= n - 1

- iix.- xf

The hot dog vendor pays 30 cents for each hot dog and sells it for 45 cents. Thus for every hot dog sold, he makes a profit of 15 cents, and for every hot dog unsold, he loses 30 cents. What are the expected value and variance of his profit if the number of hot dogs he orders is: (a) 1100; (b) 1200; and (c) 1300? If he wants to maximize his expected profits, how many hot dogs should he order?

16. An instructor wishes to "grade on the curve." The students scores seem to be normally distributed with mean 70 and standard deviation 8. If the instructor wishes to give 20 percent As, what should be the dividing line between an A grade and a grade?

17. Suppose that you replace every observation : by = : + 7 and the mean p. by Tl = 3x -b 7. What happens to the r-value you use?

18. (Reading the N, t, x\ F tables)

(a) Given that X ~ N(2, 9), find P(2 < X < 3).

(b) If Z ~ t2o, find X, and such that

P(X < X,) = 0.95 P(-x, <Z<x,) = 0.90 <X<X2) = 0.90

Note that in the last case, we have several sets of x, and Xj. Find three sets.

(c) - xo, find X, and such that P(X < x,) = 0.95 and P(X > x,) = 0.95. Find two values x, and Xj such that P(x, < X < 2) = 0.90.

Note again that we can have several sets of x, and Xj. Find three sets.

(d) If ~ F2,2o, find X, such that P(X > x,) = 0.05 and P(X > x,) = 0.01. Can you find X, if you are told that P(X > Xi) = 0.50?

19. Given that = e" is normal with mean 2 and variance 4, find the mean and variance of x.

20. Let x„ X2, . . . , x„ be a sample of size n from a normal distribution Mx, a). Consider the following point estimators of p.:

p., = X, the sample mean

P-2 = X,

= i()-----"

(a) Which of these are unbiased?

(b) Which of these are consistent?

(c) Find the relative efficiencies: x, to 1x2, Ai to 1x3, and 1x2 to (x,. What can you conclude from this?

(d) Are all unbiased estimators consistent?

(e) Is the assumption of normality needed to answer parts (a) to (d)? For what purposes is this assumption needed?

21. In exercise 20 consider the point estimators of a:

ft 1=1

a! = (X, -

Which of these estimators are

(a) Unbiased?

(b) Consistent?

Is the assumption of normality needed to answer this question? For what purposes is the normality assumption needed?

22. Suppose that a is an estimator of a derived from a sample of size T. We are given that E{a) = a + 2/ and var(a) = 4 / + -.

(a) Examine whether as an estimator of a, a is (1) unbiased, (2) consistent, (3) asymptotically unbiased, and (4) asymptotically efficient.

(b) What is the asymptotic variance of a?

23. Explain, using the estimators for \x in Exercise 20, the difference between lim £, AE, and plim. Give some examples of how they differ.

24. Examine whether the following statements are true or false. Explain your answer briefly.

(a) The null hypothesis says that the effect is zero.

(b) The alternative hypothesis says that nothing is going on besides chance variation.

(c) A hypothesis test tells you whether or not you have a useful sample.

(d) A significance level tells you how important the null hypothesis is.

(e) The P-value will tell you at what level of significance you can reject the nuU hypothesis.

(f) Calculation of P-values is useless for significance tests.

(g) It is always better to use a significance level of 0.01 than a level of 0.05.

(h) If the P-value is 0.45, the null hypothesis looks plausible.

25. Examine whether the following statements are true or false. If false, correct the statement.

(a) With small samples and large o, quite large differences may not be statistically significant but may be real and of great practical significance.

(b) The conclusions from the data cannot be summarized in the P-value. Conclusions should always have a practical meaning in terms of the problem at hand.

(c) The power of a test at the null hypothesis is equal to the significance level.

(d) In practice the alternative hypothesis , has no important role except in deciding what the nature of the rejection region should be (left-sided, right-sided, or two-sided).

(e) If you are sufficiently resourceful, you can always reject any null hypothesis.

26. Define type I error, type II error, and power of a test. What is the relation-

ship between type I error and the confidence coefficient in a confidence interval?

27. In each of the following cases, set up the null hypothesis and the alternative: Explain how you will proceed testing the hypothesis.

(a) A biscuit manufacturer is packaging 16-oz. packages. The production manager feels that something is wrong with the packaging and that the packages contain too many biscuits.

(b) A tire manufacturer advertises that its tires last for at least 30,000 miles. A consumer group does not believe it.

(c) A manufacturer of weighing scales believes that something is going wrong in the production process and the scale does not show the correct weight.

In each of the problems above, can you identify the costs of mistaken decisions if we view the hypothesis-testing problem as a decision problem?

28. An examination of sample items from a shipment showed that 51% of the items were good and 49% were defective. The company president asked the statistician, "What is the probability that over half the items are good?" The statistician replied that the question cannot be answered from the data. Is this correct? Does the question make sense? Explain why.

29. A local merchant owns two grocery stores at opposite ends of town. He wants to determine if the variability in business is the same at both locations. Two independent random samples yield

n, = 16 days «2 = 16 days s, = $200 52 = $300

(a) Is there enough evidence that the two stores have different variability in sales?

(b) The merchant reads in a trade magazine that stores similar to his have a population standard deviation in daily sales of $210 and that stores with higher variability are badly managed. Is there any evidence to suggest that either one of the two stores the merchant owns has a standard deviation of sales greater than $210?

30. A stockbroker who wants to compare mean returns and risk (measured by variance) of two stocks and gets the following results:

First stock Second stock

= 31 «2 = 31

X, = 0.45 = 0.35

5, = 0.60 S2 = 0.40

Are there any significant differences in the mean returns and risks? (Assume that daily price changes are normally distributed.)

31. If p has a uniform distribution in the range (0, 1) show that -2 log p has a x-distribution with degrees of freedom 2. If there are independent tests,

each with a -\ \ , then X = - 2 2 log p, has a x distribution with d.f.