, 36% or less have engaged in distracted driving.
Solution:
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If np and nq are greater than 5, we can approximate the distribution with a normal distribution. a) What are the mean and standard deviation for the distribution?
= =
b) What is the probability that percentage of 244 distracted drivers will be 36% or less? Continuity Correction:
Interpretive Statement:
8