1. An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. After examining the dosage, we feel that her claims regarding the effectiveness of the dosage are inflated. In an attempt to disprove her claim, we administer her prescribed dosage to 10 insomniacs and we observe Y, the number for which the drug dose induces sleep. We wish to test the hypothesis H0: π = .8 versus the alternative, H1: π < .8. Assume that the rejection region {y ≤ 6} is used.
(a) In terms of this problem, what is a type I error?
(b) Find α.
(c) In terms of this problem, what is a type II error?
(d) Find β when π = .4.
Hint: You cannot use the normal approximation in this question. The random variable Y is a Binomial random variable with n = 10 and success probability π.
(e) Find the p value if Y = 4.
2. An econometric professor's rule of thumb is that students should expect to spend 2 hours outside of class on coursework for each hour in class. For a 3 hour per week class, this means that students are expected to do 6 hours of work outside class. The professor randomly selects eight students from a class, and asks how many hours they studied econometrics during the past week. The sample values are 1, 3, 4, 4, 6, 6, 8, 12.
(a) Assuming that the population is normally distributed; can the professor conclude at the 0.05 level of significance that the students are studying on average at least 6 hours per week?
(b) Construct a 90% confidence interval for the population mean number of hours studied per week.
3. The administrators for a hospital wished to estimate the average number of days required for in-patient treatment of patients between the ages of 25 and 34. A random sample of 500 hospital patients between these ages produced a mean and standard deviation equal to 5.4 and 3.1 respectively. A federal regulatory agency hypothesizes that the average length of stay is in excess of 5 days.
(a) State the hypotheses and the decision rule implied by α = 0.05. Test the hypothesis put forward by the federal agency.
(b) Using the rejection region found in part (a), calculate the probability of a Type II Error when the mean under the alternative, µa = 5.5.
(c) How large should the sample size be if we require that α = 0.01, and β = P{Type II Error} = 0.05, when the mean under the alternative, µa = 5.5.
4. Suppose that Y1, Y2, Y3 is a random sample from N(µ, σ2) population. To estimate µ consider the weighted estimator
Y˜ = 1/2 Y1 + 1/3 Y2 + 1/6 Y3
(a) Show that Y˜ is a linear estimator.
(b) Show that Y˜ is an unbiased estimator.
(c) Find the variance of Y˜ and compare it to the variance of the sample mean Y¯.
(d) Is Y˜ as good an estimator as Y¯?
(e) If σ2 = 9, calculate the probability that each estimator is within 1 unit on either side of µ.
5. You are given the following observations of x and y.
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6
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y
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11
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(a) Using the linear regression formulas, compute using only a hand calculator, the least squares estimates of the slope and the intercept. Plot this line on your graph.
(b) Obtain the sample means y¯ = i=1∑n yi/N and x¯ = i=1∑n xi/N. Obtain the predicted value of y for x = x¯ and plot it on your graph. What do you observe about this predicted value?
(c) Using the least squares estimates from (a), compute the least squares residuals u^i. Find their sum.
(d) Calculate i=1∑n xiuˆi.