2. A pumpkin farmer weighed a simple random sample of size n= 20 pumpkins from his main patch, with
9.6, 8.8, 5.1, 9.7, 9.1, 8.9, 8, 9.2, 2.7, 9.1, 8.5, 7.3, 9.3, 9.6, 4.1, 9.9, 7.6, 9, 7.2, 8.5
(a) Create QQ plot and histogram of the weights. Do you think it is reasonable to assume that the
population distribution is normal? Explain your answer
(b) Regardless of your answer to (a), use R to perform the bootstrap with 3000 resamplings to create
a 90% CI for μ (set.seed(1) before performing the bootstrap).
(c) Now construct a 90% t CI for μ by hand and compare it to that which you found via bootstrap.
Which would you tell a scientist to use?
(d) Suppose last years pumpkins had a mean weight of 8.2 lbs. Perform a bootstrap test at alpha= 0.1 to
see if there is evidence that the mean weight has changed (again with set.seed(1) called before your bootstrap). Make sure to specify your null and alternative hypothesis, report p values, and draw your conclusions in context. How does this conclusion compare to what was found in (b)?
(e) Would a two-sided t test with alpha= 0.1 reject the null specified in (d)? Compare your answer to
what you found in (c). Find an approximate p value for the t test by hand.
(f) If there is strong evidence that the median weight of his pumpkins from the jumbo patch is different
from 15, then he feels like he will need to give specific directions to his staff on how to sort them.
Let M be the population median. Use the sign test to test: Ho : M = 15 vs HA : M is not equal to 15 at
alpha = 0.05. Compute the p value and make a conclusion in the context of the problem.
He sampled 8 pumpkins from his jumbo patch and found weights of:
12.6, 12.9, 14.8, 14.3, 19.1, 10.2, 11.4, 9.3
What advice would you give the farmer in light of the statistics - what is a limitation to our test?
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