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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
Suppose that, in the past, 40% of all adults favored capital punishment. Do we have reason to believe that the proportion of adults favoring capital punishment has increased if, in a random sample of 15 adults, 8 favor capital punishment? Use a 0.05 level of significance.
A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna.If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert’s claim?Use a 0.05 level of significance.
Nine subjects were used in an experiment to determine if exposure to carbon monoxide has an impact on breathing capability. The data were collected by personnel in the Health and Physical Education Department at Virginia Tech and were analyzed in the Statistics Consulting Center at Hokie Land. The
A study was conducted at the Department of Veterinary Medicine at Virginia Tech to determine if the “strength” of a wound from surgical incision is affected by the temperature of the knife. Eight dogs were used in the experiment. “Hot” and “cold” incisions were made on the abdomen of
For testing H0: μ = 14, H1: μ = 14, an α = 0.05 level t-test is being considered. What sample size is necessary in order for the probability to be 0.1 of falsely failing to reject H0 when the true population mean differs from 14 by 0.5? From a preliminary sample we estimate σ to be 1.25.
How large a sample is required in Exercise 10.22 if the power of the test is to be 0.8 when the true mean meditation time exceeds the hypothesized value by 1.2σ? Use α = 0.05.
How large should the samples be in Exercise 10.31 if the power of the test is to be 0.95 when the true difference between thread types A and B is 8 kilograms?
How large a sample is required in Exercise 10.24 if the power of the test is to be 0.95 when the true average height differs from 162.5 by 3.1 centimeters? Use α = 0.02.
If the distribution of life spans in Exercise 10.19 is approximately normal, how large a sample is required in order that the probability of committing a type II error be 0.1 when the true mean is 35.9 months?Assume that σ = 5.8 months
How large a sample is required in Exercise 10.20 if the power of the test is to be 0.90 when the true mean is 5.20? Assume that σ = 0.24.
In Review Exercise 9.91 on page 313, use the tdistribution to test the hypothesis that the diet reduces a woman’s weight by 4.5 kilograms on average against the alternative hypothesis that the mean difference in weight is less than 4.5 kilograms. Use a P-value.
A taxi company manager is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted
In a study conducted by the Department of Human Nutrition and Foods at Virginia Tech, the following data were recorded on sorbic acid residuals, in parts per million, in ham immediately after dipping in a sorbate solution and after 60 days of storage:Assuming the populations to be normally
According to published reports, practice under fatigued conditions distorts mechanisms that govern performance. An experiment was conducted using 15 college males, who were trained to make a continuous horizontal right-to-left arm movement from a microswitch to a barrier, knocking over the barrier
Five samples of a ferrous-type substance were used to determine if there is a difference between a laboratory chemical analysis and an X-ray fluorescence analysis of the iron content. Each sample was split into two subsamples and the two types of analysis were applied. Following are the coded data
A study was conducted by the Department of Zoology at Virginia Tech to determine if there is a significant difference in the density of organisms at two different stations located on Cedar Run, a secondary stream in the Roanoke River drainage basin.Sewage from a sewage treatment plant and overflow
In a study conducted at Virginia Tech, the plasma ascorbic acid levels of pregnant women were compared for smokers versus nonsmokers. Thirty-two women in the last three months of pregnancy, free of major health disorders and ranging in age from 15 to 32 years, were selected for the study. Prior to
The following data represent the running times of films produced by two motion-picture companies:Company Time (minutes)1 102 86 98 109 92 2 81 165 97 134 92 87 114 Test the hypothesis that the average running time of films produced by company 2 exceeds the average running time of films produced by
A UCLA researcher claims that the average life span of mice can be extended by as much as 8 months when the calories in their diet are reduced by approximately 40% from the time they are weaned. The restricted diets are enriched to normal levels by vitamins and protein. Suppose that a random sample
In Exercise 9.42 on page 295, test the hypothesis that the fuel economy of Volkswagen mini-trucks, on average, exceeds that of similarly equipped Toyota mini-trucks by 4 kilometers per liter. Use a 0.10 level of significance.
