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probability statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions
In a study to estimate the proportion of smokers among the residents in a certain city and its suburbs, it is found that 24 of 104 urban residents are smokers, while 34 of 108 suburban residents are smokers. Is there a significant difference between the proportions of smokers among urban and
In a controlled laboratory experiment, scientists at the University of Minnesota discovered that 25% of a certain strain of rats subjected to a 20% coffee bean diet and then force-fed a powerful cancer-causing chemical later developed cancerous tumors. Would we have reason to believe that the
In a winter of an epidemic flu, the parents of 2000 babies were surveyed by researchers at a well-known pharmaceutical company to determine if the company’s new medicine was effective after two days. Among 120 babies who had the flu and were given the medicine, 29 were cured within two days.Among
On an enquiry related to the frequent reporting of mouth cancer cases, it is estimated that more than 60% of the patients chewed tobacco. Does this seem to be a valid estimate if, in a random sample of 120 people, 84 were found to be tobacco users? Use a 0.05 level of significance in your
Two-thirds of the employees of a town prefer using the public transport system for their travel. Do we have reason to believe this claim if, in a random sample of 900 employees, 582 use public transport? Use a P-value in your conclusion.
It is believed that at least 90% of the residents of a certain city and its suburbs did not support the idea of constructing a nuclear power plant near the city. What conclusion would you draw if only 9 in a sample 180 residents favored the idea? Use a 0.05 level of significance.
A new radar device is being considered for a certain missile defense system. The system is checked by experimenting with aircraft in which a kill or a no kill is simulated. If, in 300 trials, 250 kills occur, accept or reject, at the 0.04 level of significance, the claim that the probability of a
According to the records of a past survey conducted among the students of an engineering college, 30% of the students use their own vehicle to travel to and from college.Recently, 26 students were surveyed at random and it was found that 11 were using their own vehicle for travel. Is it 0correct to
A soft-drink producing company believes that 30% of its customers prefer the mango flavor. Out of the 24 customers tested, 10 bought mango-flavored items. What you can conclude about the company’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 Department of Human Nutrition, Health and Exercise at Virginia Tech and were analyzed in the Statistics Consulting Center at Hokie
A study was conducted 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 each dog, and the strength was measured.
For testing H : μ = 10 and H : μ ≠ 10, the level of significance, α = 0.05 is being considered. What sample size is necessary in order for the probability to be 0.1 of falsely failing to reject H when the true population mean differs from 10 by 0.5? From a preliminary sample, we estimate the
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 a sample is required in Exercise 10.31 if the power of the test is to be 0.95 when the true difference between the scores of the students of institute ‘A’ and ‘B’ is 8?
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 distance differs from 8.4 kilometers by 0.8 kilometers? Use α = 0.05.
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 950 grams?Assume that σ = 70 grams.
In Review Exercise 9.91 on page 333, use the t-distribution 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.
The operations division manager of a logistics company is trying to identify whether the use of nitrogen to fill tires instead of normal air improves fuel economy. The tires of 12 trucks were filled with nitrogen and tested over a prescribed course.Without changing drivers, the tires of the same
In a study conducted by the Department of Human Nutrition, Food and Exercise 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
A sample of five individuals was considered to determine if their systolic blood pressure varied when measured with analog and digital equipment. Following are the readings on their systolic blood pressures (in mmHg).Assuming that the populations are normal, test at the 0.05 level of significance
A study was conducted by the Department of Biological Sciences 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
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:Test the hypothesis that the average running time of films produced by company 2 exceeds the average running time of films produced by company 1 by 10 minutes against the onesided alternative that the
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 315, 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.
Two varieties of government-approved cattle feed are available in the market, each claiming it can produce more milk. The local milk society has to decide which variety to supply to the farmers. As an experiment, two groups of 12 identical cattle each are provided with the different varieties of
To find out whether Medicine X is more effective at managing insomnia than Medicine Y, 11 patients are selected. 6 receive Medicine X, and 5 receive Medicine Y. The time taken to fall asleep, in minutes, is as follows:At the 0.05 level of significance, can medicine Y be said to be more effective?
