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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
The following data were obtained in a study of the relationship between the weight and chest size of (a) Calculate r.(b) Test the null hypothesis that ρ = 0 against the alternative that ρ > 0 at the 0.01 level of significance.(c) What percentage of the variation in infant chest sizes is explained
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 400, assume a bivariate normal distribution for x and y.(a) Calculate r.(b) Test the null hypothesis that ρ = −0.5 against the alternative that ρ < −0.5 at the 0.025 level of significance.(c) Determine the percentage of the variation in the amount of
With reference to Exercise 11.1 on page 398, assume that x and y are random variables with a bivariate normal distribution.(a) Calculate r.(b) 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:Mathematics grade 70 92 80 74 65 83 English grade 74 84 63 87 78 90
For a particular variety of plant, researchers wanted to develop a formula for predicting the quantity of seeds (in grams) as a function of the density of plants. They conducted a study with four levels of the factor x, the number of plants per plot. Four replications were used for each level of x.
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
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:(a) Fit a linear regression model.(b) 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 furnace
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 Organophosphate Pesticides on Wildlife Species, by the Department of Fisheries and Wildlife at Virginia Tech, an
Transistor gain between emitter and collector in an integrated circuit device (hFE) is related to two variables (Myers, Montgomery and Anderson-Cook, 2009) that can be controlled at the deposition process, emitter drive-in time (x1, in minutes) and emitter dose(x2, in ions × 1014). Fourteen
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 β1 = 0 against the alternative hypothesis β1 = 0 in Exercise 11.5 on page 398 at the 0.05 level of significance.
Suppose we have a linear equation through the origin (Exercise 11.28) μY |x = βx.(a) Estimate the regression line passing through the origin for the following data:
Test for linearity of regression in Exercise 11.8.on page 399. Comment.
Test for linearity of regression in Exercise 11.3.on page 398. Use a 0.05 level of significance. Comment.
For the data in Exercise 11.29, find a 95% prediction interval at x = 25.
Use the data set(a) Plot the data.(b) Fit a regression line through the origin.(c) Plot the regression line on the graph with the data.(d) Give a general formula (in terms of the yi and the slope b1) for the estimator of σ2.(e) Give a formula for Var(ˆyi), i = 1, 2,...,n, for this case.(f) Plot
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 Yi = β1xi + i, i = 1, 2, . . . , n, and only a simple parameter requires estimation. The model is
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.(a) Estimate the mileage for a vehicle weighing 4000
With reference to Exercise 11.3. on page 398, use the value of s2 found in Exercise 11.19(a) to compute(a) a 99% confidence interval for the average amount of chemical that will dissolve in 100 grams of water at 50◦C;(b) a 99% prediction interval for the amount of chemical that will dissolve in
Using the value of s2 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 398.
Using the value of s2 found in Exercise 11.17(a), graph the regression line and the 95% confidence bands for the mean response μY |x for the data of Exercise 11.5. on page 398.
With reference to Exercise 11.6. on page 399, use the value of s2 found in Exercise 11.18(a) to compute(a) a 95% confidence interval for the mean shear resistance when x = 24.5;(b) a 95% prediction interval for a single predicted value of the shear resistance when x = 24.5.
Using the value of s2 found in Exercise 11.16(a), construct a 95% confidence interval for μY |85 in Exercise 11.2. on page 398.
Test the hypothesis that β1 = 6 in Exercise 11.9 on page 399 against the alternative that β1 < 6.Use a 0.025 level of significance.
Test the hypothesis that β0 = 10 in Exercise 11.8 on page 399 against the alternative that β0 < 10.Use a 0.05 level of significance.
With reference to Exercise 11.3 on page 398,(a) evaluate s2;(b) construct a 99% confidence interval for β0;(c) construct a 99% confidence interval for β1.
With reference to Exercise 11.6 on page 399,(a) evaluate s2;(b) construct a 99% confidence interval for β0;(c) construct a 99% confidence interval for β1.
With reference to Exercise 11.5 on page 398,(a) evaluate s2;(b) construct a 95% confidence interval for β0;(c) construct a 95% confidence interval for β1.
