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Statistics Probability Inference And Decision 1st Edition Robert L. Winkler, William L. Hays - Solutions

In Exercise 43, are the variances significantly different at the .01 level?
In Exercise 41, are the variances significantly different at the .05 level?
The mean daily temperatures in a midwestern city was found for a sample of 30 days in the summers of 1930-1945. These temperatures showed s = 4.8 degrees Fahrenheit. Another random sample of 20 daily temperatures taken
A random sample of 25 U.S. men born in 1935 showed a distribution of height with a standard deviation of 3.1 inches. On the other hand, a random sample of 43 U.S. men born in 1945 showed a standard deviation s of 2.5 inches. Is this evidence (at the .05 level) that these two generations of men have
National norms for the test mentioned in Exercise 53 exist and show that the adult male population has a standard deviation of 28 on this test. Can we say, with a = .05, that junior high-school boys are more variable in their test performance than the population of adult males? How about the senior
A randomly selected group of 30 boys of junior high-school age and an in¬dependent group of 30 senior high-school boys were each given the same test of mechanical ability. The results were as follows:Junior High Senior High Ni = 30 N2 = 30 Mi = 168 M2 = 173 Si = 39 s2 = 33 At the .01 level of
In Exercise 51, suppose also that the manufacturer decides that it is too costly to reduce the variance below 0.5 gram, so that he wants to test H0:a2 = .5 against Hi'.o2 ^ .5.Find the p-value. If a = .20, would you accept or reject Ho? If a = .01?
A manufacturer makes parts under a government contract which specifies a mean weight of 100 grams and a tolerance of 1 gram. That is, the govern¬ment will reject parts unless the weight is between 99 and 101 grams. To meet these tolerances, the manufacturer wants the variance of weight to be no
For many years a program of psychotherapy has been conducted at a state hospital. It has been found that among patients treated successfully in the past, the average time under treatment was 220 hours, with a standard deviation s of 40 hours. A new treatment has just been developed, and for a
The government is interested in changes in research and development expend¬itures in the past decade. They randomly select 8 large manufacturing com¬panies and collect the following data on expenditures on research and develop¬ment (in units of $100,000):
An educator has developed a new IQ test and would like to compare it to a certain widely used IQ test. He gives both tests to a randomly chosen group of 5 students, with the following results:
The same test is given to two classes of students in statistics. The first class(41 students) has a mean score of 75 with a sample variance of 100; the second class (21 students) has a mean score of 80 with a sample variance of 121. Did one class perform substantially better than the other? State
Suppose that you open 16 soft-drink bottles and find that the contents average 11.6 ounces, with a sample variance of 1.0. What can you say about the statement “contents 12 oz.” which is on each bottle? State the relevant hypotheses and determine the p-value. What assumptions did you make in
A detergent manufacturer (the White Soap Co.) is concerned about claims made by a rival firm (Bright Detergent, Inc.) that Bright detergent results in cleaner clothes than White detergent. Twelve dirty sheets are randomly assigned to the two detergents and washed (in identical washing
In Exercise 43, suppose that you decide that the mean output of the new machine would have to be at least 5 units higher than the mean output of the old machine in order to justify considering its purchase. Wbat are the hypotheses of interest and what is the p-value?
Suppose that the output of each of two machines is normally distributed. If the new machine has a higher mean output than the old machine, you will purchase the new machine. Thus, you wish to test Ho'.^n < Mo against Hi'-hn > Mo- You are willing to assume that the variability of output is the same
If in Exercise 41 you are given the additional information that a = 3.5, how would your answers change? Under what conditions might it be reasonable for you to know a before taking the samples?
