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descriptive statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying Ye - Solutions
A random sample of 20 students yielded a mean of ¯x = 72 and a variance of s2 = 16 for scores on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for σ2.
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% confidence interval for σ2 and decide if the manufacturer’s claim that σ2 = 1 is
According to USA Today (March 17, 1997), women made up 33.7% of the editorial staff at local TV stations in the United States in 1990 and 36.2% in 1994. Assume 20 new employees were hired as editorial staff.(a) Estimate the number that would have been women in 1990 and 1994, respectively.(b)
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% confidence interval for the difference between the proportions of students favoring team A in the
In the study Germination and Emergence of Broccoli, conducted by the Department of Horticulture at Virginia Tech, a researcher found that at 5◦C, 10 broccoli seeds out of 20 germinated, while at 15◦C, 15 out of 20 germinated. Compute a 95% confidence interval for the difference between the
A clinical trial was conducted to determine if a certain type of inoculation has an effect 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
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% confidence interval for the difference between the proportions of women in these two fields of engineering. Is there a
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
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% confident that the estimate is within 0.04 of the true proportion of
A study is to be made to estimate the percentage of citizens in a town who favor having their water fluoridated. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 1% of the true percentage?
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 sufficient fluoride to protect your teeth from decay. How large a sample is
How large a sample is needed in Exercise 9.52 if we wish to be 98% confident that our sample proportion will be within 0.05 of the true proportion defective?
How large a sample is needed if we wish to be 99% confident 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?
How large a sample is needed if we wish to be 96% confident that our sample proportion in Exercise 9.53 will be within 0.02 of the true fraction of the voting population?
In the newspaper article referred to in Exercise 9.57, 32% of the 1600 adults polled said the U.S. space program should emphasize scientific exploration. How large a sample of adults is needed for the poll if one wishes to be 95% confident that the estimated percentage will be within 2% of the true
(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% confidence interval for the proportion of American adults who think the space shuttle program is
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
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
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% confidence interval for the proportion of MP3
(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
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.
s2 = 0.35 Assume that the variances are equal. Give a 95% confidence interval for the difference in the true average insulin-binding capacity between the two samples. 9.51 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
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 =
Two different 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:
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
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.
The following data represent the running times of films 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% confidence interval for the difference between the average running times of films produced by the two companies. Assume
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:
Referring to Exercise 9.43, find a 99% confidence interval for μ1 − μ2 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
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: ¯x1 = 36, 300 kilometers, s1 =
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% confidence interval for the difference between the average kilometers per liter for these two mini-trucks.Assume that the distances per liter for the
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 90-kilometer-per-hour steady-paced trials. If the 12 Volkswagen trucks averaged 16 kilometers per liter
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:
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
Students may choose between a 3-semester-hour physics course without labs and a 4-semester-hour course with labs. The final written examination is the same for each section. If 12 students in the section with
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
A study was conducted to determine if a certain treatment has any effect 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
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
A random sample of size n1 = 25, taken from a normal population with a standard deviation σ1 = 5, has a mean ¯x1 = 80. A second random sample of size n2 = 36, taken from a different normal population with a standard deviation σ2 = 3, has a mean ¯x2 = 75. Find a 94% confidence interval for μ1
Consider Exercise 9.33. Use the MSE discussed in Exercise 9.28 to determine which estimator is more efficient. Write out MSE(S2)MSE(S2)
Compare S2 and S2 (see Exercise 9.29), the two estimators of σ2, to determine which is more efficient. Assume these estimators are found using X1, X2,...,Xn, independent random variables from n(x; μ, σ). Which estimator is more efficient considering only the variance of the estimators? [Hint:
Show that the estimator P of Exercise 9.31(b)becomes unbiased as n → ∞.
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.
Consider S2, the estimator of σ2, from Exercise 9.29. Analysts often use S2 rather than dividing ni=1(Xi − X¯)2 by n − 1, the degrees of freedom in the sample.(a) What is the bias of S2?(b) Show that the bias of S2 approaches zero as n →∞.
Let us define S2 = n i=1(Xi − X¯)2/n. Show that E(S2) = [(n − 1)/n]σ2, and hence S2 is a biased estimator for σ2.
In Section 9.3, we emphasized the notion of“most efficient estimator” by comparing the variance of two unbiased estimators Θˆ 1 and Θˆ 2. However, this does not take into account bias in case one or both estimators are not unbiased. Consider the quantity MSE = E(Θˆ − θ), where MSE
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
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 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.
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
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?
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
In a study conducted by the Department of Zoology at Virginia Tech, fifteen 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
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.
A random sample of 25 tablets of buffered 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 buffered aspirin. Assume that the aspirin content is normally
Referring to Exercise 9.13, construct a 95% tolerance interval containing 90% of the measurements.
Consider Exercise 9.9. Compute a 95% prediction interval for the sugar content of the next single serving of Alpha-Bits.
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.
Referring to Exercise 9.5, construct a 99% prediction interval for the kilometers traveled annually by an automobile owner in Virginia.
The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint:Assuming that the measurements represent a random sample from a normal population, find a 95% prediction interval for the drying time for the next trial of the paint.
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,
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
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% confidence interval for the mean diameter of pieces from this machine, assuming an
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
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
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 σ = 40
How large a sample is needed in Exercise 9.3 if we wish to be 95% confident that our sample mean will be within 0.0005 inch of the true mean?
How large a sample is needed in Exercise 9.2 if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean?
