Question

8. Dr. Grieves was wondering what proportion of his extended family preferred Star Wars to Star Trek. “Surely, the proportion is more than 75%,” he thought. So he took a random sample of his extended family and did the following: • He picked a random sample and calculated pb = 0.82. • He constructed a 95% confidence interval. • He conducted a hypothesis test with H0 : p = 0.75 vs. Ha : p > 0.75 While he was having all that fun, his wife simply contacted everyone in his extended family and found out the actual proportion of all Dr. Grieves’s extended family that prefer Star Wars is p = 0.86.

(a) Dr. Grieves made a bootstrap distribution to make his confidence interval. What is the center of his bootstrap distribution?

(b) Dr. Grieves made a randomization distribution to calculate a p-value for the hypothesis test. What is the center of his randomization distribution?

(c) If Dr. Grieves really wanted to have fun and waste time, he could repeatedly take samples and calculate a pb each time and make a sampling distribution. What would be the center of his sampling distribution?

(d) Suppose the 95% confidence interval Dr. Grieves calculated was (0.76, 0.88). What do you know about the p-value of his hypothesis test?

Answer 1

(a)

the center of his bootstrap distribution = p_b = 0.82

So,

Answer is:

0.82

(b)

the center of his randomization distribution = = 0.86

So,

Answer is:

0.86

(c)

the center of his sampling distribution = p_b = 0.82

So,

Answer is:

0.82

(d)

Since all values in the confidence interval (0.76, 0.88) are greater than 0.75, we conclude that the difference is significant. Reject null hypothesis.

So,

We know that the p-value of his hypothesis test is significant. p - value is greater than significance level.