Suppose that you are interested in comparing the gasoline mileage of two brands of gasoline. You hypothesize that there is no difference in the mileages, and you assume that the mileage for each brand is normally distributed with unknown variance cr2. A sample is taken from each brand, with the
A statistician is contemplating the use of two independent groups of equal size in a study. He wants his test to have power .90 for detecting a difference of ±
A random sample of ten independent observations from a normal distribu¬tion produced the following values:51, 50, 49, 43, 56, 46, 45, 30, 55, 52.If a = .20, test the hypothesis that the mean is 50 against the alternative that it is not 50. Also, determine a 80 percent confidence interval for the
Suppose that a statistician wants to test the hypothesis that the mean of a normally distributed population is .536 against the alternative that the mean is not equal to .536. He decides to use a = .01, and he wants /3 = .05 when the true value of the mean is .540. Approximately how large should
In Exercise 35, if a = .10, find the region of rejection and determine the power of the test if the true average temperature of healthy persons taking the drug is 99.0. Draw the power curve for the test.
How would your answers to Exercise 35 change if N = 20 but everything else was the same?
A new drug manufacturer is concerned over the possibility that a new medi¬cation might have the undesirable side-effect of elevating a person’s body temperature. He wants to market the product only if he could be quite sure
Why is it that a t test cannot be applied to test a hypothesis about a pro¬portion?
Suppose that a = .05 and that K separate tests of hypotheses are made, quite independently of each other. Show that when all of the null hypotheses are true, the probability of one or more significant results by chance alone(that is, even if all of the hypotheses being tested are exactly true) is
List the assumptions underlying the small sample test of a hypothesis about a single mean and the small sample test of a difference between two means.Evaluate the apparent importance of these assumptions.
A psychologist in a public school system is asked by the superintendent to provide evidence to the school board that the students will profit from more intensive training. There are 4000 students in the city school system and 794 in this particular high school. The psychologist decides to test the
In Exercise 15, find the p-value if(a) M = 2.9 (b) M = 3.1 (c) M = 3.5 (d) M = 2.7.
In Exercise 12, find the p-value if(a) M = 75.3 (b) M = 76.0 (c) M = 74.5 (d) M = 75.0.
Suppose that you want to test H0:n = 300 against Hi ^ 300, where the population in question is normal with variance 10,000. You plan to take a sample of size 4. Consider the following two rejection regions:Region 1: M < 200 or M > 400.Region 2: 297 < M < 303.For each of these two rejection regions,
In Exercise 16, suppose that M = 21. What is the p-value associated with this Ml If you decided that a = .01, would you reject H0 in favor of Hi?If someone else decided that a = .35, would he reject Ho in favor of Hi on the basis of the same data?
Discuss the statement: “Ideally, the choice of a should be based on the relative seriousness of the losses associated with the two possible errors. The larger the loss associated with a Type I error is, or the smaller the loss associ¬ated with a lype II error is, the smaller a should be, all
Discuss the relationship between predetermined significance levels (values ofa) and the use of p-values in hypothesis testing. What are the advantages of the use of p-values?
Explain the relationship between confidence intervals and two-tailed tests of hypotheses. Could you develop “one-sided” confidence intervals to relate to one-tailed tests? Explain.
In a random sample of 400 consumers in a given city, 250 preferred Brand A cars to Brand B cars. Find an 80 percent confidence interval for the true proportion of consumers who prefer Brand A to Brand B. Use this to test the hypothesis Ho'.p = .50 against Hi'.p ^ .50 with a = .20.
An experimenter found that in a random sample of 300 women students in college, 35 percent smoked cigarettes. Estimate the 95 percent confidence interval for the true proportion of women college students who smoke ciga¬rettes, based on this data. Can one reject the hypothesis Ho'.p — .40 in
Suppose that X is normally distributed with mean p and variance 400, and you want to test Ho'.p = 50 against Hi'.p = 0>Q.If a — .10 and N — 4, find the critical value of M and compute /?. Now suppose that you wanted to test Ho against H\ :p = 80. For the same values of a and N, find the
Draw the “ideal” power curve for the test of Ho'-p < Mo against H\:p > mo-(That is, consider the best possible situation as far as power is concerned;for example, what would you like the power to be against any /x > po?)20.Suppose that X has an exponential distribution with parameter X:m =\e Xl
For the test in Exercise 16,(a) draw the power curve(b) draw the operating characteristic curve(c) draw the error curve.Do the same for the test in Exercise 17. Explain the relationship among the three types of curves in each case.
