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John E Freunds Mathematical Statistics With Applications 8th Edition Irwin Miller, Marylees Miller - Solutions

In a random sample of 250 television viewers in a large city, 190 had seen a certain controversial program. Construct a 99% confidence interval for the corresponding true proportion using (a) The large-sample confidence- interval formula of Theorem 11.6; (b) The confidence limits of Exercise 11.11.
Among 100 fish caught in a certain lake, 18 were inedible as a result of chemical pollution. Construct a 99% confidence interval for the corresponding true proportion.
In a random sample of 120 cheerleaders, 54 had suffered moderate to severe damage to their voices. With 90% confidence, what can we say about the maximum error if we use the sample proportion 54/120 = 0.45 as an estimate of the true proportion of cheerleaders who are afflicted in this way?
In a random sample of 300 persons eating lunch at a department store cafeteria, only 102 had dessert. If we use 102/300 = 0.34 as an estimate of the corresponding true proportion, with what confidence can we assert that our error is less than 0.05?
Use the result of Exercise 11.13 to rework Exercise 11.45, given that the poll has reason to believe that the true proportion does not exceed 0.30.
In a random sample of visitors to a famous tourist attraction, 84 of 250 men and 156 of 250 women bought souvenirs. Construct a 95% confidence interval for the difference between the true proportions of men and women who buy souvenirs at this tourist attraction.
Show that among all (1 €“ Î±) 100% confidence intervals of the formThe one with k = 0.5 is the shortest.
Among 500 marriage license applications chosen at random in a given year, there were 48 in which the woman was at least one year older than the man, and among 400 marriage license applications chosen at random six years later, there were 68 in which the woman was at least one year older than the
With reference to Exercise 11.50, what can we say with 98% confidence about the maximum error if we use the difference between the observed sample proportions as an estimate of the difference between the corresponding true proportions?
With reference to Exercise 11.30, construct a 95% confidence interval for the true variance of the skull length of the given species of bird. In exercise The length of the skulls of 10 fossil skeletons of an extinct species of bird has a mean of 5.68 cm and a standard deviation of 0.29 cm. Assuming
With reference to Exercise 11.32, construct a 90% confidence interval for the standard deviation of the population sampled, that is, for the percentage of impurities in the given brand of peanut butter.
With reference to Exercise 11.24, use the large-sample confidence- interval formula of Exercise 11.19 to construct a 99% confidence interval for the standard deviation of the annual growth of the given kind of cactus. In exercise A study of the annual growth of certain cacti showed that 64 of them,
With reference to Exercise 11.25, use the large-sample confidence- interval formula of Exercise 11.19 to construct a 98% confidence interval for the standard deviation of the time it takes a mechanic to perform the given task. In exercise A food inspector, examining 12 jars of a certain brand of
With reference to Exercise 11.34, construct a 98% confidence interval for the ratio of the variances of the two populations sampled. In exercise A study of two kinds of photocopying equipment shows that 61 failures of the first kind of equipment took on the average 80.7 minutes to repair with a
With reference to Exercise 11.35, construct a 98% confidence interval for the ratio of the variances of the two populations sampled. In exercise Twelve randomly selected mature citrus trees of one variety have a mean height of 13.8 feet with a standard deviation of 1.2 feet, and 15 randomly
With reference to Exercise 11.36, construct a 90% confidence interval for the ratio of the variances of the two populations sampled.In exerciseThe following are the heat- producing capacities of coal from two mines (in millions of calories per ton):Assuming that the data constitute independent
Show that if x is used as a point estimate of µ and s is known, the probability is 1 €“ Î± that | €“ µ|, the absolute value of our error, will not exceed a specified amount e when(If it turns out that n
Twenty pilots were tested in a flight simulator, and the time for each to complete a certain corrective action was measured in seconds, with the following results:Use a computer program to find a 95% confidence interval for the mean time to take corrective action.
The following are the compressive strengths (given to the nearest 10 psi) of 30 concrete samples.Use a computer program to find a 90% confidence interval for the standard deviation of these compressive strengths.
Modify Theorem 11.1 so that it can be used to appraise the maximum error when σ2 is unknown. This method can be used only after the data have been obtained.
State a theorem analogous to Theorem 11.1, which enables us to appraise the maximum error in using 1 – 2 as an estimate of µ1 – µ2 under the conditions of Theorem 11.4.
Show that S2p is an unbiased estimator of σ2 and find its variance under the conditions of Theorem 11.5.
