10.1 – Confidence Intervals: The basics

Most Common Z* Values

Level of Confidence (C) Upper tail probability Z* Value 90%

95%

99%

Example #1 Serum Cholesterol-Dr. Paul Oswick wants to estimate the true mean serum HDL cholesterol for all of his 20-29 year old female patients. He randomly selects 15 patients and computes the sample mean to be 50.67. Assume from past records, the population standard deviation for the serum HDL cholesterol for 20-29 year old female patients is =13.4.

a. Construct a 95% confidence interval for the mean serum HDL cholesterol for all of Dr. Oswick’s 20-29 year old female patients.

b. If the US National Center for Health Statistics reports the mean serum HDL cholesterol for females between 20-29 years old to be =53, do Dr. Oswick’s patients appear to have a different serum level compared to the general population? Explain.

c. What two things could you do to decrease your margin of error? Example #2 Suppose your class is investigating the weights of Snickers 1-ounce Fun-Size candy bars to see if customers are getting full value for their money. Assume that the weights are Normally distributed with standard deviation s = 0.005 ounces. Several candy bars are randomly selected and weighed with sensitive balances borrowed from the physics lab. The weights are 0.95 1.02 0.98 0.97 1.05 1.01 0.98 1.00 ounces. Determine a 90% confidence interval for the true mean, µ. Can you say that the bars weigh 1oz on average? Example #3 A statistician calculates a 95% confidence interval for the mean income of the depositors at Bank of America, located in a poverty stricken area. The confidence interval is $18,201 to $21,799.

a. What is the sample mean income?

b. What is the margin of error?

Example #4 A researcher wishes to estimate the mean number of miles on four-year-old Saturn SCI’s. How many cars should be in a sample in order to estimate the mean number of miles within a margin of error of  1000 miles with 99% confidence assuming =19,700. 10.2 – Estimating a Population Mean

Example #1 Determine the degrees of freedom and use the t-table to find probabilities for each of the following: DF Picture Probability

P(t > 1.093) n = 11

P(t < 1.093) n = 11

P(t > 0.685) n = 24

P(t < -0.685) n = 24

P(0.70 < t < 1.093) n = 11 Example #2 Practice finding t* n Degrees of Freedom Confidence Interval t* n = 10 99% CI n = 20 90% CI n = 40 95% CI n = 30 99% CI

Example #3 As part of your work in an environmental awareness group, you want to estimate the mean waste generated by American adults. In a random sample of 20 American adults, you find that the mean waste generated per person per day is 4.3 pounds with a standard deviation of 1.2 pounds. Calculate a 99% confidence interval for  and explain it’s meaning to someone who doesn’t know statistics. Matched Pairs t-procedures

Example #4 Archaeologists use the chemical composition of clay found in pottery artifacts to determine whether different sites were populated by the same ancient people. They collected five random samples from each of two sites in Great Britain and measured the percentage of aluminum oxide in each. Based on these data, do you think the same people used these two kiln sites? Use a 95% confidence interval for the difference in aluminum oxide content of pottery made at the sites and assume the population distribution is approximately normal. Can you say there is no difference between the sites?

New Forrest 20.8 18 18 15.8 18.3 Ashley Trails 19.1 14.8 16.7 18.3 17.7 Difference Example #5 The National Endowment for the Humanities sponsors summer institutes to improve the skills of high school language teachers. One institute hosted 20 Spanish teachers for four weeks. At the beginning of the period, the teachers took the Modern Language Association’s listening test of understanding of spoken Spanish. After four weeks of immersion in Spanish in and out of class, they took the listening test again. (The actual spoken Spanish in the two tests was different, so that simply taking the first test should not improve the score on the second test.) Below is the pretest and posttest scores. Give a 90% confidence interval for the mean increase in listening score due to attending the summer institute. Can you say the program was successful?

Subject Pretest Posttest Subject Pretest Posttest 1 30 29 11 30 32 2 28 30 12 29 28 3 31 32 13 31 34 4 26 30 14 29 32 5 20 16 15 34 32 6 30 25 16 20 27 7 34 31 17 26 28 8 15 18 18 25 29 9 28 33 19 31 32 10 20 25 20 29 32 10.3 – Estimating a Population Proportion

Example #1 A news release by the IRS reported 90% of all Americans fill out their tax forms correctly. A random sample of 1500 returns revealed that 1200 of them were correctly filled out. Calculate a 95% confidence interval for the proportion of Americans who correctly fill out their tax forms. Is the IRS correct in their report? Example #2 You wish to estimate with 95% confidence; the proportion of computers that need repairs or have problems by the time the product is three years old. Your estimate must be accurate within 3.5% of the true proportion.

a. Find the sample size needed if a prior study found that 19% of computers needed repairs or had problems by the time the product as three years old.

b. If no preliminary estimate is available, find the most conservative sample size required.

c. Compare the results from a and b.