With reference to Exercise 11.2 on page 398,(a) evaluate s2;(b) construct a 95% confidence interval for β0;(c) construct a 95% confidence interval for β1.
With reference to Exercise 11.1 on page 398,(a) evaluate s2;(b) test the hypothesis that β1 = 0 against the alternative that β1 = 0 at the 0.05 level of significance and interpret the resulting decision.
A professor in the School of Business in a university polled a dozen colleagues about the number of professional meetings they attended in the past five years (x) and the number of papers they submitted to refereed journals (y) during the same period. The summary data are given as follows:Fit a
A study of the amount of rainfall and the quantity of air pollution removed produced the following
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.(a) Plot the data.(b) 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.(a) Plot the data.(b) 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 car w years old. Fit a curve of the form μz|w = γδw by means of the nonlinear sample regression equation ˆz = cdw. [Hint: Write ln ˆz = ln c + (ln d)w = b0 + b1w.]
A study was made by a retail merchant to determine the relation between weekly advertising expenditures and sales(a) Plot a scatter diagram.(b) Find the equation of the regression line to predict weekly sales from advertising expenditures.(c) 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
The following is a portion of a classic data set called the “pilot plot data” in Fitting Equations to Data by Daniel and Wood, published in 1971. The response y is the acid content of material produced by titration, whereas the regressor x is the organic acid content produced by extraction and
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:(a) Estimate the regression line μY |x = β0 + β1x.(b) Estimate the shear resistance for a
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:(a) Estimate the linear regression line.(b) Estimate the mean amount of converted sugar produced when the coded temperature is 1.75.(c) Plot the residuals
The following data were collected to determine the relationship between pressure and the corresponding scale reading for the purpose of calibration(a) Find the equation of the regression line.(b) The purpose of calibration in this application is to estimate pressure from an observed scale
The amounts of a chemical compound y that dissolved in 100 grams of water at various temperatures x were recorded as follows:
The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows:(a) Estimate the linear regression line.(b) 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 weightlifting test in which weight was
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 × 10−4 gram, with a standard deviation of 4.8 × 10−4 gram, is the mean concentration of
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 means.
In a study conducted by the Water Resources Center and analyzed by the Statistics Consulting Center 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 is located where the median
In a study conducted by the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center 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, and the “bounciness” was
A study was conducted at the Department of Health and Physical Education at Virginia Tech to determine if 8 weeks of training truly reduces the cholesterol levels of the participants. A treatment group consisting of 15 people was given lectures twice a week on how to reduce cholesterol level.
A study was conducted at the VirginiaMaryland Regional College of Veterinary Medicine Equine Center to determine if the performance of a certain type of surgery on young horses had any effect on certain kinds of blood cell types in the animal. Fluid samples were taken from each of six foals before
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
If one can containing 500 nuts is selected at random from each of three different distributors of mixed nuts and there are, respectively, 345, 313, and 359 peanuts in each of the cans, can we conclude at the 0.01 level of significance that the mixed nuts of the three distributors contain equal
State the null and alternative hypotheses to be used in testing the following claims, and determine generally where the critical region is located:(a) At most, 20% of next year’s wheat crop will be exported to the Soviet Union.(b) On the average, American homemakers drink 3 cups of coffee per
In a study analyzed by the Statistics Consulting Center at Virginia Tech, a group of subjects was asked to complete a certain task on the computer. The response measured was the time to completion. The purpose of the experiment was to test a set of facilitation tools developed by the Department of
Consider the situation of Exercise 10.54 on page 360. Oxygen consumption in mL/kg/min, was also measured.