Decide in each case whether the hypothesis is simple or composite: (a) The hypothesis that a random variable has a gamma distribution with α = 3 and β = 2; (b) The hypothesis that a random variable has a gamma distribution with α = 3 and β ≠ 2; (c) The hypothesis that a random variable has
Show that if µ1 < µ0 in Example 12.4, the Neyman-Pearson lemma yields the critical region
A random sample of size n from an exponential population is used to test the null hypothesis θ = θ0 against the alternative hypothesis θ = θ1 > θ0. Use the Neyman-Pearson lemma to find the most powerful critical region of size α, and use the result of Example 7.16 on page 222 to indicate how
Use the Neyman-Pearson lemma to indicate how to construct the most powerful critical region of size α to test the null hypothesis θ = θ0, where θ is the parameter of a binomial distribution with a given value of n, against the alternative hypothesis θ = θ1 < θ0.
With reference to Exercise 12.12, if n = 100, θ0 = 0.40, θ1 = 0.30, and a is as large as possible with-out exceeding 0.05, use the normal approximation to the binomial distribution to find the probability of committing a type II error. In exercise Use the Neyman-Pearson lemma to indicate how to
A single observation of a random variable having a geometric distribution is to be used to test the null hypothesis that its parameter equals θ0 against the alter-native that it equals θ1 > θ0. Use the Neyman-Pearson lemma to find the best critical region of size α.
Given a random sample of size n from a normal population with µ = 0, use the Neyman-Pearson lemma to construct the most powerful critical region of size α to test the null hypothesis σ = σ0 against the alternative σ = σ1 > σ0.
Suppose that in Example 12.1 the manufacturer of the new medication feels that the odds are 4 to 1 that with this medication the recovery rate from the disease is 0.90 rather than 0.60. With these odds, what are the probabilities that he will make a wrong decision if he uses the decision
With reference to Exercise 12.3, suppose that we had wanted to test the null hypothesis k ≤ 2 against the alternative hypothesis k > 2. Find the probabilities of (a) Type I errors for k = 0, 1, and 2; (b) Type II errors for k = 4, 5, 6, and 7. Also plot the graph of the corresponding power
With reference to Example 12.5, suppose that we reject the null hypothesis if x ≤ 15 and accept it if x > 15. Calculate µ(θ) for the same values of θ as in the table on page 341 and plot the graph of the power function of this test criterion. Example 12.5 With reference to Example 12.1,
In the solution of Example 12.6, verify the step that led to
Decide in each case whether the hypothesis is simple or composite: (a) The hypothesis that a random variable has a Poisson distribution with λ = 1.25; (b) The hypothesis that a random variable has a Poisson distribution with λ > 1.25; (c) The hypothesis that a random variable has a normal
The number of successes in n trials is to be used to test the null hypothesis that the parameter Î¸ of a binomial population equals 12 against the alternative that it does not equal 12.(a) Find an expression for the likelihood ratio statistic.(b) Use the result of part (a) to show that
A random sample of size n is to be used to test the null hypothesis that the parameter Î¸ of an exponential population equals Î¸0 against the alternative that it does not equal Î¸0.(a) Find an expression for the likelihood ratio statistic.(b) Use the result of
A random sample of size n from a normal population with unknown mean and variance is to be used to test the null hypothesis µ = µ0 against the alternative µ ‰ µ0. Using the simultaneous maximum likelihood estimates of µ and Ïƒ2 obtained in Example
For the likelihood ratio statistic of Exercise 12.22, show that €“2 ˆ™ ln Î» approaches t2 as n †’ ˆž.In exerciseA random sample of size n from a normal population with unknown mean and variance is to be used to test the null hypothesis µ =
Given a random sample of size n from a normal population with unknown mean and variance, find an expression for the likelihood ratio statistic for testing the null hypothesis σ = σ0 against the alternative hypothesis σ ≠ σ0.
Independent random samples of sizes n1, n2, . . . , and nk from k normal populations with unknown means and variances are to be used to test the null hypothesis Ïƒ21 = Ïƒ22 = · · · = Ïƒ2k against the alternative that these variances are not all
Show that for k = 2 the likelihood ratio statistic of Exercise 12.25 can be expressed in terms of the ratio of the two sample variances and that the likelihood ratio test can, therefore, be based on the F distribution.
