Question 1

Which of the following best describes the null hypothesis ($$H_0) $$for comparing the median earnings of counties that 'went Republican' versus those that 'went Democrat' in the 2016 U.S. Presidential Election?

Accepted Answer

$$H_0$$: \mu_{Rep} = \mu_{Dem}$$

Question 2

Given the following summary statistics for the cloud seeding experiment: $$\bar{x}_{Seed} = 441.98$$, $$s_{Seed} = 650.79$$, $$\bar{x}_{Unseed} = 164.59$$, $$s_{Unseed} = 278.43$$, and both groups have $$n = 26$$, what is the formula for the test statistic to compare the means?

Accepted Answer

$$T = \frac{\bar{x}_{Seed} - \bar{x}_{Unseed}}{\sqrt{\frac{s_{Seed}^2}{n_{Seed}} + \frac{s_{Unseed}^2}{n_{Unseed}}}}$$

Question 3

Suppose you computed a P-value of 0.03 for the hypothesis test in the cloud seeding experiment. What does this P-value represent?

Accepted Answer

The probability of observing a difference as extreme as the one found, assuming the null hypothesis is true.

Question 4

Based on the histograms of median county earnings, which group appears to have a higher median income: counties that 'went Democrat' or counties that 'went Republican'?

Accepted Answer

Counties that 'went Democrat'

Question 5

In the chocolate and student ratings experiment, what is the appropriate alternative hypothesis ($$H_A) if $$you want to test whether giving chocolate increases instructor ratings?

Accepted Answer

$$H_A: \mu_{Choc} \u003e \mu_{NoChoc}$$

Question 6

If the 95% confidence interval for $$\mu_{Rep} - \mu_{Dem} is$$ $$(-8000, -2000)$$, what does this interval suggest?

Accepted Answer

Counties that 'went Republican' have significantly lower median earnings than those that 'went Democrat'.

Question 7

Which of the following is a correct interpretation of a 95% confidence interval for $$\mu_{Seed} - \mu_{Unseed}$$ that does not include zero?

Accepted Answer

There is evidence at the 5% significance level that seeding clouds changes the mean rainfall.

Question 8

A student claims that the distribution of ratings for students who received chocolate is more positively skewed than for those who did not. Based on the histograms, is this claim supported?

Accepted Answer

No, the ratings for students who received chocolate are more concentrated at the higher end (5), indicating less skewness.

Question 9

Which of the following best describes a Type I error in the context of the cloud seeding experiment?

Accepted Answer

Concluding that seeding increases rainfall when it actually does not.

Question 10

Suppose you want to test if the mean rating for students who received chocolate is different from those who did not. Which statistical test is most appropriate?

Accepted Answer

Two-sample t-test for independent means

Question 11

If the P-value for the test comparing $$\mu_{Choc}$$ and $$\mu_{NoChoc} is 0.001$$, what is the correct conclusion at the 5% significance level?

Accepted Answer

Reject the null hypothesis; there is evidence that chocolate increases ratings.

Question 12

Which of the following is NOT an assumption of the two-sample t-test for comparing means?

Accepted Answer

The two samples are paired.

Question 13

Given the following data: $$\bar{x}_1 = 25,000$$, $$s_1 = 5$$,000$$, n_1$ = 100$$; $$\bar{x}_2 = 30,000$$, $$s_2 = 7$$,000$$, n_2$ = 100$$, calculate the standard error for the difference in means.

Accepted Answer

$$SE = \sqrt{\frac{5000^2$}{100} + \frac{7000^2$}{100}} = \sqrt{250000 + 490000} = \sqrt{740000} \approx 860.23$$

Question 14

If the 95% confidence interval for $$\mu_{Choc} - \mu_{NoChoc} is$$ $$(0.2, 0.8)$$, what does this mean in the context of the student ratings experiment?

Accepted Answer

Students who received chocolate rated the instructor, on average, between 0.2 and 0.8 points higher than those who did not.

Question 15

Which of the following scenarios would most likely violate the independence assumption in the cloud seeding experiment?

Accepted Answer

If the same cloud was seeded and unseeded on different days.

Step-by-Step Guidance for Comparing Means in Statistics

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