Engineers at a large automobile manufacturing company are trying to decide whether to purchase brand A or brand B tires for the company’s new models. To help them arrive at a decision, an experiment is conducted using 12 of each brand. The tires are run
To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of the disease, are selected. Five mice receive the treatment and 4 do not. Survival times, in years, from the time the experiment commenced are as follows:At the 0.05 level of significance, can the serum be
A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were randomly selected to participate in either a 3-semesterhour course without labs or a 4-semester-hour course with labs. In the section with labs, 11
A study was conducted to see if increasing the substrate concentration has an appreciable effect on the velocity of a chemical reaction. With a substrate concentration of 1.5 moles per liter, the reaction was run 15 times, with an average velocity of 7.5 micromoles per 30 minutes and a standard
Amstat News (December 2004) lists median salaries for associate professors of statistics at research institutions and at liberal arts and other institutions in the United States. Assume that a sample of 200 associate professors from research institutions has an average salary of $70,750 per year
kilograms with a standard deviation of 5.61 kilograms. Test the manufacturer’s claim using a 0.05 level of significance.
kilograms with a standard deviation of 6.28 kilograms, while type B thread had an average tensile strength of
A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by at least 12 kilograms. To test this claim, 50 pieces of each type of thread were tested under similar conditions. Type A thread had an average tensile strength of
A random sample of size n1 = 25, taken from a normal population with a standard deviation σ1 = 5.2, has a mean ¯x1 = 81. A second random sample of size n2 = 36, taken from a different normal population with a standard deviation σ2 = 3.4, has a mean ¯x2 = 76.Test the hypothesis that μ1 = μ2
minutes to complete this test with a standard deviation of 4.3 minutes, test the hypothesis, at the 0.05 level of significance, that μ = 35 minutes against the alternative that μ < 35 minutes.
Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a mean of 35 minutes. If a random sample of 20 high school seniors took an average of
According to Chemical Engineering, an important property of fiber is its water absorbency. The average percent absorbency of 25 randomly selected pieces of cotton fiber was found to be 20 with a standard deviation of 1.5. A random sample of 25 pieces of acetate yielded an average percent of 12 with
A study at the University of Colorado at Boulder shows that running increases the percent resting metabolic rate (RMR) in older women. The average RMR of 30 elderly women runners was 34.0% higher than the average RMR of 30 sedentary elderly women, and the standard deviations were reported to be
According to a dietary study, high sodium intake may be related to ulcers, stomach cancer, and migraine headaches. The human requirement for salt is only 220 milligrams per day, which is surpassed in most single servings of ready-to-eat cereals. If a random sample of 20 similar servings of a
It is claimed that automobiles are driven on average more than 20,000 kilometers per year. To test this claim, 100 randomly selected automobile owners are asked to keep a record of the kilometers they travel.Would you agree with this claim if the random sample showed an average of 23,500 kilometers
The average height of females in the freshman class of a certain college has historically been 162.5 centimeters with a standard deviation of 6.9 centimeters.Is there reason to believe that there has been a change in the average height if a random sample of 50 females in the present freshman class
Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters.Use a 0.01 level of significance and assume that the distribution of contents
In the American Heart Association journal Hypertension, researchers report that individuals who practice Transcendental Meditation (TM) lower their blood pressure significantly. If a random sample of 225 male TM practitioners meditate for 8.5 hours per week with a standard deviation of 2.25 hours,
An electrical firm manufactures light bulbs that have a lifetime that is approximately normally distributed with a mean of 800 hours and a standard deviation of 40 hours. Test the hypothesis that μ = 800 hours against the alternative, μ = 800 hours, if a random sample of 30 bulbs has an average
A random sample of 64 bags of white cheddar popcorn weighed, on average, 5.23 ounces with a standard deviation of 0.24 ounce. Test the hypothesis that μ = 5.5 ounces against the alternative hypothesis,μ < 5.5 ounces, at the 0.05 level of significance.