When we test a simple null hypothesis against a composite alternative, a critical region is said to be unbiased if the corresponding power function takes on its minimum value at the value of the parameter assumed under the null hypothesis. In other words, a critical region is unbiased if the
An airline wants to test the null hypothesis that 60 percent of its passengers object to smoking inside the plane. Explain under what conditions they would be committing a type I error and under what conditions they would be committing a type II error.
A doctor is asked to give an executive a thorough physical checkup to test the null hypothesis that he will be able to take on additional responsibilities. Explain under what conditions the doctor would be committing a type I error and under what conditions he would be committing a type II error.
A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the probabilities of
The average drying time of a manufacturer’s paint is 20 minutes. Investigating the effectiveness of a modification in the chemical composition of her paint, the manufacturer wants to test the null hypothesis µ = 20 minutes against a suitable alternative, where µ is the average drying time of
A city police department is considering replacing the tires on its cars with a new brand tires. If µ1 is the average number of miles that the old tires last and µ2 is the average number of miles that the new tires will last, the null hypothesis to be tested is µ1 = µ2. (a) What alternative
A botanist wishes to test the null hypothesis that the average diameter of the flowers of a particular plant is 9.6 cm. He decides to take a random sample of size n = 80 and accept the null hypothesis if the mean of the sample falls between 9.3 cm and 9.9 cm; if the mean of this sample falls
An education specialist is considering the use of instructional material on compact discs for a special class of third-grade students with reading disabilities. Students in this class are given a standardized test in May of the school year, and µ1 is the average score obtained on these tests after
Suppose that we want to test the null hypothesis that an antipollution device for cars is effective. (a) Explain under what conditions we would commit a type I error and under what conditions we would commit a type II error. (b) Whether an error is a type I error or a type II error depends on how
Rework Example 12.3 with (a) β = 0.03; (b) β = 0.01. 12.38. Example 12.3With reference to Example 12.2, determine the minimum sample size needed to test the null hypothesis µ0 = 10 against the alternative hypothesis µ1 = 11 with β ≤ 0.06.
A single observation is to be used to test the null hypothesis that the mean waiting time between tremors recorded at a seismological station (the mean of an exponential population) is θ = 10 hours against the alternative that θ ≠ 10 hours. If the null hypothesis is to be rejected if and only
With reference to Example 12.1, what would have been the probabilities of type I and type II errors if the acceptance region had been x > 16 and the corresponding rejection region had been x ≤ 16? Example 12.1 Suppose that the manufacturer of a new medication wants to test the null hypothesis θ
A random sample of size 64 is to be used to test the null hypothesis that for a certain age group the mean score on an achievement test (the mean of a normal population with σ2 = 256) is less than or equal to 40.0 against the alternative that it is greater than 40.0. If the null hypothesis is to
The sum of the values obtained in a random sample of size n = 5 is to be used to test the null hypothesis that on the average there are more than two accidents per week at a certain intersection (that λ > 2 for this Poisson population) against the alternative hypothesis that on the average the
Verify the statement on page 343 that 57 heads and 43 tails in 100 flips of a coin do not enable us to reject the null hypothesis that the coin is perfectly balanced (against the alternative that it is not perfectly balanced) at the 0.05 level of significance.
To compare the variations in weight of four breeds of dogs, researchers took independent random samples of sizes n1 = 8, n2 = 10, n3 = 6, and n4 = 8, and got σ21 = 16, σ22 = 25, σ23 = 12, and σ24 = 24. Assuming that the populations sampled are normal, use the formula of part (b) of Exercise
The times to failure of certain electronic components in accelerate environment tests are 15, 28, 3, 12, 42, 19, 20, 2, 25, 30, 62, 12, 18, 16, 44, 65, 33, 51, 4, and 28 minutes. Looking upon these data as a random sample from an exponential population, use the results of Exercise 12.21 and Theorem
A single observation of a random variable having a geometric distribution is used to test the null hypothesis θ = θ0 against the alternative hypothesis θ = θ1 > θ0. If the null hypothesis is rejected if and only if the observed value of the random variable is greater than or equal to the
A single observation of a random variable having an exponential distribution is used to test the null hypothesis that the mean of the distribution is θ = 2 against the alternative that it is θ = 5. If the null hypothesis is accepted if and only if the observed value of the random variable is less
Let X1 and X2 constitute a random sample from a normal population with σ2 = 1. If the null hypothesis µ = µ0 is to be rejected in favor of the alternative hypothesis µ = µ1 > µ0 when > µ0 + 1, what is the size of the critical region?
