AP Stats Confidence Intervals for Means

A Census Bureau report on the income of Americans says that with 90% confidence the median income of all U.S. households in a recent year was $57,005 with a margin of error of $742. Which of the following is the most appropriate conclusion?

• 90% of all possible samples of this same size would result in a sample median that falls within $742 of $57,005.

• 90% of the households in the sample interviewed by the Census Bureau had incomes in the interval $57,005 ± $742.

• 90% of all households had incomes in the interval $57,005 ± $742.

• The Census Bureau got the result $57,005 ± $742 using a method that will capture the true median income 90% of the time when used repeatedly.

• We can be sure that the median income for all households in the country lies in the interval $57,005 ± $742.

Answer explanation

This interval does not tell us where the population median is for sure, nor does it tell us where individual incomes are. It only provides a set of plausible values for the population median.

You want to calculate a 98% confidence interval for a population mean with a sample of n = 20. What is the appropriate t*?

• 2.326

• 2.528

• 2.539

• 2.518

Mrs. Soto wants to find the sample size needed to find with 95% confidence the mean amount of time her stats students spent on homework during the snow days. The population is approximately normal and the known standard deviation is 6 minutes. How large should the sample size be if she wants a margin of error of +- .5 minutes?

• 553

• 551

• 600

• 601

A researcher wants to find a confidence interval estimate of the mean time it takes to complete an order over the phone at a call center for a large retail company. He has selected a random sample of 18 orders and recorded the time to complete the order. The sample mean time was 4.27 minutes and the sample standard deviation was 0.78 minutes. Assuming the conditions needed for inference are met, which of the following is the appropriate 98% confidence interval?

• 4.27 ± 2.326 ⋅ (0.78/√18)

• 4.27 ± 2.054 ⋅ (0.78/√18)

• 4.27 ± 2.552 ⋅ (0.78/√18)

• 4.27 ± 2.567 ⋅ (0.78/√18)

• 4.27 ± 2.567 ⋅ (0.78/√17)

Which of the following would not decrease the width of a confidence interval?

• I only

• II only

• III only

• I and II only

• II and III only

An irate patient complained that the cost of a doctor's visit was too high.  She randomly surveyed 20 other patients and found that the mean amount of money they spent on each doctor's visit was $44.80.  The standard deviation of the sample was $3.53.  Find the 95% confidence interval of the population mean.

• $43.15 - $46.45

• $44.53 - $48.21

• $42.65 - $48.25

• $43.15 - $48.25

2000 children were surveyed and the report that followed said, "With 98% confidence, between 20% and 50% of all parents make their children eat breakfast." What does 98% confidence mean in this situation?

• We can be 98% confident that the method used to get the intervals always gives the correct answer

• 98% of all parents will make their kids eat breakfast

• When used repeatedly, the method used to find the interval produces intervals which include the true proportion about 98% of the time

• The kids were 98% confident that they ate breakfast that day.

Mr. Reinecke wants to estimate the proportion of seniors who will go to college next year with 96% confidence and a margin of error of no more than 5%. What size sample does he need to take?

• 422

• 419

• 410

• 400

The heights of 5-year old boys are approximately normal. The mean height of a random sample of 37 5-year old boys is found to be 30 inches with a standard deviation of 3 inches. What is the standard error of x̄?

• .493

• 1.32

• 5.77

• 1.96

A survey of a random sample of 2500 adults who have a credit card found that 570 had a balance of $1000 or more on their credit card.

What would be the margin of error if you had a 99% confidence?

• 0.022

• 0.0046

• 0.008

• 0.016