AP Stats Confidence Intervals for Means

A research group wishes to estimate the mean amount of time (in hours) that members of a fitness center spend exercising each week. They want to estimate the mean within a margin of error of 0.5 hours with a 95% level of confidence. Previous data suggests that the standard deviation of the population is 2.2. Which of the following is the smallest sample size they could use?

• 60

• 75

• 90

• 180

An IQ test was given to a simple random sample of 75 students at a certain college. The sample mean score was 105.2. Scores on this test are known to have a standard deviation of 10. It is desired to construct a 90% confidence interval for the mean IQ scores of the students at the college. Find the critical value.

• 0.90

• 1.155

• 1.645

• ±1.645

A 90% confidence interval for the mean speed driven by cars on Due West Road is given by (21, 45). What was the sample mean of this study?

• 12

• 24

• 33

• None of these.

A 90% confidence interval for the mean speed driven by cars on Due West Road is given by (21, 45). What was the margin of error of this interval?

• 12

• 24

• 33

• None of these.

A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was $139,048 to $154,144. Give the interpretation of the interval.

• 90% of the sampled CEOs have salaries that fell in the interval $139,048 to $154,144

• We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,048 to $154,144.

• There is a 90% chance that CEOs in the electronics industry have salaries that fall between $139,048 to $154,144

• We are 90% confident that the mean salary of the sampled CEOs falls in the interval $139,048 to $154,144.

If the 95% Confidence Interval for a population mean is (114.56, 125.54), then is it likely that the true population mean µ is greater than 130?

• Yes

• No

• Not enough information

• I have no idea what you're asking of me

You want to compute a 90% confidence interval for the mean difference in height for mothers and their adult daughters using a random sample of 30 mothers who have an adult daughter. What critical value should you use for this interval? (Enter 3 decimal places)

Answer explanation

The 𝑡* value for 90% confidence and df = 29 is 𝑡* = 1.699.

A quiz question gives random samples of 𝑛 = 10 observations from each of two Normally distributed populations. Tom uses a table of 𝑡 distribution critical values and 9 degrees of freedom to calculate a 95% confidence interval for the difference in the two population means. Janelle uses her calculator’s two-sample 𝑡 interval with 16.87 degrees of freedom to compute the 95% confidence interval. Assume that both students calculate the intervals correctly. Which of the following is true?

• Tom’s confidence interval is wider.

• Janelle made a mistake; degrees of freedom has to be a whole number.

• There is insufficient information to determine which confidence interval is wider.

• Both confidence intervals are the same width.

• Janelle’s confidence interval is wider.

Answer explanation

Because Tom's interval is less precise with smaller degrees of freedom and a larger 𝑡∗, it will be wider than one calculated using technology.

The makers of a specialty brand of bottled water claim that their “mini” bottles contain 8 ounces of water. To investigate this claim, a consumer advocate randomly selected a sample of 10 bottles and carefully measured the amount of water in each bottle. The mean volume was 7.98 ounces and the 95% confidence interval for true mean volume is 7.93 to 8.03 ounces.

Complete this statement.

Because ​​ (a)   ​ (b)   the interval, there ​​ (c)   convincing evidence that their claim is incorrect.

• 8

• is

• is not

• 0

• 7.98

Answer explanation

Because 8 is in the interval, there is not convincing evidence that their claim is incorrect.

Anne claims that a store-brand fertilizer works better than homemade compost as a soil enhancement when growing tomatoes. To test her theory, she plants two tomato plants in each of five planters. One plant in each planter is grown in soil with storebrand fertilizer and the other plant is grown in soil with homemade compost, with the choice of soil determined at random. In three months, she will harvest and weigh the tomatoes from each plant. What is the correct confidence interval Anne should use to analyze these data?

• The correct interval cannot be determined without the data

Answer explanation

Notice that Anne has 5 pairs of tomato plants: one in each pair is randomly assigned to each of two different soil types.