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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

For the simple linear regression model, prove that E(s ) =σ .
Suppose a scientist postulates a model:0 1 1 1 1. What is the appropriate least squares estimator of β ? Justify your answer.2. What is the variance of the slope estimator?
Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during strenuous exercise. A study was conducted on 24 middle-aged men to determine the influence on oxygen uptake of the time
Observations on the yield of a chemical reaction taken at various temperatures were recorded as follows:1. Plot the data.2. Does it appear from the plot as if the relationship is linear?3. Fit a simple linear regression and test for lack of fit.4. Draw conclusions based on your result in (c).
Consider the vehicle data from Consumer Reports in Figure 11.30 on page 460. Weight is in tons, mileage in miles per gallon, and drive ratio is also indicated. A regression model was fitted relating weight x to mileage y. A partial SAS printout in Figure 11.30 on page 460 shows some of the results
The business section of the Washington Times in March of 1997 listed 21 different used computers and printers and their sale prices. Also listed was the average hover bid. Partial results from regression analysis using SAS software are shown in Figure 11.29 on page 459.1. Explain the difference
The following data represent the chemistry grades for a random sample of 12 freshmen at a certain college along with their scores on an intelligence test administered while they were still seniors in high school.1. Compute and interpret the sample correlation coefficient.2. State necessary
An experiment was designed for the Department of Materials Science and Engineering at Virginia Tech to study hydrogen embrittlement properties based on electrolytic hydrogen pressure measurements. The solution used was 0.1 N NaOH, and the material was a certain type of stainless steel. The cathodic
With reference to Exercise 11.9 on page 419, construct 1. a 95% confidence interval for the average weekly sales when$45 is spent on advertising;2. a 95% prediction interval for the weekly sales when $45 is spent on advertising.
The number of cigarettes consumed per day and the rate of coronary heart disease per 10000 people of the population is recorded for two states.0 1 1 1 1. Estimate the linear regression line.2. Test at the 0.05 level of significance and state whether the linear model is adequate.
The Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech analyzed data on normal woodchucks for the Department of Veterinary Medicine. The variables of interest were body weight in grams and heart weight in grams. It was desired to develop a linear regression equation in order to
With reference to Exercise 11.8 on page 419, construct 1. a 95% confidence interval for the average course grade of students who make a 35 on the placement test;2. a 95% prediction interval for the course grade of a student who made a 35 on the placement test.
The following data were obtained in a study of the relationship between the weight and chest size of infants at birth.1. Calculate r.2. Test the null hypothesis that ρ = 0 against the alternative thatρ > 0 at the 0.01 level of significance.3. What percentage of the variation in infant chest sizes
Test the hypothesis that ρ = 0 in Exercise 11.43 against the alternative that ρ ≠ 0. Use a 0.05 level of significance.
With reference to Exercise 11.13 on page 420, assume a bivariate normal distribution for x and y.1. Calculate r.2. Test the null hypothesis that ρ = – 0.5 against the alternative that ρ < –0.5 at the 0.025 level of significance.3. Determine the percentage of the variation in the amount of
With reference to Exercise 11.1 on page 418, assume that x and y are random variables with a bivariate normal distribution.1. Calculate r.2. Test the hypothesis that ρ = 0 against the alternative that ρ ≠ 0 at the 0.05 level of significance.