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-semester-hour course without labs or a 4-semesterhour 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
A training institute for competitive examinations, ‘A’, claims that they perform better than the training institute ‘B’ in the same city, and that their students score at least 10 marks more than those scored by students of ‘B’. To test this claim, the marks obtained by 50 students of
A random sample of size n = 26, taken from a normal population with a standard deviation σ = 4.9, has a mean =78. A second random sample of size n = 38, taken from a different normal population with a standard deviation σ = 3.2, has a mean = 74. Test the hypothesis that μ = μ against the
The average quantity of sugar purchased by customers at a supermarket is 2.3 kilograms, and it follows normal distribution. If a random sample of 20 customers came to purchase sugar and their average purchase was noted as 2.15 kilograms, with a standard deviation of 0.34 kilograms, test the
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 10.5
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 by an automobile company that the average mileage for a particular model of their diesel car is more than 22 kilometers/liter. To test this claim, 100 randomly selected car owners were asked to keep a record of the mileage of their cars.Would you agree with this claim if the average
The average distance between the college and homes of a group of students was 8.4 kilometers, with a standard deviation of 1.4 kilometers. Is there reason to believe that there has been a change in the average distance if a random sample of 54 students studying at the college lives at an average
Test the hypothesis that a coffee-vending machine spills an average of 240 milliliters of coffee in a single operation. Ten cups of coffee are examined and are found to contain 245, 238, 246, 242, 237, 241, 236, 238, 240, and 241 milliliters of coffee, respectively. Use a 0.05 level of significance
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 8.7 hours per week with a standard deviation of 2.25 hours,
An automobile industry uses a particular brand of automobile batteries that have their lifetimes approximately normally distributed, with a mean of 3.5 years and standard deviation of 0.5 years. Test the hypothesis that μ = 3.5 years against the alternative, μ ≠ 3.5 years, if a random sample of
A supermarket has been selling discounted apples in bundles of five at their counters. A random sample of 49 bundles weighs 960 grams on average, with a standard deviation of 70 grams. Test the hypothesis that μ = 1000 grams against the alternative hypothesis of μ < 1000 grams at the 0.05 level
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 H corresponding to various alternatives for μ (including the value specified by H ) 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.0
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 logistics company transporting freights from station A to station B claims a mean transportation time of 7 hours with a standard deviation of 0.5 hours. To test the hypothesis that μ = 7 hours against the alternative μ > 7 hours, a random sample of 40 trucks will be tested. The critical region
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.1. 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 = 0.5. 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 =
Repeat Exercise 10.4 but assume that 100 orders are selected and the critical region is defined to be x ≤ 50, 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 4 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.1. Find the
A bank is being charged for non-compliance with regulator’s policy.1. What hypothesis is being tested if a jury commits a type I error by finding the bank guilty?2. What hypothesis is being tested if a jury commits a type II error by finding the bank guilty?
The Department of Motor Vehicles is concerned about the effectiveness of an awareness program designed to get more drivers to strictly obey traffic rules while driving.1. What is the hypothesis being tested if the researchers commit a type I error by erroneously concluding that the awareness
Suppose that a health professional wishes to test the assumption that at least 5% of the residents of a village are addicted to smoking. Explain how the health professional could commit:1. a type I error;2. a type II error.
A leading manufacturing company produces high quality, durable galvanized bolt nuts. The company claims that the proportion of the defective bolts produced is 0.02. A construction company that uses the bolts made by this firm randomly tested 500 bolts and found 13 of them to be defective.1. Compute
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 , the pooled estimate of σdiscussed in Section 9.8. It is used when one is willing to assume that . Show that the estimator is unbiased for σ [i.e., M MF FM F 22 show that ]. You may make use of results from any theorem or example in this chapter.