A single observation of a random variable having a uniform density with α = 0 is used to test the null hypothesis β = β0 against the alternative hypothesis β = β0 + 2. If the null hypothesis is rejected if and only if the random variable takes on a value greater than β0 + 1, find the
Let X1 and X2 constitute a random sample of size 2 from the population given byIf the critical region x1x2 ≥ 3/4 is used to test the null hypothesis θ = 1 against the alternative hypothesis θ = 2, what is the power of this test at θ = 2?
Given a random sample of size n from a normal population with the known variance σ2, show that the null hypothesis µ = µ0 can be tested against the alternative hypothesis µ ≠ µ0 with the use of a one- tailed criterion based on the chi- square distribution.
With reference to Exercise 13.9, use Table II on page 492 to find values corresponding to k0.025 and k'0.025 to test the null hypothesis λ = 3.6 against the alternative hypothesis λ ≠ 3.6 on the basis of five observations. Use the 0.05 level of significance. In exercise Modify the critical
For k = 2, show that the θ2 formula on page 369 can be written as
Given large random samples from two binomial populations, show that the null hypothesis θ1 = θ2 can be tested on the basis of the statisticWhere
Show that the square of the expression for z in Exercise 13.12 equalsSo that the two tests are actually equivalent when the alternative hypothesis is Î¸1 ‰ Î¸2. The test described in Exercise 13.12, but not the one based on the Î¸2 statistic, can be
Verify that if the expected cell frequencies are calculated in accordance with the rule on page 372, their sum for any row or column equals the sum of the corresponding observed frequencies.
Show that the rule on page 372 for calculating the expected cell frequencies applies also when we test the null hypothesis that we are sampling r populations with identical multinomial distributions.
Show that the following computing formula for x2 is equivalent to the formula on page 372:
Use the formula of Exercise 13.16 to recalculate x2 for Example 13.10.
If the analysis of a contingency table shows that there is a relationship between the two variables under consideration, the strength of this relationship may be measured by means of the contingency coefficientWhere x2 is the value obtained for the test statistic, and f is the grand total as
Suppose that a random sample from a normal population with the known variance Ïƒ2 is to be used to test the null hypothesis µ = µ0 against the alternative hypothesis µ = µ1, where µ1 > µ0, and that the probabilities of type I and type II errors are to
In the test of a certain hypothesis, the P-value corresponding to the test statistic is 0.0316. Can the null hypothesis be rejected at the (a) 0.01 level of significance; (b) 0.05 level of significance; (c) 0.10 level of significance?
With reference to Example 13.1, verify that the P-value corresponding to the observed value of the test statistic is 0.0046. Example 13.1 Suppose that it is known from experience that the standard deviation of the weight of 8- ounce packages of cookies made by a certain bakery is 0.16 ounce. To
With reference to Example 13.2, verify that the P-value corresponding to the observed value of the test statistic is 0.0808. Example 13.2 Suppose that 100 high-performance tires made by a certain manufacturer lasted on the average 21,819 miles with a standard deviation of 1,295 miles. Test the
With reference to Example 13.3, use suitable statistical software to find the P-value that corresponds to t = –0.49, where t is a value of a random variable having the t distribution with 4 degrees of freedom. Use this P-value to rework the example. Example 13.3 The specifications for a certain
Test at the 0.05 level of significance whether the mean of a random sample of size n = 16 is “significantly less than 10” if the distribution from which the sample was taken is normal, = 8.4, and σ = 3.2. What are the null and alternative hypotheses for this test?
According to the norms established for a reading comprehension test, eighth graders should average 84.3 with a standard deviation of 8.6. If 45 randomly selected eighth graders from a certain school district averaged 87.8, use the four steps on page 354 to test the null hypothesis µ = 84.3 against
The security department of a factory wants to know whether the true average time required by the night guard to walk his round is 30 minutes. If, in a random sample of 32 rounds, the night guard averaged 30.8 minutes with a standard deviation of 1.5 minutes, determine whether this is sufficient
In 12 test runs over a marked course, a newly designed motorboat averaged 33.6 seconds with a standard deviation of 2.3 seconds. Assuming that it is reason-able to treat the data as a random sample from a normal population, use the four steps on page 354 to test the null hypothesis µ = 35 against
With reference to the preceding exercise, find the required size of the sample when σ = 9, µ0 = 15, µ1 = 20, α = 0.05, and β = 0.01.