Compute and interpret the correlation coefficient for the following grades of 6 students selected at random:
A non-athletic group of adolescents from a college was examined. Researchers identified 16 students, four each with four different weights. They wanted to estimate a formula for predicting the weight of a student from the information about the weight each could lift. The weight lifted by the
Evaluating nitrogen deposition from the atmosphere is a major role of the National Atmospheric Deposition Program(NADP), a partnership of many agencies. NADP is studying atmospheric deposition and its effect on agricultural crops, forest surface waters, and other resources. Nitrogen oxides may
The data for total sales (x) of a company and its expenditure towards advertisement (y) for 2014 is given as follows:1. Fit a linear regression model relating to sales(x) and the advertisement cost (y). Test H : β = 0.2. Perform a lack-of-fit. Is the linear model appropriate for this based on the
A regression model is desired relating temperature and the proportion of impurities passing through solid helium.Temperature is listed in degrees centigrade. The data are as follows:1. Fit a linear regression model.0 1 2. Does it appear that the proportion of impurities passing through helium
Heat treating is often used to carburize metal parts such as gears. The thickness of the carburized layer is considered an important feature of the gear, and it contributes to the overall reliability of the part. Because of the critical nature of this feature, a lab test is performed on each
Organophosphate (OP) compounds are used as pesticides.However, it is important to study their effect on species that are exposed to them. In the laboratory study Some Effects of 12 0 1 1 Organophosphate Pesticides on Wildlife Species, by the Department of Fisheries and Wildlife Conservation at
The dataset consists of variables relating to blood pressures of 15 Peruvians (n = 15) who have moved from rural, high-altitude areas to urban, lower-altitude areas. The variables Y|x 0 1 0 01 1 in this dataset are: systolic blood pressure (Y), weight (X ), height (X ), and pulse.1. Determine if
The following data are a result of an investigation as to the effect of reaction temperature x on percent conversion of a chemical process y. (See Myers, Montgomery and Anderson-Cook, 2009.) Fit a simple linear regression, and use a lack-of-fit test to determine if the model is adequate. Discuss.
Use an analysis-of-variance approach to test the hypothesis that β = 0 against the alternative hypothesis β ≠ 0 in Exercise 11.5 on page 418 at the 0.05 level of significance.
Suppose we have a linear equation through the origin(Exercise 11.28) μ = βx.1. Estimate the regression line passing through the origin for the following data:Y|x 2. Suppose it is not known whether the true regression should pass through the origin. Estimate the linear model μ = β +β x and test
Test for linearity of regression in Exercise 11.8 on page 419. Comment.
Test for linearity of regression in Exercise 11.3 on page 418. Use a 0.05 level of significance. Comment.
For the data in Exercise 11.29, find a 95% prediction interval at x = 25.1 1 1 1 i 1 2i Figure 11.13: SAS printout for Exercise 11.27.
Use the data set 1. Plot the data.2. Fit a regression line through the origin.3. Plot the regression line on the graph with the data.4. Give a general formula (in terms of the y and the slope b )for the estimator of σ .5. Give a formula for Var( ), i = 1, 2, …, n, for this case.6. Plot 95%
There are important applications in which, due to known scientific constraints, the regression line must go through the origin (i.e., the intercept must be zero). In other words, the model should read and only a simple parameter requires estimation. The model is often called the regression through
Consider the regression of mileage for certain automobiles, measured in miles per gallon (mpg) on their weight in pounds (wt). The data are from Consumer Reports(April 1997). Part of the SAS output from the procedure is shown in Figure 11.13.1. Estimate the mileage for a vehicle weighing 4000
With reference to Exercise 11.3 on page 418, use the value of s found in Exercise 11.19(a) to compute 1. a 99% confidence interval for the average amount of chemical that will dissolve in 100 grams of water at 50°C;2. a 99% prediction interval for the amount of chemical that will dissolve in 100
Using the value of s found in Exercise 11.17(a), construct a 95% confidence interval for the amount of converted sugar corresponding to x = 1.6 in Exercise 11.5 on page 418.
Using the value of s found in Exercise 11.17(a), graph the regression line and the 95% confidence bands for the mean response μ for the data of Exercise 11.5 on page 418.
With reference to Exercise 11.6 on page 419, use the value of s found in Exercise 11.18(a) to compute 1. a 95% confidence interval for the mean shear resistance when x = 24.5;0 12 01 00 11 2Y|85 22. a 95% prediction interval for a single predicted value of the shear resistance when x = 24.5.
Using the value of s found in Exercise 11.16(a), construct a 95% confidence interval for μ in Exercise 11.2 on page 418.
Test the hypothesis that β = 6 in Exercise 11.9 on page 419 against the alternative that β < 6. Use a 0.025 level of significance.
Test the hypothesis that β = 10 in Exercise 11.8 on page 419 against the alternative that β < 10. Use a 0.05 level of significance.
With reference to Exercise 11.3 on page 418, 1. evaluate s ;2. construct a 99% confidence interval for β ;3. construct a 99% confidence interval for β .
With reference to Exercise 11.6 on page 419, 1. evaluate s ;2 01 22. construct a 99% confidence interval for β ;3. construct a 99% confidence interval for β .
