Unit 5 MC Review

Sue and Javier are working on a statistics project to estimate the proportion of students at their school who have a pet dog. Sue selects a random sample of 81 students from 2,400 students at their school, and Javier selects a separate random sample of 64 students. They will both construct a 90 percent confidence interval from their estimates. Consider the situation in which the sample proportion from Sue's sample is equal to the sample proportion from Javier's sample. Which of the following statements correctly describes their intervals?

• Javier's interval will have a greater degree of confidence than Sue's interval will.

• Sue's interval will have a greater degree of confidence than Javier's interval will

• The width of Sue's interval will be the same as the width of Javier's interval

• The width of Sue's interval will be wider than the width of Javier's interval.

• The width of Sue's interval will be narrower than the width of Javier's interval.

Answer explanation

When all other things remain the same, the width of the confidence interval will decrease as the sample size increases. Sue's sample size is greater than Javier's sample size, so Sue's confidence interval is narrower.

A recent national survey indicated that 73 percent of respondents try to include locally grown foods in their diets. A 95 percent confidence interval for the proportion of all people in the country who try to include locally grown foods in their diets is given as (0.70, 0.76)

Assume all conditions for inference are met. Based on the confidence interval, which of the following claims is true?

• Less than half of all people in the country try to include locally grown foods in their diets

• Less than 70 percent of all people in the people in the country try to include locally grown foods in their diets.

• Less than 75 percent of all people in the people in the country try to include locally grown foods in their diets.

• Less than 80 percent of all people in the people in the country try to include locally grown foods in their diets.

• Less than 95 percent of all people in the people in the country try to include locally grown foods in their diets.

Answer explanation

A claim that the actual percent is less than 80 percent is supported by the interval. The interval represents plausible values for the population proportion, and all values contained in the interval are less than 0.80

To study the effectiveness of a certain adult reading program, researchers will select a random sample of adults who are eligible for the program. The selected adults will be given a pretest before beginning the program and a posttest after completing the program. The difference in the number of correct answers on the pretest and the number of correct answers on the posttest will be recorded for each adult in the sample.

Which of the following is the most appropriate inference procedure for the researchers to use to analyze the results?

• A one-sample z-interval for a population proportion

• A one-sample t-interval for a sample mean difference

• A matched-pairs t-interval for a population mean difference

• A matched-pairs t-interval for a sample mean difference

• A two-sample t-interval for a difference between means

Answer explanation

Two measurements, the pretest and posttest scores, will be recorded for each person in the program. Each person had both treatments, and their individual scores was subtracted before the interval was created.

The mean and standard deviation of a random sample of 7 baby orca whales were calculated as 430 pounds and 26.9 pounds, respectively. Assuming all conditions for inference are met, which of the following is a 90 percent confidence interval for the mean weight of all baby orca whales?

Go to: https://apcentral.collegeboard.org/media/pdf/statistics-formula-sheet-and-tables-2020.pdf to access the formula sheet

• 26.9 ± 1.895(430/√7)

• 26.9 ± 1.943(430/√7)

• 430 ± 1.440(26.9/√7)

• 430 ± 1.895(26.9/√7)

• 430 ± 1.943(26.9/√7)

Answer explanation

The appropriate interval formula is

x-bar ± t*(s/√n). The point estimate is

x-bar = 430. With a sample size of 7, there are n-1 = 7-1 = 6 degree of freedom, and the correct critical value (t*) with 90% confidence and 6 df is 1.943

Researchers studying the sticky droplets found on a spider webs will measure the widths of a random sample of droplets. From the sample, the researchers will construct a 95 percent confidence interval to estimate the mean width of all such droplets. Which of the following statements about a 95 percent confidence interval for the mean width is correct?

• The interval will be narrower if the researchers increase the level of confidence to 99 percent.

• The interval will be narrower if the researchers increase the sample size of droplets.

• The interval will be wider if the researchers decrease the level of confidence to 90 percent

• The interval will be wider if the researchers increase the sample size of droplets.

• The width of the interval will not be affected if the researchers increase or decrease the number of droplets in the sample.

Answer explanation

With all else remaining the same, the width of the interval decreases as the sample size increases.

During the winter months, outside temperatures at the Starneses’ cabin in Colorado can stay well below freezing (32°F, or 0°C) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50°F. The manufacturer claims that the thermostat allows variation in home temperature that follows a Normal distribution with 𝜎 = 3°F. To test this claim, Mrs. Starnes programs her digital thermometer to take an SRS of 𝑛= 10 readings during a 24-hour period. Suppose the thermostat is working properly and that the actual temperatures in the cabin vary according to a Normal distribution with mean 𝜇 = 50°F and standard deviation 𝜎 = 3°F.

The dotplot shows the results of taking 300 SRSs of 10 temperature readings from a Normal population with 𝜇 = 50 and 𝜎 = 3 and recording the sample minimum each time. Suppose that the minimum of an actual sample is 40°F.

What would you conclude about the thermostat manufacturer’s claim? Explain your reasoning.

• A sample minimum of 40ºF provides convincing evidence that the manufacturer’s claim is false because only 2 of the 300 simulated SRSs had a minimum that small.

• A sample minimum of 40ºF does not provide convincing evidence that the manufacturer’s claim is false because 2 of the 300 simulated SRSs had a minimum that small.

• A sample minimum of 40ºF provides convincing evidence that the manufacturer’s claim is true because 2 of the 300 simulated SRSs had a minimum that small.

• A sample minimum of 40ºF provides convincing evidence that the manufacturer’s claim is true because 40 is only a few degrees below the center of this distribution.

• A sample minimum of 40ºF provides inconclusive evidence about the manufacturer’s claim because none of the simulated sample minimum temperatures in the dotplot were 40ºF.

Choose the correct definition of a sample distribution.

The sampling distribution of a statistic of size n is

• the distribution of the statistic in all possible samples of size n from a given population

• the distribution of the statistic in one simple random sample of size n from a given population

• t

• t

• t

Which confidence would result in a wider interval (assuming nothing else changed), 91% or 97%?

• 97%, z* is larger for higher confidence intervals

• 97%, the standard error is larger for higher confidence intervals

• 91%, z* is larger for lower confidence intervals

• 91%, the standard error is smaller for lower confidence intervals

Tags

CCSS.HSS.ID.A.4

Which of the following does NOT have an affect on the width of a confidence interval?

• Sample mean
• t*
• Sample standard deviation
• Sample size

Tags

CCSS.HSS.IC.B.4

A quality control specialist at a glass factory must estimate the mean clarity rating for a new batch of glass using a sample of 18 glass sheets from the batch. Past investigations show that clarity ratings are normally distributed. The specialist decides to use a t-distribution rather than a z-distribution because ...

• The t distribution is more accurate than a z distribution.

• Clarity ratings for the entire bacth are normally distributed.

• The t-distribution will create a narrower interval than z.

• The standard deviation for the population is unknown.

Tags

CCSS.HSS.IC.A.1

CCSS.HSS.IC.B.4