Solved: The Department of Civil Engineering at Virginia

Problem 9E The article “Evaluation of a Ventilation Strategy to Prevent Barotrauma in Patients at High Risk for Acute Respiratory Distress Syndrome” (New Engl. J. of Med., 1998: 355–358) reported on an experiment in which 120 patients with similar clinical features were randomly divided into a control group and a treatment group, each consisting of 60 patients. The sample mean ICU stay (days) and sample standard deviation for the treatment group were 19.9 and 39.1, respectively, whereas these values for the control group were 13.7 and 15.8. a. Calculate a point estimate for the difference between true average ICU stay for the treatment and control groups. Does this estimate suggest that there is a significant difference between true average stays under the two conditions? b. Answer the question posed in part (a) by carrying out a formal test of hypotheses. Is the result different from what you conjectured in part (a)? c. Does it appear that ICU stay for patients given the ventilation treatment is normally distributed? Explain your reasoning. d. Estimate true average length of stay for patients given the ventilation treatment in a way that conveys information about precision and reliability.

Problem 15E a.? ?Show for the upper-tailed test with ?1 and ?2 known that as either ?m ?or ?n increases, ? decreases when . b.? ?For the case of equal sample sizes (m = n) and fixed ? , what happens to the necessary sample size ?n ?as ? is decreased, where ? is the desired type II error probability at a fixed alternative?

A UCLA researcher claims that the life span of mice can be extended by as much as 25% when the calories in their diet are reduced by approximately 40% from the time they are weaned. The restricted diet is enriched to normal levels by vitamins and protein. Assuming that it is known from previous studies that =5 .8 months, how many mice should be included in our sample if we wish to be 99% condent that the mean life span of the sample will be within 2 months of the population mean for all mice subjected to this reduced diet?

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm.

Many cardiac patients wear an implanted pacemaker to control their heartbeat. A plastic connector module mounts on the top of the pacemaker. Assuming a standard deviation of 0.0015 inch and an approximately normal distribution, nd a 95% condence interval for the mean of the depths of all connector modules made by a certain manufacturing company. A random sample of 75 modules has an average depth of 0.310 inch.

The heights of a random sample of 50 college students showed a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. (a) Construct a 98% condence interval for the mean height of all college students. (b) What can we assert with 98% condence about the possible size of our error if we estimate the mean height of all college students to be 174.5 centimeters?

A random sample of 100 automobile owners in the state of Virginia shows that an automobile is driven on average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Assume the distribution of measurements to be approximately normal. (a) Construct a 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia. (b) What can we assert with 99% confidence about the possible size of our error if we estimate the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year?

How large a sample is needed in Exercise 9.2 if we wish to be 96% condent that our sample mean will be within 10 hours of the true mean?

How large a sample is needed in Exercise 9.3 if we wish to be 95% condent that our sample mean will be within 0.0005 inch of the true mean?

An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will she need to be 95% confident that her sample mean will be within 15 seconds of the true mean? Assume that it is known from previous studies that \(\sigma = 40\) seconds.

Regular consumption of presweetened cereals contributes to tooth decay, heart disease, and other degenerative diseases, according to studies conducted by Dr. W. H. Bowen of the National Institute of Health and Dr. J. Yudben, Professor of Nutrition and Dietetics at the University of London. In a random sample consisting of 20 similar single servings of Alpha-Bits, the average sugar content was 11.3 grams with a standard deviation of 2.45 grams. Assuming that the sugar contents are normally distributed, construct a 95% confidence interval for the mean sugar content for single servings of Alpha-Bits.

A random sample of 12 graduates of a certain secretarial school typed an average of 79.3 words per minute with a standard deviation of 7.8 words per minute. Assuming a normal distribution for the number of words typed per minute, find a 95% confidence interval for the average number of words typed by all graduates of this school.

A machine produces metal pieces that are cylindrical in shape. A sample of pieces is taken, and the diameters are found to be 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99% condence interval for the mean diameter of pieces from this machine, assuming an approximately normal distribution.

A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean calorie content of this brand of energy bar. Assume that the distribution of the calorie content is approximately normal.

A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the pin head. Measurements on the Rockwell hardness are made for each of the 12, yielding an average value of 48.50 with a sample standard deviation of 1.5. Assuming the measurements to be normally distributed, construct a 90% confidence interval for the mean Rockwell hardness.

The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint: 3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 3.7 2.8 4.4 4.0 5.2 3.0 4.8 Assuming that the measurements represent a random sample from a normal population, nd a 95% prediction interval for the drying time for the next trial of the paint.

Referring to Exercise 9.5, construct a 99% prediction interval for the kilometers traveled annually by an automobile owner in Virginia.

Consider Exercise 9.10. Compute the 95% prediction interval for the next observed number of words per minute typed by a graduate of the secretarial school.