A study was made to determine whether more Italians than Americans prefer white champagne to pink champagne at weddings. Of the 300 Italians selected at random, 72 preferred white champagne, and of the 400 Americans selected, 70 preferred white champagne. Can we conclude that a higher proportion of
A geneticist is interested in the proportions of males and females in a population who have a certain minor blood disorder. In a random sample of 100 males, 31 are found to be afflicted, whereas only 24 of 100 females tested have the disorder. Can we conclude at the 0.01 level of significance that
State the null and alternative hypotheses to be used in testing the following claims and determine generally where the critical region is located:(a) The mean snowfall at Lake George during the month of February is 21.8 centimeters.(b) 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:
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 areas
A random sample of 200 married men, all retired, was classified according to education and number of children:Number of Children Education 0–1 2–3 Over 3 Elementary 14 37 32 Secondary 19 42 17 College 12 17 10 Test the hypothesis, at the 0.05 level of significance, that the size of a family is
A random sample of 90 adults is classified according to gender and the number of hours of television watched during a week:
In an experiment to study the dependence of hypertension on smoking habits, the following data were taken on 180 individuals:Non- Moderate Heavy smokers Smokers Smokers Hypertension 21 36 30 No hypertension 48 26 19 Test the hypothesis that the presence or absence of hypertension is independent of
For Exercise 1.19 on page 31, 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 31, 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 ex periment 256 times, we obtained the following results:x 1 2 3 4 5678 f 136 60 34 12 9 1 3 1 Test the hypothesis, at the 0.05 level of significance, that the observed distribution of X may be fitted by
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 180 times with the following results:x 123456 f 28 36 36 30 27 23 Is this a balanced die? Use a 0.01 level of significance.
The grades in a statistics course for a particular semester were as follows:Grade ABCDF f 14 18 32 20 16 Test the hypothesis, at the 0.05 level of significance, that the distribution of grades is uniform.
A machine is supposed to mix peanuts, hazelnuts, cashews, and pecans in the ratio 5:2:2:1. A can containing 500 of these mixed nuts was found to have 269 peanuts, 112 hazelnuts, 74 cashews, and 45 pecans.At the 0.05 level of significance, test the hypothesis that the machine is mixing the nuts in
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.Production line 1:
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 358, test the hypothesis that σ2 1 = σ2 2 against the alternative that σ2 1 = σ2 2, where σ2 1 and σ2 2 are the variances for the running times of films produced by company 1 and company 2, respectively. Use a P-value.
For Exercise 10.41 on page 358, test the hypothesis at the 0.05 level of significance that σ2 1 = σ2 2against the alternative that σ2 1 = σ2 2, where σ2 1 andσ2 2 are the variances of the number of organisms per square meter of water at the two different locations on Cedar Run.
A study is conducted to compare the lengths of time required by men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women is approximately normal but the variance of the times for women is less than that for men. A random sample of
Large-Sample Test of σ2 = σ2 0: When n ≥30, we can test the null hypothesis that σ2 = σ2 0, orσ = σ0, by computing which is a value of a random variable whose sampling distribution is approximately the standard normal distribution.
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 thatσ2 = 4.2 ppm against the alternative that σ2 = 4.2 ppm. Use a P-value in your conclusion.
Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a standard deviation of 6 minutes. Test the hypothesis that σ = 6 against the alternative that σ < 6 if a random sample of the test times of 20 high school
The content of containers of a particular lubricant is known to be normally distributed with a variance of 0.03 liter. Test the hypothesis that σ2 = 0.03 against the alternative that σ2 = 0.03 for the random sample of 10 containers in Exercise 10.23 on page 356.Use a P-value in your conclusion.
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
An urban community would like to show that the incidence of breast cancer is higher in their area than in a nearby rural area. (PCB levels were found to be higher in the soil of the urban community.) If it is found that 20 of 200 adult women in the urban community have breast cancer and 10 of 150
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 wives
In a study to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant, it is found that 63 of 100 urban residents favor the construction while only 59 of 125 suburban residents are in favor. Is there a significant difference between
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 wellknown 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
At a certain college, it is estimated that at most 25% of the students ride bicycles to class. Does this seem to be a valid estimate if, in a random sample of 90 college students, 28 are found to ride bicycles to class? Use a 0.05 level of significance.
A fuel oil company claims that one-fifth of the homes in a certain city are heated by oil. Do we have reason to believe that fewer than one-fifth are heated by oil if, in a random sample of 1000 homes in this city, 136 are heated by oil? Use a P-value in your conclusion.
It is believed that at least 60% of the residents in a certain area favor an annexation suit by a neighboring city. What conclusion would you draw if only 110 in a sample of 200 voters favored the suit? 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
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