AP Stats and Power

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decides to carry out a significance test of 𝐻₀:𝜇𝑠𝑢𝑏𝑢𝑟𝑏𝑎𝑛=𝜇𝑐𝑖𝑡𝑦 versus a two-sided alternative.

Which is the correct standardized test statistic?

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Answer explanation

A test for the difference between two means with population standard deviations unknown is a 𝑡 test.

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decides to carry out a significance test of 𝐻0:𝜇suburban = 𝜇city versus a two-sided alternative.

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence that the average time spent on extracurricular activities by students in the suburban and city school districts is the same.

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• reject 𝐻0 because 0.048<𝛼=0.050. There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

Answer explanation

With the 𝑃-value less than 0.05, we reject 𝐻₀ and have convincing evidence for the alternative hypothesis.

Are TV commercials louder than their surrounding programs? To find out, researchers collected data on 50 randomly selected commercials in a given week. With the television’s volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first 30 seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is

• a two-sample 𝑧 test for a difference in proportions.

• a two-sample 𝑡 interval for a difference in means.

• a two-sample 𝑡 test for a difference in means.

• a paired 𝑡 interval for a mean difference.

• a paired 𝑡 test for a mean difference.

Answer explanation

For each commercial selected, there were two measurements being made. These are unlikely to be independent because they were on the same channel at nearly the same time.

Researchers want to evaluate the effect of a natural product on reducing blood pressure. They plan to carry out a randomized experiment to compare the mean reduction in blood pressure of a treatment (natural product) group and a placebo group. Then they will use the data to perform a test of 𝐻₀:𝜇𝑇−𝜇𝑃=0 versus 𝐻_a:𝜇𝑇−𝜇𝑃>0, where 𝜇𝑇= the true mean reduction in blood pressure when taking the natural product and 𝜇𝑃=the true mean reduction in blood pressure when taking a placebo for subjects like the ones in the experiment. The researchers would like to detect whether the natural product reduces blood pressure by at least 7 points more, on average, than the placebo. If groups of size 50 are used in the experiment, a two-sample 𝑡 test using 𝛼 = 0.01 will have a power of 80% to detect a 7-point difference in mean blood pressure reduction. If the researchers want to be able to detect a 5-point difference instead, then the power of the test

• would be greater than 80%.

• would be less than 80%.

• would still be 80%.

• could be either less than or greater than 80%.

• would vary depending on the standard deviation of the data.

Answer explanation

If we are trying to detect a smaller difference, the power will be smaller because it will be harder to detect the difference.

The table above shows the height, in inches, and the arm span, in inches, for 10 randomly selected high school students. Which of the following significance tests should be used to determine whether a linear relationship exists between height and arm span, provided the assumptions of the test are met?

• Two-sample z-test

• Two-sample t-test

• Chi-square test of independence

• Chi-square goodness-of-fit test

• t-test for slope of regression line

An independent research firm conducted a study of 100 randomly selected children who were participating in a program advertised to improve mathematics skills. The results showed no statistically significant improvement in mathematics skills, using a = 0.05. The program sponsors complained that the study had insufficient statistical power. Assuming that the program is effective, which of the following would be an appropriate method for increasing power in this context.

• Use a two-sided test instead of a one-sided test.

• Use a one-sided test instead of a two-sided test.

• Use a = 0.01 instead of a = 0.05.

• Decrease the sample size to 50 children.

• Increase the sample size to 200 children.

• (A) I only

• (B) I and II only

• (C) I and III only

• (D) II and III only

• (E) I, II, and III

A game of chance is played in which X, the number of points scored in each game, has the distribution shown above. Which of the following is true for the sampling distribution of the sum, Y, of the scores when the game is played twice?

• Y takes on values 0, 1, 2 with respective probabilities 0.3, 0.4, and 0.3.

• Y takes on values 0, 2, 4 according to a binomial distribution with mean equal to 2.

• Y takes on values 0, 2, 4 with respective probabilities 0.3, 0.4, and 0.3.

• Y takes on values 0, 1, 2, 3, 4 according to a binomial distribution with mean equal to 2.

• Y takes on values 0, 1, 2, 3, 4 with respective probabilities 0.09, 0.24, 0.34, 0.24, and 0.09.

Random variable X is normally distributed with mean 10 and standard deviation 3, and random variable Y is normally distributed with mean 9 and standard deviation 4. If X and Y are independent, which of the following describes the distribution of Y - X?

• (A) Normal with mean 1 and standard deviation -1

• (B) Normal with mean -1 and standard deviation -1

• (C) Normal with mean -1 and standard deviation 5

• (D) Normal with mean 1 and standard deviation 7

• (E) Normal with mean -1 and standard deviation 7

• (A)  90% confidence

• (B)  93% confidence

• (C)  95% confidence

• (D)  98% confidence

• (E)  99% confidence