Do Exercise 16, considering H0:n< 20 against Hi :/x > 20.
Suppose that you are interested in X, the gasoline mileage (in miles per gallon) of a particular brand of gasoline. You decide to test H0:n = 20 against Hii/x 20.Assuming that X is normally distributed with variance 16, that N = 25, and that a = .01, find the region of rejection. What is the power
Suppose that you want to test the hypothesis Hoifi = 3 against Hi'-n ^ 3 for a normally distributed population with variance 1.44. Given a = .10 and N = 36, how powerful would this test be if m = 4? Given a = .10 and X = 16, how powerful would this test be if /x = 4? If the statistician wanted his
In Exercise 12, find the critical value of M and draw the power curve if a = .05 and you are testing Hq'-h > 75 against I7r./x< 75.Also, find the region of rejection and draw the power function if a = .05 and you are testing Ho:fi = 75 against Hi'.fi ^ 75.
In Exercise 12, find the critical value of M and draw the power curve if a = .20.Compare this power curve with the one obtained in Exercise 12.
Suppose that X is the height, in inches, of a population consisting of male basketball players, and you are interested in testing H0‘-n < 75 against Hx:n> 75.Assume that X is normally distributed with a variance of 9. A sample of size 100 is to be drawn from the population. If a = .05, what is
Suppose that you know that exactly one of the following two hypotheses must be true:Ho'p = 100,
If everything else is held constant, why does making a smaller lead to a larger/3, and vice versa? In what ways can the experimenter make both a and /3 smaller?
In Exercise 8, suppose that you stand to lose 5 cents if you decide that the bookbag contains 70R — 30B when it actually contains 70B — 30R, and you stand to lose 10 cents if the opposite error is made. Construct a table showing the two expected loss values, and consider the following
Suppose that a bookbag were filled with 100 poker chips. You know that either 70 of the chips are red and the remainder blue, or that 70 are blue and the remainder red. Ten chips are to be drawn successively with replace¬ment. You must decide whether the composition of the bookbag is 70R — 30B
Discuss the following proposition: “Much of the controversy surrounding the use of conventions such as that given in Section 7.9 can be explained in terms of whether hypothesis testing is viewed as a “signaling device,” as discussed in Section 7.11, or as a formal decision-making procedure.”
Some statisticians claim that one must never, under any circumstances, accept the “null” hypothesis as true. One can reject the null hypothesis with suffi¬cient evidence, they agree, but apparently one is in some sort of limbo when a result fails to be significant. Comment on this stand. Does
Distinguish between the two types of errors which are possible in hypothesis testing. In general, is it reasonable to assume that one type of error is always more serious than the other? If there are two types of error, why does the“classical” convention presented in Section 7.9 only involve
Discuss the propositions “Personal probabilities of the statistician have no place in rigorous scientific investigation”, and “All probabilities are inter¬pretable as personal probabilities.”
Make up an example of a decision problem in your area of interest which is somewhat similar to the example in Section 7.6. Formulate several decision rules and try to decide among them on the basis of(a) the minimax-loss criterion(b) the expected loss criterion(c) the subjective expected loss
In Section 7.5, Decision-rule 3 was declared to be inadmissible. Suppose that this rejection region or its equivalent were used in a test of H0: M = 0 against Hi: p 0. What property or properties of a “good” test would be violated, according to Section 7.19?
Explain the difference between a statistical hypothesis and what is generally thought of as a scientific hypothesis.
For the data in Exercise 35, find 90 percent and 98 percent confidence intervals for the ratio of the variances of weights from the two machines.
If X and Y are normally distributed and we let Z = cX + d and W = bY + g, discuss the relationship between the F ratio for ox2 and cry2 and the F ratio for crz2 and aw2- Consider the following two cases:(a) c = b and d = g(b) c b and d ^ g.How would the t distribution for the difference between
Given the fact that the expected value of an F variable with vi and v2 degrees of freedom is equal to v2/\(V2 — 2), for v2 > 2, prove that the variance of a t variable with v degrees of freedom is v/ (v — 2).
