Chapters 20-21 Summative Assessment Part 1

Question

Researchers polled residents in Kent and Sussex counties and asked if they were in favor of increasing funding for the arts. In a random sample of 100 residents from Sussex County, 22% supported the increase and in a random sample of 100 residents from Kent County, 26% supported the increase. A 95% confidence interval for the difference (p_s - p_k) was calculated as -0.04 ± 0.12. Which of the following is the best interpretation of the interval?

±

 −0.04±0.12

a	We are 95% confident that the difference in the sample proportions of voters in districts Sussex County and Kent County who favor an increase in state spending for the arts is between −0.16 and 0.08.
b	We are 95% confident that the difference in the proportions of all voters in Sussex County and Kent County who favor an increase in state spending for the arts is between −0.16 and 0.08.
c	We are 95% confident that the majority of all voters in the state favor an increase in state spending for the arts.
d	We are 95% confident that the proportion of all voters in the state who favor an increase in state spending for the arts is between −0.16 and 0.08.
e	We are 95% confident that less than half of all voters in the state favor an increase in state spending for the arts.

In a lightbulb factory, an administrator selects a random sample of bulbs produced on assembly line A and a random sample of bulbs produced on assembly line B. The administrator calculates the proportion of malfunctioning bulbs produced by each assembly line and finds that the difference between them (A - B) is 0.008. A researcher conducted a hypothesis test with the following hypotheses:

H0: The proportion of malfunctioning bulbs from assembly line A is the sample as the proportion of malfunctioning bulbs from assembly line B.

HA: The proportion of malfunctioning bulbs from assembly line A is greater than the proportion of malfunctioning bulbs from assembly line B.

She found a P-value of 0.016. What is the best interpretation of this P-value?

a

If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at least 0.008 is 0.016.

b

If there is a difference of 0.016 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.008.

c

If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference equal to 0.008 is 0.016.

d

If there is a difference of 0.008 in the proportions of all defective parts made on the two assembly lines, the probability of observing that difference is 0.016.

e

If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at most 0.008 is 0.016.

Answer 1

Hi!

For your first question, the answer would be b) We are 95% confident that the difference in the proportions of all voters in Sussex County and Kent County who favor an increase in state spending for the arts is between −0.16 and 0.08. The entire purpose of confidence intervals is to predict the TRUE proportion, so we need to take ALL voters in both counties into consideration in this case.

For your second question - we need to remember that the P-VALUE represents the probability of the ALTERNATE HYPOTHESIS could've happened due to NATURAL SAMPLING VARIABILITY given that the NULL HYPOTHESIS is true. So first, we're looking at an answer that starts with "if there was NO DIFFERENCE" (null hypothesis is true). So we're down to options A, C, and E. Now, we need to figure out what the alternate hypothesis actually is. The p-value represents the probability that the prop. of malfunctioning bulbs from A is GREATER THAN B. Therefore, the difference needs to be at LEAST 0.008, NOT AT MOST. Therefore, the correct answer is A :)