In a research report, Richard H. Weindruch of the UCLA Medical School claims that mice with an average life span of 32 months will live to be about 40 months old when 40% of the calories in their diet are replaced by vitamins and protein. Is there any reason to believe that μ < 40 if 64 mice that
If we plot the probabilities of failing to reject H0 corresponding to various alternatives for μ (including the value specified by H0) and connect all the points by a smooth curve, we obtain the operating characteristic curve of the test criterion, or simply the OC curve. Note that the probability
A new curing process developed for a certain type of cement results in a mean compressive strength of 5000 kilograms per square centimeter with a standard deviation of 120 kilograms. To test the hypothesis that μ = 5000 against the alternative that μ < 5000, a random sample of 50 pieces of cement
Repeat Exercise 10.15 for samples of size n =25. Use the same critical region.
A soft-drink machine at a steak house is regulated so that the amount of drink dispensed is approximately normally distributed with a mean of 200 milliliters and a standard deviation of 15 milliliters.The machine is checked periodically by taking a sample of 9 drinks and computing the average
A manufacturer has developed a new fishing line, which the company claims has a mean breaking strength of 15 kilograms with a standard deviation of 0.5 kilogram. To test the hypothesis that μ = 15 kilograms against the alternative that μ < 15 kilograms, a random sample of 50 lines will be tested.
Suppose, in Exercise 10.12, we conclude that 60% of the voters favor the gasoline sales tax if more than 214 but fewer than 266 voters in our sample favor it. Show that this new critical region results in a smaller value for α at the expense of increasing β.
A random sample of 400 voters in a certain city are asked if they favor an additional 4% gasoline sales tax to provide badly needed revenues for street repairs.If more than 220 but fewer than 260 favor the sales tax, we shall conclude that 60% of the voters are for it.(a) Find the probability of
Repeat Exercise 10.8 but assume that 70 patients are given the mussel extract and the critical region is defined to be x < 24, where x is the number of osteoarthritic patients who receive relief.
Repeat Exercise 10.9 but assume that 100 spots are treated and the critical region is defined to be x > 82, where x is the number of spots removed.
A dry cleaning establishment claims that a new spot remover will remove more than 70% of the spots to which it is applied. To check this claim, the spot remover will be used on 12 spots chosen at random. If fewer than 11 of the spots are removed, we shall not reject the null hypothesis that p =
In Relief from Arthritis published by Thorsons Publishers, Ltd., John E. Croft claims that over 40%of those who suffer from osteoarthritis receive measurable relief from an ingredient produced by a particular species of mussel found off the coast of New Zealand.To test this claim, the mussel
Repeat Exercise 10.6 but assume that 200 adults are selected and the fail-to-reject region is defined to be 110 ≤ x ≤ 130, where x is the number of college graduates in our sample. Use the normal approximation.
The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6.To test this hypothesis, a random sample of 15 adults is selected. If the number of college graduates in the sample is anywhere from 6 to 12, we shall not reject the null hypothesis that p = 0.6;
Repeat Exercise 10.4 but assume that 50 orders are selected and the critical region is defined to be x ≤ 24, where x is the number of orders in the sample that arrived late. Use the normal approximation.
A fabric manufacturer believes that the proportion of orders for raw material arriving late is p = 0.6.If a random sample of 10 orders shows that 3 or fewer arrived late, the hypothesis that p = 0.6 should be rejected in favor of the alternative p < 0.6. Use the binomial distribution.(a) Find the
A large manufacturing firm is being charged with discrimination in its hiring practices.(a) What hypothesis is being tested if a jury commits a type I error by finding the firm guilty?(b) What hypothesis is being tested if a jury commits a type II error by finding the firm guilty?
A sociologist is concerned about the effectiveness of a training course designed to get more drivers to use seat belts in automobiles.(a) What hypothesis is she testing if she commits a type I error by erroneously concluding that the training course is ineffective?(b) What hypothesis is she testing
Suppose that an allergist wishes to test the hypothesis that at least 30% of the public is allergic to some cheese products. Explain how the allergist could commit(a) a type I error;(b) a type II error.
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
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
Consider the statistic S2 p, the pooled estimate of σ2 discussed in Section 9.8. It is used when one is willing to assume that σ2 1 = σ2 2 = σ2. Show that the estimator is unbiased for σ2 [i.e., show that E(S2 p) = σ2].You may make use of results from any theorem or example in this chapter.