In Exercise 34, find a 95 percent confidence interval for the ratio of the vari¬ance of the freshman population to the variance of the senior population(
Using the data in Exercise 33, find a 90 percent confidence interval for a2 based on(a) the first sample only(b) the second sample only(c) both samples.Repeat the process under the assumption that the sample sizes were 10 and 5, respectively.
Would the sampling distribution in Exercise 42 change if the samples were drawn from normal populations with different means but with the same vari¬ance? Explain.
Suppose that two independent samples from the same normal population were pooled to form the following estimator of the population variance:_ NlSi2 + AW~~ Nx + N2 - 2 'Discuss the sampling distribution of this pooled estimator of a2.
In Exercise 20, if s2 is the sample variance of the sample of size 25, find(a) PCs2 < 198.40)(b) P (304.64 < s2 < 687.68)(c) P(531.20 < s2)(d) E (s2).
Suppose that a sample of size 8 is chosen randomly from a normally distrib¬uted population with mean 24 and variance 9. The standardized value cor¬responding to each observed value is computed by subtracting 24 and dividing the result by 3. Let T denote the sum of the squares of the 8
For samples from normal populations, the variance of the sampling distri¬bution of s2, the unbiased estimator of cr2, is 2 cr4/(N — 1). Prove this, using the fact that the mean of a chi-square random variable is equal to the number of degrees of freedom. [Hint: Remember that the variance of the
Under what circumstances might one be interested in the lower tail of a chi-square distribution?
Show that the fact that [N/ (N — l)]s2 is an unbiased estimator of a2 and that iVs2/a2 is a chi-square variable implies that the expectation of a chi-square random variable with IV — 1 degrees of freedom is simply N — 1.
Assuming that the variances of two normally distributed populations are equal, develop a formula for finding a confidence interval for the sum of the means. [Hint: Consider the sampling distribution of mi + Pi-~\ Use this to
Independent random samples are taken from the output of two machines on a production line. The weight of each item is of interest. From the first ma¬chine, a sample of size 36 is taken, with sample mean weight of 120 grams and a sample variance of 4 grams. From the second machine, a sample of size
An IQ test is given to a randomly selected group of 10 freshmen at a given university and also to a randomly selected group of 5 seniors at the same university. For the freshmen, the sample mean is 120 and the (unbiased)sample variance is 196. For the seniors, the sample mean is 128 and
Suppose that daily changes in the Dow-Jones Average of Industrial Stocks are normally distributed and that the change on any given day is independent of the change on any other day. A random sample of 81 daily changes is obtained, with sample mean .20 and sample variance (s2) 1.50. Find a 90
In Exercise 19, assume that the variance of X is unknown and find 80 percent and 95 percent confidence intervals for p. Compare these with the intervals computed in parts (d) and (e) of Exercise 19.
Does it seem reasonable that the t distribution should have “fatter tals”than the normal distribution? Why?
Why is it that the probability that (M — p)/cj is greater than 1.65 is only.05 for samples of size 5 from a normal distribution, whereas the probability that (M — ya)/est. cr is greater than 1.65 is approximately .09 for samples of the same size from the same population?
In a sample of 100 items from a production process, 4 were found to be defec¬tive. Assuming that the production process behaves like a Bernoulli process, determine 95 percent and 99.7 percent confidence intervals for p, the prob¬ability of obtaining a defective item on any single trial.
Discuss the following statement: “The determination of sample size in an experiment involves a balancing of the cost of taking the sample and the desired precision of the results of the sample.”
In determining a (1 —a) percent confidence interval for the mean of a normal population, assuming that the variance is known, how large a sample is needed to make the confidence interval 1/3 as large as it is when the sample size is A? What are the implications of this result?
Suppose that we take a sample of size 1 from an exponential distribution with parameter X. Can you find a 90 percent confidence interval for X based on this single sample observation?