With reference to Exercise 11.5 on page 418, 1. evaluate s ;2. construct a 95% confidence interval for β ;3. construct a 95% confidence interval for β .
With reference to Exercise 11.2 on page 418, 1. evaluate s ;2. construct a 95% confidence interval for β ;3. construct a 95% confidence interval for β .2 1 1 20 1Figure 11.12: Confidence and prediction intervals for the chemical oxygen demand reduction data; inside bands indicate the confidence
With reference to Exercise 11.1 on page 418, 0 0 0 xx 0.025 00 1. evaluate s ;2. test the hypothesis that β = 0 against the alternative that β ≠0 at the 0.05 level of significance and interpret the resulting decision.
A professor in the school of mathematics in a university polled a dozen colleagues about the number of academic workshops they attended in the previous five years (x) and the number of papers they submitted to refereed journals (y)during the same period. The summary of the data are given as
A study of the amount of rainfall and the quantity of air pollution removed produced the following data:1. Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall.2. Estimate the amount of particulate removed when the daily rainfall is x = 4.8
A study was done to study the effect of ambient temperature x on the electric power consumed by a chemical plant y. Other factors were held constant, and the data were collected from an experimental pilot plant.1. Plot the data.2. Estimate the slope and intercept in a simple linear regression
The thrust of an engine (y) is a function of exhaust temperature (x) in °F when other important variables are held constant. Consider the following data.1. Plot the data.2. Fit a simple linear regression to the data and plot the line through the data.
The following data are the selling prices z of a certain make and model of used motorcycle w years old. Fit a curve of the form by means of the nonlinear sample regression equation . [Hint: Write .]
A study was made by a retail merchant to determine the relation between weekly advertising expenditures and sales.1. Plot a scatter diagram.2. Find the equation of the regression line to predict weekly sales from advertising expenditures.3. Estimate the weekly sales when advertising costs are
A mathematics placement test is given to all entering freshmen at a small college. A student who receives a grade below 35 is denied admission to the regular mathematics course and placed in a remedial class. The placement test scores and the final grades for 20 students who took the regular course
In a certain type of metal test specimen, the normal stress on a specimen is known to be functionally related to the shear resistance. The following is a set of coded experimental data on the two variables:1. Estimate the regression line μ = β + β x.2. Estimate the shear resistance for a normal
A study was made on the amount of converted sugar in a certain process at various temperatures. The data were coded and recorded as follows:1. Estimate the linear regression line.2. Estimate the mean amount of converted sugar produced when the coded temperature is 1.75.3. Plot the residuals versus
The following data were collected from 10 individuals to determine the relationship between age and systolic blood pressure for the purpose of a medical study.1. Find the equation of the regression line.2. The purpose of study is to estimate the age from an observed systolic blood pressure of the
The amounts of a chemical compound y that dissolved in 100 grams of water at various temperatures x were recorded as follows:1. Find the equation of the regression line.2. Graph the line on a scatter diagram.3. Estimate the amount of chemical that will dissolve in 100 grams of water at 50°C.
The grades of a class of 9 students on a midterm report (x)and on the final examination (y) are as follows:0 1 Y|x 0 1Y|30 1. Estimate the linear regression line.2. Estimate the final examination grade of a student who received a grade of 85 on the midterm report.
A study was conducted at Virginia Tech to determine if certain static arm-strength measures have an influence on the“dynamic lift” characteristics of an individual. Twenty-five individuals were subjected to strength tests and then were asked to perform a weight-lifting test in which weight was
z-Value for Testing p – p = d : To test the null hypothesis H that p – p = d , where d ≠ 0, we base our decision on which is a value of a random variable whose distribution approximates the standard normal distribution as long as n and n are both large. With reference to Example 10.11 on page
Studies show that the concentration of PCBs is much higher in malignant breast tissue than in normal breast tissue.If a study of 50 women with breast cancer reveals an average PCB concentration of 22.8 × 10 gram, with a standard deviation of 4.8 × 10 gram, is the mean concentration of PCBs less
The following data show the numbers of defects in 100,000 lines of code in a particular type of software program developed in the United States and Japan. Is there enough evidence to claim that there is a significant difference between the programs developed in the two countries? Test on
In a study conducted by the Water Resources Research Center and analyzed by the Laboratory for Interdisciplinary Statistical Consulting at Virginia Tech, two different wastewater treatment plants are compared. Plant A is located where the median household income is below $22,000 a year, and plant B
In a study conducted by the Department of Mechanical Engineering and analyzed by the Laboratory for Interdisciplinary Statistical Consulting at Virginia Tech, steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company,
A study was made to determine whether there is a difference between the proportions of parents in the states of Maryland (MD), Virginia (VA), Georgia (GA), and Alabama (AL)who favor placing Bibles in the elementary schools. The responses of 100 parents selected at random in each of these states are
In a random sample of 90 students recruited from a college, 36, 42, and 39 students were from departments A, B, and C, respectively. Can we conclude at a 0.05 level of confidence that the placement of students from all three departments was in equal proportion?