Consider Exercise 9.9. Compute a 95% prediction interval for the sugar content of the next single serving of Alpha-Bits.

Referring to Exercise 9.13, construct a 95% tolerance interval containing 90% of the measurements.

A random sample of 25 tablets of buered aspirin contains, on average, 325.05 mg of aspirin per tablet, with a standard deviation of 0.5 mg. Find the 95% tolerance limits that will contain 90% of the tablet contents for this brand of buered aspirin. Assume that the aspirin content is normally distributed.

Consider the situation of Exercise 9.11. Estimation of the mean diameter, while important, is not nearly as important as trying to pin down the location of the majority of the distribution of diameters. Find the 95% tolerance limits that contain 95% of the diameters.

In a study conducted by the Department of Zoology at Virginia Tech, fteen samples of water were collected from a certain station in the James River in order to gain some insight regarding the amount of orthophosphorus in the river. The concentration of the chemical is measured in milligrams per liter. Let us suppose that the mean at the station is not as important as the upper extreme of the distribution of the concentration of the chemical at the station. Concern centers around whether the concentration at the extreme is too large. Readings for the fteen water samples gave a sample mean of 3.84 milligrams per liter and a sample standard deviation of 3.07 milligrams per liter. Assume that the readings are a random sample from a normal distribution. Calculate a prediction interval (upper 95% prediction limit) and a tolerance limit (95% upper tolerance limit that exceeds 95% of the population of values). Interpret both; that is, tell what each communicates about the upper extreme of the distribution of orthophosphorus at the sampling station.

A type of thread is being studied for its tensile strength properties. Fifty pieces were tested under similar conditions, and the results showed an average tensile strength of 78.3 kilograms and a standard deviation of 5.6 kilograms. Assuming a normal distribution of tensile strengths, give a lower 95% prediction limit on a single observed tensile strength value. In addition, give a lower 95% tolerance limit that is exceeded by 99% of the tensile strength values.

Refer to Exercise 9.22. Why are the quantities requested in the exercise likely to be more important to the manufacturer of the thread than, say, a confidence interval on the mean tensile strength?

Refer to Exercise 9.22 again. Suppose that specifications by a buyer of the thread are that the tensile strength of the material must be at least 62 kilograms. The manufacturer is satisfied if at most 5% of the manufactured pieces have tensile strength less than 62 kilograms. Is there cause for concern? Use a one-sided 99% tolerance limit that is exceeded by 95% of the tensile strength values.

Consider the drying time measurements in Exercise 9.14. Suppose the 15 observations in the data set are supplemented by a 16th value of 6.9 hours. In the context of the original 15 observations, is the 16th value an outlier? Show work.

Consider the data in Exercise 9.13. Suppose the manufacturer of the shearing pins insists that the Rockwell hardness of the product be less than or equal to 44.0 only 5% of the time. What is your reaction? Use a tolerance limit calculation as the basis for your judgment.

Consider the situation of Case Study 9.1 on page 281 with a larger sample of metal pieces. The diameters are as follows: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 1.01, 1.03, 0.99, 1.00, 1.00, 0.99, 0.98, 1.01, 1.02, 0.99 centimeters. Once again the normality assumption may be made. Do the following and compare your results to those of the case study. Discuss how they are dierent and why. (a) Compute a 99% condence interval on the mean diameter. (b) Compute a 99% prediction interval on the next diameter to be measured. (c) Compute a 99% tolerance interval for coverage of the central 95% of the distribution of diameters.

In Section 9.3, we emphasized the notion of most ecient estimator by comparing the variance of two unbiased estimators 1 and 2. However, this does not take into account bias in case one or both estimators are not unbiased. Consider the quantity MSE = E( ), where MSE denotes mean squared error. The MSE is often used to compare two estimators 1 and 2 of when either or both is unbiased because (i) it is intuitively reasonable and (ii) it accounts for bias. Show that MSE can be written MSE = E[ E()]2 +[E( )]2 = Var() + [Bias()]2.

Let us define \(S’^{2} = \sum _{i=1} ^n (X_i - \overline{X})^2 /n\). Show that \(E(S’^{2}) = [(n - 1)/n] \sigma^2\), and hence \(S’^{2}\) is a biased estimator for \(\sigma^2\).

Consider \(S’^{2}\), the estimator of \(\sigma^2\), from Exercise 9.29. Analysts often use \(S’^{2}\) rather than dividing \(\sum _{i=1} ^n (X_i-\overline{X})^2\) by n - 1, the degrees of freedom in the sample. (a) What is the bias of \(S’^{2}\)? (b) Show that the bias of \(S’^{2}\) approaches zero as \(n \rightarrow \infty\).

If X is a binomial random variable, show that (a)  P = X/n is an unbiased estimator of p; (b) P = X+n/2 n+n is a biased estimator of p.