Given a sample mean of 45, a population variance of 25, and a sample of size 30 drawn from a normal population, establish a 99 percent confidence interval for the population mean based on this sample. Can you find another pair of limits for which it will also be true that the probability of these
In the process of estimating the mean of a population, a statistician wants the probability to be .95 that his estimate will come within ,2
Suppose that you find a 90 percent confidence interval for the parameter n, based on a particular sample, and that the confidence limits are 50 and 60.In terms of the long-run frequency interpretation of probability, how should this confidence statement be interpreted? Might a different
An educator is interested in the mean IQ in a given population (say, the freshman class at a certain university). He cannot afford to test the entire population, but he is willing to assume that IQ is normally distributed with variance 121 in the population of interest. He wants to take a random
Based on historical data, the variability of the diameter of a part produced by a particular machine can be represented by a standard deviation of .1 inch. However, nothing is known about the population mean. A sample of size 100 is taken, and the sample mean diameter is 8.4 inches. Find 95 percent
Suppose that in a certain population of men, the mean weight is p and the variance is 400. If it is assumed that weight is normally distributed in this population, and if a random sample (with replacement) of 25 persons from the population yields a sample mean weight of 168, find 50 percent and 98
The random variable X is normally distributed with unknown mean p and known variance a2 = 225. A sample of size N = 9 from X is observed, with the sample values 42, 56, 68, 56, 48, 36, 45, 71, and 64.(a) Compute the sample mean and sample variance.(b) Name three intuitively reasonable point
If you have two independent sample proportions drawn from the same Bernoulli process, determine a pooled estimate of the Bernoulli parameter p which is unbiased. Denote the two sample proportions by P\ and P2, and assume that they are based on samples of size Ni and N2, respectively.
In the text, formula (6.9.1) gives a way to form a pooled estimate of the mean based on two nonoverlapping samples. Try to arrive at a similar ex¬pression that applies when the first sample, of size Ni, and the second sample, of size N2, have exactly N3 cases in common. {Hint: A Venn diagram may
On the basis of a sample of size N from a Poisson process, determine an esti¬mator of the Poisson parameter X by using (a) the principle of maximum likelihood, (b) the method of moments.
For a sample of size N from an exponential distribution with parameter X, find the maximum-likelihood estimator of X. Suppose that X is the intensity per hour of the arrival of cars at a toll booth. If a sample of size ten is taken, with the following amounts of time between arrivals, compute the
Using the sampling distributions of the sample mean and sample median determined in Exercises 28 and 29 in Chapter 5, compare the efficiency of these two estimators of the population mean for the given population dis¬tribution.
Prove that the kth sample moment is an unbiased estimator of the &th popu¬lation moment of a random variable (provided, of course, that the kth pop¬ulation moment exists).
For samples of three independent observations from the distribution given in Exercise 11, find (by enumeration) the sampling distribution of the sample variance (s2), the modified sample variance (s2), and the sample range (the difference between the largest and smallest values in the sample).(a)
Given the following very simple population distribution, construct the theo¬retical sampling distribution for the sample mean based on three independent observations from the population.x P(X = x)1 1/3 2 1/3 3 1/3[Hint: Enumerate the possible combinations of events for samples of size 3, and then
Consider a distribution of the random variable X such that the density function is given by?
Given that E{X) = X and var {X) = X for the Poisson distribution, show that sample means based on samples of N independent observations from a Poisson process are(a) unbiased estimators of X(b) distributed with variance \/N(c) consistent estimators of X(d) relatively more efficient as estimators of
Suppose that we have a sample of N independent observations from a certain population. We want to estimate the population mean, p. Under what con¬ditions is the linear combination NG = £ alXi i=1 an unbiased estimator of n? {Hint: Find the conditions which the a; must satisfy for E(G) = /x-3
If G is an unbiased estimator of 6, under what conditions would it be possible for G2 to be an unbiased estimator of 02? Explain your answer.
Using the definition of bias given by Equation (6.4.4), find the bias of s2 as an estimator of a2.

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