A behavioral therapist is interested in studying the driving behavior among male and female college students. A test is conducted, in which the data for the maximum distance covered by students in a single drive are recorded. For the 18 boys who participate in this study, the mean distance μ = 146
A sugar packaging machine helps the manufacturer pack various quantities of sugar in leak-proof pouches. The machine is capable of functioning at an average speed of 120 pouches per minute, with a standard deviation of 6 pouches. However, there is some concern within the company about the proper
A bank has implemented a new online banking system in which it is observed that the waiting time for 18 out of 120 random actions is more than 15 seconds. In order to increase the service performance, the server capacity has been increased. Now, it is found that the waiting time for 6 out of 120
Refer to Review Exercise 9.106. Suppose there is concern about whether the point estimate = 8/30 is accurate enough because the confidence interval around p is not sufficiently narrow. Using as the estimate of p, how many companies would need to be sampled in order to have a 95% confidence interval
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 σ = σ = $4000. Are the sample sizes in Review Exercise 9.103 sufficient to produce a 95% confidence interval on μ – μ having a width of only $1000? Show all work.1
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
Specify the moment estimators for α and β for the gamma distribution.2 22
Specify the moment estimators for μ and σ for the lognormal distribution.
Specify the moment estimators for μ and σ 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 and sample variance s and solving for the parameters, the results being the moment estimators. In the case of a single
Based on the scores in the entrance examination for an engineering college, it is argued that the students from school A perform better than those from school B. The scores of 6 randomly selected students from these two schools are:°1 2 Assuming equal variance, what conclusions can you
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
A national daily conducted a reader poll on whether “bike riders without a helmet should be heavily fined”. Suppose that independent samples are taken from urban and rural responses, and it is found that 38 out of 140 people from urban areas and 24 out of 120 people from rural areas are against
An experiment is conducted to determine whether a particular medicine administered to patients with neurological disorders has an effect on their systolic blood pressure.Theoretically, the presence of some of the chemicals in the medicine should result in an increase of blood pressure. A random
The Department of Civil Engineering and Environmental 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
A study claims that a particular training regimen can improve memory retention capacity among students. In an experiment to test the claim, a group of 8 randomly selected high school students was asked to repeat, from memory, certain digits after hearing them once. These students then underwent the
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.Test the claim about the diet by computing a 95% confidence interval for the mean
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 σ for a sample x , x , …, x , which is drawn from a normal distribution with mean μ and 1 2 n 1 2 n 21 2 n variance σ . The estimators are the unbiased estimator and the maximum likelihood estimator. Discuss the variance properties of these two estimators.
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.
A sample consisting of the values, 9.6, 13.5, 5.4, 7.9, 4.8, 5.1,–1.2, and 6.8 comes from a population with the density function Obtain the maximum likelihood estimator of the parameter, θ.
Consider the independent observations x , x , …, x from the gamma distribution discussed in Section 6.6.1. Write out the likelihood function.2. Write out a set of equations that, when solved, give the maximum likelihood estimators of α and β.
Consider a random sample, x , x , …, x from a Poisson distribution, P(λ), with an unknown parameter, λ, where λ > 0.Determine the maximum likelihood estimator of λ.
Consider a random sample, x , x , …, x , from a population, with the parameter, θ, and a density function 1. Write out the likelihood function.2. Write out the equation that, when solved, will give the maximum likelihood estimator of θ.1 2 n 21 n 1 2 n
Consider a random sample of x , …, x 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 parameter β.
Consider the lognormal distribution with the density function given in Section 6.9. Suppose we have a random sample x , x , …, x from a lognormal distribution.1. Write out the likelihood function.2. Develop the maximum likelihood estimators of μ and σ .
Suppose that there are n trials x , x , …, x from a Bernoulli process with parameter p, the probability of a success. That is, 1 2 n the probability of r successes is given by . Work out the maximum likelihood estimator for the parameter p.
Construct a 95% confidence interval for in Exercise 9.49 on page 315. Should the equal-variance assumption be used?
Construct a 90% confidence interval for in Exercise 9.46 on page 315. Should we have assumed in constructing our confidence interval for μ – μ ?
Construct a 90% confidence interval for in Exercise 9.43 on page 315. Were we justified in assuming that when we constructed the confidence interval for μ – μ ?2 22 1 2 1 2 1 2
Construct a 98% confidence interval for σ /σ in Exercise 9.42 on page 315, where σ and σ are, respectively, the standard deviations for the distances traveled per liter of fuel by the Volkswagen and Toyota mini-trucks.
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