Consider the situation of Exercise 10.54 on page 380.Oxygen consumption in mL/kg/min, was also measured.It is conjectured that oxygen consumption should be higher in an environment relatively free of CO. Do a significance test and discuss the conjecture.
A study was conducted to determine whether the proportion of female professionals in the Indian software industry from rural areas was more than that of their male counterparts. Of the 420 female professionals selected at random, 174 were from the rural areas, whereas, out of the 380 male
In a socio-economic survey conducted among a random sample of 1200 villagers from a coastal village A, 744 earn their living by fishing. In a nearby village B, 672 out of 1200 earn living by fishing. Can we conclude that at 0.05 level of significance, the proportion of the villagers in village A
State the null and alternative hypotheses to be used in testing the following claims and determine generally where the critical region is located:1. The mean snowfall at Lake George during the month of February is 21.8 centimeters.2. No more than 20% of the faculty at the local university
In a study to estimate the proportion of wives who regularly watch soap operas, it is found that 52 of 200 wives in Denver, 31 of 150 wives in Phoenix, and 37 of 150 wives in Rochester watch at least one soap opera. Use a 0.05 level of significance to test the hypothesis that there is no difference
A survey was conducted in two Virginia cities to determine voter sentiment about two gubernatorial candidates in an upcoming election. Five hundred voters were randomly selected from each city and the following data were recorded:At the 0.05 level of significance, test the null hypothesis that
A survey was conducted in Indiana, Kentucky, and Ohio to determine the attitude of voters concerning school busing. A poll of 200 voters from each of these states yielded the following results:At the 0.05 level of significance, test the null hypothesis that the proportions of voters within each
To determine current attitudes about prayer in public schools, a survey was conducted in four Virginia counties. The following table gives the attitudes of 200 parents from Craig County, 150 parents from Giles County, 100 parents from Franklin County, and 100 parents from Montgomery County:Test for
A college infirmary conducted an experiment to determine the degree of relief provided by three cough remedies. Each cough remedy was tried on 50 students and the following data recorded:Test the hypothesis that the three cough remedies are equally effective. Use a P-value in your conclusion.
The following responses concerning the standard of living at the time of an independent opinion poll of 1000 households versus one year earlier seem to be in agreement with the results of a study published in Across the Board (June 1981):Test the hypothesis that the proportions of households within
According to a Johns Hopkins University study published in the American Journal of Public Health, widows live longer than widowers. Consider the following survival data collected on 100 widows and 100 widowers following the death of a spouse:Can we conclude at the 0.05 level of significance that
A criminologist conducted a survey to determine whether the incidence of certain types of crime varied from one part of a large city to another. The particular crimes of interest were assault, burglary, larceny, and homicide. The following table shows the numbers of crimes committed in four of the
A random sample of 200 married men, all retired, was classified according to education and number of children:Test the hypothesis, at the 0.05 level of significance, that the size of a family is independent of the level of education attained by the father.
A study selects 60 boys and 40 girls from a school and classifies them according to their ability in mathematics.Use a 0.05 level of significance and test the hypothesis that gender and ability in mathematics are independent.
In an experiment to study the dependence of hypertension on smoking habits, the following data were taken on 180 individuals:Test the hypothesis that the presence or absence of hypertension is independent of smoking habits. Use a 0.05 level of significance.