Five measurements of the tar content of a certain kind of cigarette yielded 14.5, 14.2, 14.4, 14.3, and 14.6 mg/cigarette. Assuming that the data are a random sample from a normal population, use the four steps on page 354 to show that at the 0.05 level of significance the null hypothesis µ = 14.0
With reference to Exercise 13.30, use suitable statistical software to find the P-value that corresponds to the observed value of the test statistic. Use this P-value to rework the exercise. In exercise Five measurements of the tar content of a certain kind of cigarette yielded 14.5, 14.2, 14.4,
If the same hypothesis is tested often enough, it is likely to be rejected at least once, even if it is true. A professor of biology, attempting to demonstrate this fact, ran white mice through a maze to determine if white mice ran the maze faster than the norm established by many previous tests
An epidemiologist is trying to discover the cause of a certain kind of cancer. He studies a group of 10,000 people for five years, measuring 48 different “factors” involving eating habits, drinking habits, smoking, exercise, and so on. His object is to determine if there are any differences in
With reference to Example 13.4, for what values of x1 – x2 would the null hypothesis have been rejected? Also find the probabilities of type II errors with the given criterion if (a) µ1 – µ2 = 0.12; (b) µ1 – µ2 = 0.16; (c) µ1 – µ2 = 0.24; (d) µ1 – µ2 = 0.28. Example 13.4 An
A study of the number of business lunches that executives in the insurance and banking industries claim as deductible expenses per month was based on random samples and yielded the following results: n1 = 40 x1 = 9.1 s1 = 1.9 n2 = 50 x2 = 8.0 s2 = 2.1 Use the four steps on page 354 and the 0.05
Rework Exercise 13.36, basing the decision on the P-value corresponding to the observed value of the test statistic. In exercise A study of the number of business lunches that executives in the insurance and banking industries claim as deductible expenses per month was based on random samples and
Sample surveys conducted in a large county in a certain year and again 20 years later showed that originally the average height of 400 ten-year-old boys was 53.8 inches with a standard deviation of 2.4 inches, whereas 20 years later the average height of 500 ten-year- old boys was 54.5 inches with
Rework Exercise 13.38, basing the decision on the P-value corresponding to the observed value of the test statistic. In exercise Sample surveys conducted in a large county in a certain year and again 20 years later showed that originally the average height of 400 ten-year-old boys was 53.8 inches
Suppose that independent random samples of size n from two normal populations with the known variances Ïƒ21 and Ïƒ22 are to be used to test the null hypothesis µ1 €“ µ2 = ˆ‚t against the alternative hypothesis µ1 €“ µ2 = d
To find out whether the inhabitants of two South Pacific islands may be regarded as having the same racial ancestry, an anthropologist determines the cephalic indices of six adult males from each island, getting 1 = 77.4 and 2 = 72.2 and the corresponding standard deviations s1 = 3.3 and s2 =
To compare two kinds of front- end designs, six of each kind were installed on a certain make of compact car. Then each car was run into a concrete wall at 5 miles per hour, and the following are the costs of the repairs (in dollars):
With reference to Exercise 13.42, use suitable statistical software to find the P-value corresponding to the observed value of the test statistic. Use this P-value to rework the exercise.In exerciseTo compare two kinds of front- end designs, six of each kind were installed on a certain make of
In a study of the effectiveness of certain exercises in weight reduction, a group of 16 persons engaged in these exercises for one month and showed the following results:Use the 0.05 level of significance to test the null hypothesis µ1 €“ µ2 = 0 against the alternative
The following are the average weekly losses of work- hours due to accidents in 10 industrial plants before and after a certain safety program was put into operation: 45 and 36, 73 and 60, 46 and 44, 124 and 119, 33 and 35, 57 and 51, 83 and 77, 34 and 29, 26 and 24, and 17 and 11 Use the four
Nine determinations of the specific heat of iron had a standard deviation of 0.0086. Assuming that these determinations constitute a random sample from a normal population, test the null hypothesis σ = 0.0100 against the alternative hypothesis σ < 0.0100 at the 0.05 level of significance.
In a random sample, the weights of 24 Black Angus steers of a certain age have a standard deviation of 238 pounds. Assuming that the weights constitute a random sample from a normal population, test the null hypothesis σ = 250 pounds against the two- sided alternative σ ≠ 250 pounds at the 0.01

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