A consumer group is interested in comparing operating costs for two different types of automobile engines. The group is able to find 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
A machine is used to fill 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
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,
Refer to Review Exercise 9.106. Suppose there is concern about whether the point estimate ˆp = 8/30 is accurate enough because the confidence interval around p is not sufficiently narrow. Using ˆp as the estimate of p, how many companies would need to be sampled in order to have a 95% confidence
A random sample of 30 firms dealing in wireless products was selected to determine the proportion of such firms that have implemented new software to improve productivity. It turned out that 8 of the 30 had implemented such software. Find a 95% confidence interval on p, the true proportion of such
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
Consider Review Exercise 9.103. Let us assume that the data have not been collected yet and that previous statistics suggest that σ1 = σ2 = $4000. Are the sample sizes in Review Exercise 9.103 sufficient to produce a 95% confidence interval on μ1 − μ2 having a width of only $1000? Show all
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.
Specify the moment estimators for α and β for the gamma distribution.
Specify the moment estimators for μ and σ2 for the lognormal distribution.
Specify the moment estimators for μ and σ2 for the normal distribution.
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
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):
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
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
An experiment was conducted to determine whether surface finish has an effect 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 effect on the standard deviation
The Department of Civil Engineering at Virginia Tech compared a modified (M-5 hr) assay technique for recovering fecal coliforms in stormwater runoff from an urban area to a most probable number (MPN) technique. A total of 12 runoff samples were collected and analyzed by the two techniques. Fecal
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:
A study was undertaken at Virginia Tech to determine if fire 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
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.
According to the Roanoke Times, McDonald’s sold 42.1% of the market share of hamburgers. A random sample of 75 burgers sold resulted in 28 of them being from McDonald’s. Use material in Section 9.10 to determine if this information supports the claim in the Roanoke Times.
Consider two estimators of σ2 for a sample x1, x2,...,xn, which is drawn from a normal distribution with mean μ and variance σ2. The estimators are the unbiased estimator s2 = 1 n−1 ni=1(xi − x¯)2 and the maximum likelihood estimator ˆσ2 = 1 nn i=1(xi−x¯)2.Discuss the variance
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 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) = pxq1−x, x = 0, 1, where p is the probability of a success (cure) and q = 1 − p. Now, of course,
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 α and β.
Consider a random sample of x1,...,xn from a uniform distribution U(0, θ) with unknown parameterθ, where θ > 0. Determine the maximum likelihood estimator of θ.
Consider a random sample of x1, x2,...,xn observations from a Weibull distribution with parametersα and β and density function f(x) = αβxβ−1e−αxβ, x> 0, 0, elsewhere, for α, β > 0.(a) Write out the likelihood function.(b) Write out the equations that, when solved, give the maximum
Consider a random sample of x1,...,xn coming from the gamma distribution discussed in Section 6.6.Suppose the parameter α is known, say 5, and determine the maximum likelihood estimation for parameterβ.
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 μand σ2.
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 by n rpr(1 − p)n−r. Work out the maximum likelihood estimator for the parameter p.
Construct a 95% confidence interval for σ2 A/σ2 Bin Exercise 9.49. on page 295. Should the equal-variance assumption be used?
Construct a 90% confidence interval for σ2 1/σ2 2in Exercise 9.46. on page 295. Should we have assumedσ2 1 = σ2 2 in constructing our confidence interval forμI − μII ?
Construct a 90% confidence interval for σ2 1/σ2 2 in Exercise 9.43 on page 295. Were we justified in assuming that σ2 1 = σ2 2 when we constructed the confidence interval for μ1 − μ2?
Construct a 98% confidence interval for σ1/σ2 in Exercise 9.42 on page 295, where σ1 and σ2 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 σ in Exercise 9.13 on page 283.
Construct a 99% confidence interval for σ in Exercise 9.12 on page 283.
Construct a 99% confidence interval for σ2 in Exercise 9.11 on page 283.
Construct a 95% confidence interval for σ2 in Exercise 9.9 on page 283.
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