Many alpine ski centers base their projections of revenues

Question

Econ 310, Spring 2014- Week 10:

Problem 1- Reconsider the case of a new drug. Now we interchange the null and alternative hypotheses.

H₀: The drug is safe.

H₁: The drug is not safe.

Describe the Type I error and Type II error in this new test. Does this test better than the previous one?

Problem 2- Many Alpine ski centers base their projections of revenues and profits on the assumption that the average Alpine skier skis four times per year. To investigate the validity of this assumption, a random sample of 63 skiers is drawn and each is asked to report the number of times he or she skied the previous year. The average was 4.38 times per year. If we assume that the standard deviation is 2, can we infer at the 10% significance level that the assumption is wrong?

(1) Set the null and alternative hypotheses.

(2) Find the rejection region and calculate the test statistic. What is your conclusion?

(3) Calculate the 90% Confidence Interval(CI) of the population mean from the test statistic. Does the conclusion in (2) change if we use the CI for hypothesis testing?

Problem 3- A manufacturer of lightbulbs advertises that, on average, its long-life bulb will last more than 5,000 hours. To test the claim, a statistician took a random sample of 100 bulbs and measured the amount of time until each bulb burned out. The average lifespan of the lightbulbs was 4,880 hours. If we assume that the lifetime of this type of bulb has a standard deviation of 400 hours, can we conclude at the 5% significance level that the claim is true?

(1) Set the null and alternative hypotheses.

(2) Calculate the test statistic and find the rejection region. What is your conclusion?

(3) Calculate the p-value. Is your conclusion from the p-value the same with (2)?

Problem 4- A researcher formulated the following hypotheses. H₀: µ = 200 and H₁: µ < 200 and learned that x¯ = 190, n = 9, and σ = 50. (We assume that x_i is normally distributed.)

(1) Compute the p-value of the test.

(2) Find the p-value for σ = 30 and σ = 10, respectively. Comment on your findings.