Show that the estimator P’ of Exercise 9.31(b) becomes unbiased as \(n\ \rightarrow\ \infty\).

Compare \(S^2\) and \(S’^{2}\) (see Exercise 9.29), the two estimators of \(\sigma^2\), to determine which is more efficient. Assume these estimators are found using \(X_1\),\(X_2\), . . . , \(X_n\), independent random variables from \(n(x; \mu, \sigma)\). Which estimator is more efficient considering only the variance of the estimators? [Hint: Make use of Theorem 8.4 and the fact that the variance of \(X^2 _v\) is 2v, from Section 6.7.]

Consider Exercise 9.33. Use the MSE discussed in Exercise 9.28 to determine which estimator is more ecient. Write out MSE(S2) MSE(S2) .

A random sample of size \(n_1 = 25\), taken from a normal population with a standard deviation \(\sigma_1 = 5\), has a mean \(\overline{x}_1 = 80\). A second random sample of size \(n_2 = 36\), taken from a different normal population with a standard deviation \(\sigma_2 = 3\), has a mean \(\overline{x}_2 = 75\). Find a 94% confidence interval for \(\mu_1 ? \mu_2\).

Two kinds of thread are being compared for strength. Fifty pieces of each type of thread are tested under similar conditions. Brand A has an average tensile strength of 78.3 kilograms with a standard deviation of 5.6 kilograms, while brand B has an average tensile strength of 87.2 kilograms with a standard deviation of 6.3 kilograms. Construct a 95% confidence interval for the difference of the population means.

A study was conducted to determine if a certain treatment has any eect on the amount of metal removed in a pickling operation. A random sample of 100 pieces was immersed in a bath for 24 hours without the treatment, yielding an average of 12.2 millimeters of metal removed and a sample standard deviation of 1.1 millimeters. A second sample of 200 pieces was exposed to the treatment, followed by the 24-hour immersion in the bath, resulting in an average removal of 9.1 millimeters of metal with a sample standard deviation of 0.9 millimeter. Compute a 98% condence interval estimate for the dierence between the population means. Does the treatment appear to reduce the mean amount of metal removed?

Two catalysts in a batch chemical process, are being compared for their effect on the output of the process reaction. A sample of 12 batches was prepared using catalyst 1, and a sample of 10 batches was prepared using catalyst 2. The 12 batches for which catalyst 1 was used in the reaction gave an average yield of 85 with a sample standard deviation of 4, and the 10 batches for which catalyst 2 was used gave an average yield of 81 and a sample standard deviation of 5. Find a 90% confidence interval for the difference between the population means, assuming that the populations are approximately normally distributed with equal variances.

Students may choose between a 3-semester-hour physics course without labs and a 4-semester-hour course with labs. The nal written examination is the same for each section. If 12 students in the section with labs made an average grade of 84 with a standard deviation of 4, and 18 students in the section without labs made an average grade of 77 with a standard deviation of 6, nd a 99% condence interval for the dierence between the average grades for the two courses. Assume the populations to be approximately normally distributed with equal variances.

In a study conducted at Virginia Tech on the development of ectomycorrhizal, a symbiotic relationship between the roots of trees and a fungus, in which minerals are transferred from the fungus to the trees and sugars from the trees to the fungus, 20 northern red oak seedlings exposed to the fungus Pisolithus tinctorus were grown in a greenhouse. All seedlings were planted in the same type of soil and received the same amount of sunshine and water. Half received no nitrogen at planting time, to serve as a control, and the other half received 368 ppm of nitrogen in the form NaNO3. The stem weights, in grams, at the end of 140 days were recorded as follows: No Nitrogen Nitrogen 0.32 0.26 0.53 0.43 0.28 0.47 0.37 0.49 0.47 0.52 0.43 0.75 0.36 0.79 0.42 0.86 0.38 0.62 0.43 0.46 Construct a 95% condence interval for the dierence in the mean stem weight between seedlings that receive no nitrogen and those that receive 368 ppm of nitrogen. Assume the populations to be normally distributed with equal variances.

The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections: Find a 99% confidence interval for the difference \(\mu_2 ? \mu_1\) in the mean recovery times for the two medications, assuming normal populations with equal variances.

An experiment reported in Popular Science compared fuel economies for two types of similarly equipped diesel mini-trucks. Let us suppose that 12 Volkswagen and 10 Toyota trucks were tested in 90kilometer-per-hour steady-paced trials. If the 12 Volkswagen trucks averaged 16 kilometers per liter with a standard deviation of 1.0 kilometer per liter and the 10 Toyota trucks averaged 11 kilometers per liter with a standard deviation of 0.8 kilometer per liter, construct a 90% condence interval for the dierence between the average kilometers per liter for these two mini-trucks. Assume that the distances per liter for the truck models are approximately normally distributed with equal variances.

A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are Brand A: \(\overline{x}_1 = 36, 300\) kilometers, \(s_1 = 5000\) kilometers. Brand B: \(\overline{x}_2 = 38, 100\) kilometers, \(s_2 = 6100\) kilometers. Compute a 95% confidence interval for \(\mu_A ? \mu_B\) assuming the populations to be approximately normally distributed. You may not assume that the variances are equal.

Referring to Exercise 9.43, nd a 99% condence interval for 1 2 if tires of the two brands are assigned at random to the left and right rear wheels of 8 taxis and the following distances, in kilometers, are recorded: Taxi Brand A Brand B 1 34,400 36,700 2 45,500 46,800 3 36,700 37,700 4 32,000 31,100 5 48,400 47,800 6 32,800 36,400 7 38,100 38,900 8 30,100 31,500 Assume that the dierences of the distances are approximately normally distributed.

The federal government awarded grants to the agricultural departments of 9 universities to test the yield capabilities of two new varieties of wheat. Each variety was planted on a plot of equal area at each university, and the yields, in kilograms per plot, were recorded as follows: University Variety 1 2 3 4 5 6 7 8 9 1 38 23 35 41 44 29 37 31 38 2 45 25 31 38 50 33 36 40 43 Find a 95% condence interval for the mean dierence between the yields of the two varieties, assuming the dierences of yields to be approximately normally distributed. Explain why pairing is necessary in this problem.

The following data represent the running times of lms produced by two motion-picture companies. Company Time (minutes) I 103 94 110 87 98 II 97 82 123 92 175 88 118 Compute a 90% condence interval for the dierence between the average running times of lms produced by the two companies. Assume that the running-time differences are approximately normally distributed with unequal variances.

Fortune magazine (March 1997) reported the total returns to investors for the 10 years prior to 1996 and also for 1996 for 431 companies. The total returns for 10 of the companies are listed below. Find a 95% confidence interval for the mean change in percent return to investors.

An automotive company is considering two types of batteries for its automobile. Sample information on battery life is collected for 20 batteries of type A and 20 batteries of type B. The summary statistics are xA = 32 .91, xB = 30 .47, sA =1 .57, and sB =1 .74. Assume the data on each battery are normally distributed and assume A = B. (a) Find a 95% condence interval on A B. (b) Draw a conclusion from (a) that provides insight into whether A or B should be adopted.

Two dierent brands of latex paint are being considered for use. Fifteen specimens of each type of paint were selected, and the drying times, in hours, were as follows: Paint A Paint B 3.5 2.7 3.9 4.2 3.6 4.7 3.9 4.5 5.5 4.0 2.7 3.3 5.2 4.2 2.9 5.3 4.3 6.0 5.2 3.7 4.4 5.2 4.0 4.1 3.4 5.5 6.2 5.1 5.4 4.8 Assume the drying time is normally distributed with A = B. Find a 95% condence interval on B A, where A and B are the mean drying times.

Two levels (low and high) of insulin doses are given to two groups of diabetic rats to check the insulinbinding capacity, yielding the following data: Low dose: n1 = 8 x1 =1 .98 s1 =0 .51 High dose: n2 = 13 x2 =1 .30 s2 =0 .35 Assume that the variances are equal. Give a 95% condence interval for the dierence in the true average insulin-binding capacity between the two samples.

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. In a random sample of 1000 homes in a certain city, it is found that 228 are heated by oil. Find 99% confidence intervals for the proportion of homes in this city that are heated by oil using both methods presented on page 297.

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. Compute 95% confidence intervals, using both methods on page 297, for the proportion of defective items in a process when it is found that a sample of size 100 yields 8 defectives.

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. (a) A random sample of 200 voters in a town is selected, and 114 are found to support an annexation suit. Find the 96% confidence interval for the fraction of the voting population favoring the suit. (b) What can we assert with 96% confidence about the possible size of our error if we estimate the fraction of voters favoring the annexation suit to be 0.57?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A manufacturer of MP3 players conducts a set of comprehensive tests on the electrical functions of its product. All MP3 players must pass all tests prior to being sold. Of a random sample of 500 MP3 players, 15 failed one or more tests. Find a 90% condence interval for the proportion of MP3 players from the population that pass all tests.

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. A new rocket-launching system is being considered for deployment of small, short-range rockets. The existing system has p = 0.8 as the probability of a successful launch. A sample of 40 experimental launches is made with the new system, and 34 are successful. (a) Construct a 95% confidence interval for p. (b) Would you conclude that the new system is better?

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. A geneticist is interested in the proportion of African males who have a certain minor blood disorder. In a random sample of 100 African males, 24 are found to be afflicted. (a) Compute a 99% confidence interval for the proportion of African males who have this blood disorder. (b) What can we assert with 99% confidence about the possible size of our error if we estimate the proportion of African males with this blood disorder to be 0.24?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. (a) According to a report in the Roanoke Times & World-News, approximately 2/3 of 1600 adults polled by telephone said they think the space shuttle program is a good investment for the country. Find a 95% condence interval for the proportion of American adults who think the space shuttle program is a good investment for the country. (b) What can we assert with 95% condence about the possible size of our error if we estimate the proportion of American adults who think the space shuttle program is a good investment to be 2/3?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. In the newspaper article referred to in Exercise 9.57, 32% of the 1600 adults polled said the U.S. space program should emphasize scientic exploration. How large a sample of adults is needed for the poll if one wishes to be 95% condent that the estimated percentage will be within 2% of the true percentage?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. How large a sample is needed if we wish to be 96% condent that our sample proportion in Exercise 9.53 will be within 0.02 of the true fraction of the voting population?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. How large a sample is needed if we wish to be 99% condent that our sample proportion in Exercise 9.51 will be within 0.05 of the true proportion of homes in the city that are heated by oil?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. How large a sample is needed in Exercise 9.52 if we wish to be 98% condent that our sample proportion will be within 0.05 of the true proportion defective?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A conjecture by a faculty member in the microbiology department at Washington University School of Dental Medicine in St. Louis, Missouri, states that a couple of cups of either green or oolong tea each day will provide sucient uoride to protect your teeth from decay. How large a sample is needed to estimate the percentage of citizens in a certain town who favor having their water uoridated if one wishes to be at least 99% condent that the estimate is within 1% of the true percentage?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A study is to be made to estimate the percentage of citizens in a town who favor having their water uoridated. How large a sample is needed if one wishes to be at least 95% condent that the estimate is within 1% of the true percentage?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A study is to be made to estimate the proportion of residents of a certain city and its suburbs who favor the construction of a nuclear power plant near the city. How large a sample is needed if one wishes to be at least 95% condent that the estimate is within 0.04 of the true proportion of residents who favor the construction of the nuclear power plant?

In this set of exercises, for estimation concerning one proportion, use only method 1 to obtain the confidence intervals, unless instructed otherwise. A certain geneticist is interested in the proportion of males and females in the population who have a minor blood disorder. In a random sample of 1000 males, 250 are found to be afflicted, whereas 275 of 1000 females tested appear to have the disorder. Compute a 95% confidence interval for the difference between the proportions of males and females who have the blood disorder.

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. Ten engineering schools in the United States were surveyed. The sample contained 250 electrical engineers, 80 being women; 175 chemical engineers, 40 being women. Compute a 90% condence interval for the dierence between the proportions of women in these two elds of engineering. Is there a signicant dierence between the two proportions?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A clinical trial was conducted to determine if a certain type of inoculation has an eect on the incidence of a certain disease. A sample of 1000 rats was kept in a controlled environment for a period of 1 year, and 500 of the rats were given the inoculation. In the group not inoculated, there were 120 incidences of the disease, while 98 of the rats in the inoculated group contracted it. If p1 is the probability of incidence of the disease in uninoculated rats and p2 the probability of incidence in inoculated rats, compute a 90% condence interval for p1 p2.

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. In the study Germination and Emergence of Broccoli, conducted by the Department of Horticulture at Virginia Tech, a researcher found that at 5C, 10 broccoli seeds out of 20 germinated, while at 15C, 15 out of 20 germinated. Compute a 95% condence interval for the dierence between the proportions of germination at the two dierent temperatures and decide if there is a signicant dierence.

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. A survey of 1000 students found that 274 chose professional baseball team A as their favorite team. In a similar survey involving 760 students, 240 of them chose team A as their favorite. Compute a 95% condence interval for the dierence between the proportions of students favoring team A in the two surveys. Is there a signicant dierence?

For estimation concerning one proportion, use only method 1 to obtain the condence intervals, unless instructed otherwise. According to USA Today (March 17, 1997), women made up 33.7% of the editorial sta at local TV stations in the United States in 1990 and 36.2% in 1994. Assume 20 new employees were hired as editorial sta. (a) Estimate the number that would have been women in 1990 and 1994, respectively. (b) Compute a 95% condence interval to see if there is evidence that the proportion of women hired as editorial sta was higher in 1994 than in 1990.

A manufacturer of car batteries claims that the batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years, construct a 95% condence interval for 2 and decide if the manufacturers claim that 2 = 1 is valid. Assume the population of battery lives to be approximately normally distributed.

A random sample of 20 students yielded a mean of \(\overline{x} = 72\) and a variance of \(s^2 = 16\) for scores on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for \(\sigma^2\).

Construct a 95% condence interval for 2 in Exercise 9.9 on page 283.

Construct a 99% confidence interval for \(\sigma^2\) in Exercise 9.11 on page 283.

Construct a 99% condence interval for in Exercise 9.12 on page 283.

Construct a 90% condence interval for in Exercise 9.13 on page 283.

Construct a 98% condence interval for 1/2 in Exercise 9.42 on page 295, where 1 and 2 are, respectively, the standard deviations for the distances traveled per liter of fuel by the Volkswagen and Toyota mini-trucks.

Construct a 90% confidence interval for \(\sigma_1 ^2 / \sigma_2 ^2\) in Exercise 9.43 on page 295. Were we justified in assuming that \(\sigma_1 ^2\ \neq\ \sigma_2 ^2\) when we constructed the confidence interval for \(\mu_1 ? \mu_2\)?

Construct a 90% condence interval for 2 1/2 2 in Exercise 9.46 on page 295. Should we have assumed 2 1 = 2 2 in constructing our condence interval for I II?

Construct a 95% condence interval for 2 A/2 B in Exercise 9.49 on page 295. Should the equal-variance assumption be used?

Suppose that there are n trials x1,x2,...,xn from a Bernoulli process with parameter p, the probability of a success. That is, the probability of r successes is given byn rpr(1p)nr. Work out the maximum likelihood estimator for the parameter p.

Consider the lognormal distribution with the density function given in Section 6.9. Suppose we have a random sample x1,x2,...,xn from a lognormal distribution. (a) Write out the likelihood function. (b) Develop the maximum likelihood estimators of and 2.

Consider a random sample of \(x_1\), . . . , \(x_n\) coming from the gamma distribution discussed in Section 6.6. Suppose the parameter \(\alpha\) is known, say 5, and determine the maximum likelihood estimation for parameter \(\beta\).

Consider a random sample of x1,x2,...,xn ob servations from a Weibull distribution with parameters and and density function f(x)=x1ex, x > 0, 0, elsewhere, for , > 0. (a) Write out the likelihood function. (b) Write out the equations that, when solved, give the maximum likelihood estimators of and .

Consider a random sample of x1,...,xn from a uniform distribution U(0,) with unknown parameter , where >0. Determine the maximum likelihood estimator of .

Consider the independent observations x1,x2,...,xn from the gamma distribution discussed in Section 6.6. (a) Write out the likelihood function.(b) Write out a set of equations that, when solved, give the maximum likelihood estimators of and .

Consider a hypothetical experiment where a man with a fungus uses an antifungal drug and is cured. Consider this, then, a sample of one from a Bernoulli distribution with probability function \(f(x)=p^{x} q^{1-x}, \quad x=0,1,\) where p is the probability of a success (cure) and q = 1 ? p. Now, of course, the sample information gives x = 1. Write out a development that shows that \(\hat{p} = 1.0\) is the maximum likelihood estimator of the probability of a cure.

Consider the observation X from the negative binomial distribution given in Section 5.4. Find the maximum likelihood estimator for p, assuming k is known.

Consider two estimators of 2 for a sample x1,x2,...,xn, which is drawn from a normal distribution with mean and variance 2. The estimators are the unbiased estimator s2 = 1 n1 n  i=1 (xi x)2 and the maximum likelihood estimator 2 = 1 n n  i=1 (xi x)2. Discuss the variance properties of these two estimators

According to the Roanoke Times, McDonalds sold 42.1% of the market share of hamburgers. A random sample of 75 burgers sold resulted in 28 of them being from McDonalds. Use material in Section 9.10 to determine if this information supports the claim in the Roanoke Times.

It is claimed that a new diet will reduce a person’s weight by 4.5 kilograms on average in a period of 2 weeks. The weights of 7 women who followed this diet were recorded before and after the 2-week period. Test the claim about the diet by computing a 95% confidence interval for the mean difference in weights. Assume the differences of weights to be approximately normally distributed.

A study was undertaken at Virginia Tech to determine if re can be used as a viable management tool to increase the amount of forage available to deer during the critical months in late winter and early spring. Calcium is a required element for plants and animals. The amount taken up and stored in plants is closely correlated to the amount present in the soil. It was hypothesized that a re may change the calcium levels present in the soil and thus aect the amount available to deer. A large tract of land in the Fishburn Forest was selected for a prescribed burn. Soil samples were taken from 12 plots of equal area just prior to the burn and analyzed for calcium. Postburn calcium levels were analyzed from the same plots. These values, in kilograms per plot, are presented in the following table: Calcium Level (kg/plot) Plot Preburn Postburn 1 2 3 4 5 6 7 8 9 10 11 12 50 50 82 64 82 73 77 54 23 45 36 54 9 18 45 18 18 9 32 9 18 9 9 9 Construct a 95% condence interval for the mean difference in calcium levels in the soil prior to and after the prescribed burn. Assume the distribution of dierences in calcium levels to be approximately normal.

A health spa claims that a new exercise program will reduce a person’s waist size by 2 centimeters on average over a 5-day period. The waist sizes, in centimeters, of 6 men who participated in this exercise program are recorded before and after the 5-day period in the following table: By computing a 95% confidence interval for the mean reduction in waist size, determine whether the health spa’s claim is valid. Assume the distribution of differences in waist sizes before and after the program to be approximately normal.

The Department of Civil Engineering at Virginia Tech compared a modied (M-5 hr) assay technique for recovering fecal coliforms in stormwater runo from an urban area to a most probable number (MPN) technique. A total of 12 runo samples were collected and analyzed by the two techniques. Fecal coliform counts per 100 milliliters are recorded in the following table. Sample MPN Count M-5 hr Count 1 2 3 4 5 6 7 8 9 10 11 12 2300 1200 450 210 270 450 154 179 192 230 340 194 2010 930 400 436 4100 2090 219 169 194 174 274 183 Construct a 90% condence interval for the dierence in the mean fecal coliform counts between the M-5 hr and the MPN techniques. Assume that the count differences are approximately normally distributed.

An experiment was conducted to determine whether surface nish has an eect on the endurance limit of steel. There is a theory that polishing increases the average endurance limit (for reverse bending). From a practical point of view, polishing should not have any eect on the standard deviation of the endurance limit, which is known from numerous endurance limit experiments to be 4000 psi. An experiment was performed on 0.4% carbon steel using both unpolished and polished smooth-turned specimens. The data are as follows: Endurance Limit (psi) Polished Unpolished 0.4% Carbon 0.4% Carbon 85,500 82,600 91,900 82,400 89,400 81,700 84,000 79,500 89,900 79,400 78,700 69,800 87,500 79,900 83,100 83,400 Find a 95% condence interval for the dierence between the population means for the two methods, as suming that the populations are approximately normally distributed.

An anthropologist is interested in the proportion of individuals in two Indian tribes with double occipital hair whorls. Suppose that independent samples are taken from each of the two tribes, and it is found that 24 of 100 Indians from tribe A and 36 of 120 Indians from tribe B possess this characteristic. Construct a 95% condence interval for the dierence pB pA between the proportions of these two tribes with occipital hair whorls.

A manufacturer of electric irons produces these items in two plants. Both plants have the same suppliers of small parts. A saving can be made by purchasing thermostats for plant B from a local supplier. A single lot was purchased from the local supplier, and a test was conducted to see whether or not these new thermostats were as accurate as the old. The thermostats were tested on tile irons on the 550F setting, and the actual temperatures were read to the nearest 0.1F with a thermocouple. The data are as follows: New Supplier (F) 530.3 559.3 549.4 544.0 551.7 566.3 549.9 556.9 536.7 558.8 538.8 543.3 559.1 555.0 538.6 551.1 565.4 554.9 550.0 554.9 554.7 536.1 569.1 Old Supplier (F) 559.7 534.7 554.8 545.0 544.6 538.0 550.7 563.1 551.1 553.8 538.8 564.6 554.5 553.0 538.4 548.3 552.9 535.1 555.0 544.8 558.4 548.7 560.3 Find 95% condence intervals for 2 1/2 2 and for 1/2, where 2 1 and 2 2 are the population variances of the thermostat readings for the new and old suppliers, respectively.

It is argued that the resistance of wire A is greater than the resistance of wire B. An experiment on the wires shows the following results (in ohms): Wire A Wire B 0.140 0.135 0.138 0.140 0.143 0.136 0.142 0.142 0.144 0.138 0.137 0.140 Assuming equal variances, what conclusions do you draw? Justify your answer.

An alternative form of estimation is accomplished through the method of moments. This method involves equating the population mean and variance to the corresponding sample mean x and sample variance s2 and solving for the parameters, the results being the moment estimators. In the case of a single parameter, only the means are used. Give an argument that in the case of the Poisson distribution the maximum likelihood estimator and moment estimators are the same.

Specify the moment estimators for and 2 for the normal distribution.

Specify the moment estimators for and 2 for the lognormal distribution.

Specify the moment estimators for and for the gamma distribution.

A survey was done with the hope of comparing salaries of chemical plant managers employed in two areas of the country, the northern and west central regions. An independent random sample of 300 plant managers was selected from each of the two regions. These managers were asked their annual salaries. The results are as follows North West Central x1 = $102,300 x2 = $98,500 s1 = $5700 s2 = $3800 (a) Construct a 99% condence interval for 1 2, the dierence in the mean salaries. (b) What assumption did you make in (a) about the distribution of annual salaries for the two regions? Is the assumption of normality necessary? Why or why not? (c) What assumption did you make about the two variances? Is the assumption of equality of variances reasonable? Explain!

Consider Review Exercise 9.103. Let us assume that the data have not been collected yet and that previous statistics suggest that 1 = 2 = $4000. Are the sample sizes in Review Exercise 9.103 sucient to produce a 95% condence interval on 12 having a width of only $1000? Show all work.

A labor union is becoming defensive about gross absenteeism by its members. The union leaders had always claimed that, in a typical month, 95% of its members were absent less than 10 hours. The union decided to check this by monitoring a random sample of 300 of its members. The number of hours absent was recorded for each of the 300 members. The results were \(\overline{x} = 6.5\) hours and s = 2.5 hours. Use the data to respond to this claim, using a one-sided tolerance limit and choosing the confidence level to be 99%. Be sure to interpret what you learn from the tolerance limit calculation.

A random sample of 30 rms dealing in wireless products was selected to determine the proportion of such rms that have implemented new software to improve productivity. It turned out that 8 of the 30 had implemented such software. Find a 95% condence interval on p, the true proportion of such rms that have implemented new software.

Refer to Review Exercise 9.106. Suppose there is concern about whether the point estimate p =8 /30 is accurate enough because the condence interval around p is not suciently narrow. Using p as the estimate of p, how many companies would need to be sampled in order to have a 95% condence interval with a width of only 0.05?

A manufacturer turns out a product item that is labeled either “defective” or “not defective.” In order to estimate the proportion defective, a random sample of 100 items is taken from production, and 10 are found to be defective. Following implementation of a quality improvement program, the experiment is conducted again. A new sample of 100 is taken, and this time only 6 are found to be defective. (a) Give a 95% confidence interval on \(p_1 ? p_2\), where p1 is the population proportion defective before improvement and \(p_2\) is the proportion defective after improvement. (b) Is there information in the confidence interval found in (a) that would suggest that \(p_1\ >\ p_2\)? Explain.

A machine is used to ll boxes with product in an assembly line operation. Much concern centers around the variability in the number of ounces of product in a box. The standard deviation in weight of product is known to be 0.3 ounce. An improvement is implemented, after which a random sample of 20 boxes is selected and the sample variance is found to be 0.045 ounce2. Find a 95% condence interval on the variance in the weight of the product. Does it appear from the range of the condence interval that the improvement of the process enhanced quality as far as variability is concerned? Assume normality on the distribution of weights of product.

A consumer group is interested in comparing operating costs for two dierent types of automobile engines. The group is able to nd 15 owners whose cars have engine type A and 15 whose cars have engine type B. All 30 owners bought their cars at roughly the same time, and all have kept good records for a certain 12-month period. In addition, these owners drove roughly the same number of miles. The cost statistics are yA = $87.00/1000 miles, yB = $75.00/1000 miles, sA = $5 .99, and sB = $4 .85. Compute a 95% condence interval to estimate A B, the dierence in the mean operating costs. Assume normality and equal variances.

Consider the statistic \(S^2 _p\), the pooled estimate of \(\sigma^2\) discussed in Section 9.8. It is used when one is willing to assume that \(\sigma^2 _1\ =\ \sigma^2 _2\ =\ \sigma^2\). Show that the estimator is unbiased for \(\sigma^2\) [i.e., show that \(E(S^2 _p)\ =\ \sigma^2\)]. You may make use of results from any theorem or example in this chapter.

A group of human factor researchers are concerned about reaction to a stimulus by airplane pilots in a certain cockpit arrangement. An experiment was conducted in a simulation laboratory, and 15 pilots were used with average reaction time of 3.2 seconds with a sample standard deviation of 0.6 second. It is of interest to characterize the extreme (i.e., worst case scenario). To that end, do the following: (a) Give a particular important one-sided 99% condence bound on the mean reaction time. What assumption, if any, must you make on the distribution of reaction times? (b) Give a 99% one-sided prediction interval and give an interpretation of what it means. Must you make an assumption about the distribution of reaction times to compute this bound? (c) Compute a one-sided tolerance bound with 99% condence that involves 95% of reaction times. Again, give an interpretation and assumptions about the distribution, if any. (Note: The onesided tolerance limit values are also included in Table A.7.)

A certain supplier manufactures a type of rubber mat that is sold to automotive companies. The material used to produce the mats must have certain hardness characteristics. Defective mats are occasionally discovered and rejected. The supplier claims that the proportion defective is 0.05. A challenge was made by one of the clients who purchased the mats, so an experiment was conducted in which 400 mats are tested and 17 were found defective. (a) Compute a 95% two-sided confidence interval on the proportion defective. (b) Compute an appropriate 95% one-sided confidence interval on the proportion defective. (c) Interpret both intervals from (a) and (b) and comment on the claim made by the supplier.