For Exercise 1.19 on page 51, test the goodness of fit between the observed class frequencies and the corresponding expected frequencies of a normal distribution with μ = 1.8 and σ= 0.4, using a 0.01 level of significance.
For Exercise 1.18 on page 51, test the goodness of fit between the observed class frequencies and the corresponding expected frequencies of a normal distribution with μ = 65 and σ= 21, using a 0.05 level of significance.
A coin is thrown until a head occurs and the number X of tosses recorded. After repeating the experiment 256 times, we obtained the following results:Test the hypothesis, at the 0.05 level of significance, that the observed distribution of X may be fitted by the geometric distribution g(x; 1/2), x
Three marbles are selected from an urn containing 5 red marbles and 3 green marbles. After the number X of red marbles is recorded, the marbles are replaced in the urn and the experiment repeated 112 times. The results obtained are as follows:Test the hypothesis, at the 0.05 level of significance,
A die is tossed 240 times and shows the following results:Verify whether the die is unbiased. Use a 0.05 level of significance.
The grades in a statistics course for a particular semester were as follows:Test the hypothesis, at the 0.05 level of significance, that the distribution of grades is uniform.
The ratio of the students attending a mathematics exhibition from high school, higher secondary classes, undergraduate classes, and graduate classes is 3:5:6:2. Out of the 512 mathematics students who attended the exhibition, 98 were from high school, 162 were from the higher secondary classes, 180
Hydrocarbon emissions from cars are known to have decreased dramatically during the 1980s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), for automobiles from 1980 and 1990. Twenty cars of each model year were randomly selected, and their
An experiment was conducted to compare the alcohol content of soy sauce on two different production lines.Production was monitored eight times a day. The data are shown here.A B A B Assume both populations are normal. It is suspected that production line 1 is not producing as consistently as
Two types of instruments for measuring the amount of sulfur monoxide in the atmosphere are being compared in an air-pollution experiment. Researchers wish to determine whether the two types of instruments yield measurements having the same variability. The readings in the following table were
With reference to Exercise 10.39 on page 378, test the hypothesis that against the alternative that , where and are the variances for the running times of films produced by company 1 and company 2, respectively. Use a Pvalue.
For Exercise 10.41 on page 378, test the hypothesis at the 0.05 level of significance that against the alternative that, where and are the variances of the number of organisms per square meter of water at the two different locations on Cedar Run.
Large-Sample Test of : When n ≥ 30, we can test the null hypothesis that , or σ = σ , by computing which is a value of a random variable whose sampling distribution is approximately the standard normal distribution.1. With reference to Example 10.4, test at the 0.05 level of significance
A soft-drink dispensing machine is said to be out of control if the variance of the contents exceeds 1.15 deciliters. If a random sample of 25 drinks from this machine has a variance of 2.03 deciliters, does this indicate at the 0.05 level of significance that the machine is out of control? Assume
Past data indicate that the amount of money contributed by the working residents of a large city to a volunteer rescue squad is a normal random variable with a standard deviation of$1.40. It has been suggested that the contributions to the rescue squad from just the employees of the sanitation
Aflotoxins produced by mold on peanut crops in Virginia must be monitored. A sample of 64 batches of peanuts reveals levels of 24.17 ppm, on average, with a variance of 4.25 ppm.Test the hypothesis that σ = 4.2 ppm against the alternative thatσ ≠ 4.2 ppm. Use a P-value in your conclusion.
A study conducted by an agriculture department reveals that the water content in tender coconuts is normally distributed, with a variance of 0.075 liter. Test the hypothesis that σ = 0.075 against the alternative that σ ≠ 0.075 using the water content found in a random sample of 10 tender
Group Project: The class should be divided into pairs of students for this project. Suppose it is conjectured that at least 25% of students at your university exercise for more than two hours a week. Collect data from a random sample of 50 students. Ask each student if he or she works out for at
It is believed that the proportion of boys getting placed through campus recruitment is higher than that of girls. Among the 365 boys and 324 girls in the final semester of an engineering college, 180 and 142 got placed in various companies through the campus recruitment drive, respectively.Can we
In a study on the fertility of married women conducted by Martin O’Connell and Carolyn C. Rogers for the Census Bureau in 1979, two groups of childless wives aged 25 to 29 were selected at random, and each was asked if she eventually planned to have a child. One group was selected from among

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