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
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% confidence interval for the mean height of all college students.(b) What can we assert with 98% confidence about the possible size of our error if
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, find a 95% confidence interval for the mean of the depths of all
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
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
Group Project: The class should be divided into groups of four people. The four students in each group should go to the college gym or a local fitness center. The students should ask each person who comes through the door his or her height in inches.Each group will then divide the height data by
Consider the situation in Review Exercise 8.74.Suppose a considerable effort is conducted to “tighten”the variability in the system. Following the effort, a random sample of size 40 is taken from the new assembly line and the sample variance is s2 = 0.188 ounces2.Do we have strong numerical
Suppose a filling machine is used to fill cartons with a liquid product. The specification that is strictly enforced for the filling machine is 9 ± 1.5 oz. If any carton is produced with weight outside these bounds, it is considered by the supplier to be defective. It is hoped that at least 99% of
In Chapter 9, the concept of parameter estimation will be discussed at length. Suppose X is a random variable with mean μ and variance σ2 = 1.0.Suppose also that a random sample of size n is to be taken and ¯x is to be used as an estimate of μ. When the data are taken and the sample mean is
Given a normal random variable X with mean 20 and variance 9, and a random sample of size n taken from the distribution, what sample size n is necessary in order that P(19.9 ≤ X¯ ≤ 20.1) = 0.95?
From the information in Review Exercise 8.70, compute (assuming μB = 65%) P(X¯B ≥ 70).
The concentration of an active ingredient in the output of a chemical reaction is strongly influenced by the catalyst that is used in the reaction. It is felt that when catalyst A is used, the population mean concentration exceeds 65%. The standard deviation is known to be σ = 5%. A sample of
Two distinct solid fuel propellants, type A and type B, are being considered for a space program activity. Burning rates of the propellant are crucial. Random samples of 20 specimens of the two propellants are taken with sample means 20.5 cm/sec for propellant A and 24.50 cm/sec for propellant B.
Consider the situation of Review Exercise 8.62.If the population from which the sample was taken has population mean μ = 53, 000 kilometers, does the sample information here seem to support that claim? In your answer, compute t = x¯ − 53, 000 s/√10 and determine from Table A.4 (with 9 d.f.)
The breaking strength X of a certain rivet used in a machine engine has a mean 5000 psi and standard deviation 400 psi. A random sample of 36 rivets is taken. Consider the distribution of X¯, the sample mean breaking strength.(a) What is the probability that the sample mean falls between 4800 psi
Consider Review Exercise 8.56. Comment on any outliers in the data.
Consider Example 1.5 on page 25. Comment on any outliers.
If S2 1 and S2 2 represent the variances of independent random samples of size n1 = 25 and n2 = 31, taken from normal populations with variances σ2 1 = 10 and σ2 2 = 15, respectively, find P(S2 1 /S2 2 > 1.26).
Consider the data of Exercise 1.19 on page 31.Construct a box-and-whisker plot. Comment. Compute the sample mean and sample standard deviation.
A taxi company tests a random sample of 10 steel-belted radial tires of a certain brand and records the following tread wear: 48,000, 53,000, 45,000, 61,000, 59,000, 56,000, 63,000, 49,000, 53,000, and 54,000 kilometers. Use the results of Exercise 8.1 on page 231 to find the standard deviation of
If the number of hurricanes that hit a certain area of the eastern United States per year is a random variable having a Poisson distribution with μ = 6, find the probability that this area will be hit by(a) exactly 15 hurricanes in 2 years;(b) at most 9 hurricanes in 2 years.
A random sample of 5 bank presidents indicated annual salaries of $395,000, $521,000, $483,000,$479,000, and $510,000. Find the variance of this set.
If S2 1 and S2 2 represent the variances of independent random samples of size n1 = 8 and n2 = 12, taken from normal populations with equal variances, find P(S2 1 /S2 2 < 4.89).
on page 231 to find the standard deviation for the carbon monoxide content of a random sample of 15 cigarettes of this brand if the coded measurements are 3.8, −0.9, 5.4, 4.5, 5.2, 5.6, 2.7, −0.1, −0.3, −1.7, 5.7, 3.3, 4.4, −0.5, and 1.9.
In testing for carbon monoxide in a certain brand of cigarette, the data, in milligrams per cigarette, were coded by subtracting 12 from each observation. Use the results of Exercise
If X1, X2,...,Xn are independent random variables having identical exponential distributions with parameter θ, show that the density function of the random variable Y = X1+X2+···+Xn is that of a gamma distribution with parameters α = n and β = θ.
Consider the data displayed in Exercise 1.20 on page 31. Construct a box-and-whisker plot and comment on the nature of the sample. Compute the sample mean and sample standard deviation.
Construct a normal quantile-quantile plot of these data, which represent the diameters of 36 rivet heads in 1/100 of an inch:
Construct a quantile plot of these data, which represent the lifetimes, in hours, of fifty 40-watt, 110-volt internally frosted incandescent lamps taken from forced life tests:
Consider the following measurements of the heat-producing capacity of the coal produced by two mines (in millions of calories per ton):Mine 1: 8260 8130 8350 8070 8340 Mine 2: 7950 7890 7900 8140 7920 7840 Can it be concluded that the two population variances are equal?
Pull-strength tests on 10 soldered leads for a semiconductor device yield the following results, in pounds of force required to rupture the bond:Another set of 8 leads was tested after encapsulation to determine whether the pull strength had been increased by encapsulation of the device, with the
For an F-distribution, find(a) f0.05 with v1 = 7 and v2 = 15;(b) f0.05 with v1 = 15 and v2 = 7:(c) f0.01 with v1 = 24 and v2 = 19;(d) f0.95 with v1 = 19 and v2 = 24;(e) f0.99 with v1 = 28 and v2 = 12.
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