Question

1. The produce manager for a local grocery store is interested in estimating the proportion of apples that arrive on a shipment with major bruises. The supplier claims the rate will be about 10%. a) The produce manager wants to estimate the true proportion of bruised apples to within ±2%, with 90% confidence. How many apples should he sample? b) A random sample of 150 apples showed 12 with major bruises. Conduct a full hypothesis test to see if this store’s rate is different from the supplier’s claim. (Be sure to state your hypotheses, your test statistic, your p-value and your full conclusion.) c) You are talking to a family member and they are confused. They say, “Of course the store is different: look – it has a rate that is 2% less than the claim!” Explain why you conducted this test. In particular, explain what the p-value means. (Make reference to the following concepts in your explanation: point estimate, sample error, potential errors in the test conclusion, and statistical significance. Make sure to explain clearly so that someone who hasn’t taken statistics can understand.

Answer 1

(a) n = (z/E)2*p*(1 - p)

n = (1.645/0.02)2*0.10*(1 - 0.10)

n = 609

(b) The hypothesis being tested is:

H0: p = 0.10

Ha: p ≠ 0.10

p̂ = 12/150 = 0.08

The test statistic, z = (p̂ - p)/√p(1-p)/n = (0.08 - 0.10)/√0.10(1-0.10)/150 = -0.82

The p-value is 0.4142.

Since the p-value (0.4142) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that this store’s rate is different from the supplier’s claim.

(c) The p-value or probability value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct.