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Statistics Principles And Methods 7th Edition Richard A. Johnson, Gouri K. Bhattacharyya - Solutions

Refer to the physical fitness data in Table D.5 of the Data Bank. Find a 95% confidence interval for the mean difference of the pretest minus posttest time to complete the rowing test. Also test that the mean difference is zero versus a two-sided alternative with α = .05.
Plot the line y = 2 + 3x by locating the points for x = 1 and x = 4. What is its intercept? What is its slope?
Consider the linear regression model 7 = β0 + β1x + e Where β0 = -2,β1 = - 1, and the normal random variable e has standard deviation 3. (a) What is the mean of the response y when x = 3? When x = 5? (b) Will the response at x = 3 always be larger than that at x = 6? Explain.
Consider the following linear regression model y = β0 + β1x + e where β0 = 4, β1= 3, and the normal random variable e has the standard deviation 4. (a) What is the mean of the response y when x = 4? When x = 5? (b) Will the response at x = 5 always be larger than that at x = 4? Explain.
A student collected data on the number of large pizzas consumed, y, while x students are watching a professional football game on TV. Suppose that the data from five games are:(a) Construct a scatter diagram. (b) Calculate , Sxx, Sxy , and Syy. (d) Determine the fitted line and draw the line on
The office manager at a real estate firm makes a pot of coffee every morning. The time before it runs out, y, in hours depends on the number of persons working inside that day, x. Suppose that the pairs of ( x, y) values from six days are:(a) Plot the scatter diagram.(b) Calculate ,Sxy, and Syy
Refer to Exercise 11.12. (a) Find the residuals and verify that they sum to zero. (b) Alculate the residual sum of squares SSE by (i) Adding the squares of the residuals. (ii) Using the formula SSE = Syy - S2xy / Sxx (c) Obtain the estimate of a 2.
Refer to Exercise 11.13. (a) Find the residuals and verify that they sum to zero. (b) Calculate the residual sums of squares SSE by (i) Adding the squares of the residuals. (ii) Using the formula SSE = Syy - S2xy / Sxx (c) Obtain the estimate of σ2.
A help desk devoted to student software problems also receives phone calls. The number of persons that can be served in person, within one hour, is the response y. The predictor variable, x, is the number of phone calls answered.(a) Calculate ,Sxy, and Syy (c) Determine the fitted line. (d) Use the
A student hourly employee does small secretarial projects. The number of projects she completes in a day is the response variable y. The number of hours she works in a day is the predictor variable x.(a) Calculate ,Sxy, and Syy (c) Determine the fitted line. (d) Use the fitted line to give a point
To be able to better predict the output of an oil field, researchers2 collected data that included y = the yearly output (100,000 barrels) and the number of new well drilled (1000) in the previous year.The summary statistics are(a) Obtain the equation of the best fitting straight line. (b)
To extend known results on the social hierarchy of monkeys, researchers3 scanned n = 14 healthy persons using positron emission tomography to image dopamine type 2/3 in the brain. The social status of each person was assessed using the Bartlett Simplified Measure of Social Status. Letting x =
A store manager has determined that the monthly profit y realized from selling a particular brand of car battery is given by y = 12x - 75 where x denotes the number of these batteries sold in a month. (a) If 41 batteries were sold in a month, what was the profit? (b) At least how many batteries
The data on female wolves in Table D.9 of the Data Bank concerning body weight (lb) and body length (cm) are(a) Obtain the least squares fit of body weight to the predictor body length. (b) Calculate the residual sum of squares. (c) Estimate a 2.
Refer to the data on female wolves in Exercise 11.20.(a) Obtain the least squares fit of body length to the predictor body weight. (b) Calculate the residual sum of squares. (c) Estimate o2. (d) Compare your answer in part (a) with your answer to part (a) of Exercise 11.20. Should the two answers
(a) The predicted values are
We all typically go to the shortest line in the grocery store. Data were collected on the number of carts ahead in line and the total time to check out (minutes), including time in line, on five occasions.(b) Test H0: Î²1 = 0 versus H1: Î²1 ‰ 0 with a = .05. (c)
Refer to Exercise 11.25. Construct a 90% confidence interval for the intercept Î²1. Interpret.In Exercise 11.25We all typically go to the shortest line in the grocery store. Data were collected on the number of carts ahead in line and the total time to check out (minutes), including time
Refer to Exercise 11.25. Obtain a 95% confidence interval for Î²1. Interpret.In Exercise 11.25We all typically go to the shortest line in the grocery store. Data were collected on the number of carts ahead in line and the total time to check out (minutes), including time in line, on five
An engineer found that by adding small amounts of a compound to rechargeable batteries during manufacture, she could extend their lifetimes. She experimented with different amounts of the additive (g) and measured the hours they lasted in a laptop.Amount Lifeof
For a random sample of seven homes that were recently sold in a city suburb, the assessed values x and the selling prices y are(a) Plot the scatter diagram.(b) Determine the equation of the least squares regression line and draw this line on the scatter diagram.(c) Construct a 95% confidence
Identify the predictor variable x and the response variable y in each of the following situations. (a) A training director wishes to study the relationship between the duration of training for new recruits and their performance in a skilled job. (b) The aim of a study is to relate the carbon
Refer to the data in Exercise 11.29.(a) Estimate the expected selling price of homes that were assessed at $290,000 and construct a 95% confidence interval. Interpret.(b) For a single home that was assessed at $290,000, give a 95% prediction interval for the selling price. Interpret.Data From
One measure of the development of a country is the Human Development Index (HDI). Life expectancy, literacy, educational attainment, and gross domestic product per capita are combined into an index between 0 and 1, inclusive with 1 being the highest development. The United Nations Development
Refer to Exercise 11.31.(a) Obtain the least squares estimates by fitting a straight line to the response Internet usage using the predictor variable HDI.(b) Test, with a = .05, H(): 3 = 0 versus a two-sided alternative.(c) Obtain a 95% confidence interval for the mean Internet usage per 100
According to the computer output in Table 8:(a) What model is fitted?(b) Test, with Î± = .05, if the x term is needed in the model.Table 8
According to the computer output in Table 8:(a) Predict the mean response when x = 5000.(b) Find a 90% confidence interval for the mean response when x = 5000. You will need the additional information n = 30, = 8354, and ˆ‘ (x1 - )2 = 97,599,296.Table 8
According to the computer output in Table 9: (a) What model is fitted? (b) Test, with α = .05, if the x term is needed in the model. Table 9 Computer Output for Exercise 11.35
According to the computer output in Table 9:(a) Predict the mean response when x = 3.(b) Find a 90% confidence interval for the mean response when x = 3. You will need the additional information n = 25, = 1.793, and ˆ‘ (xi - )2 = 1.848.(c) Find a 90% confidence interval for the mean
Consider the data on male wolves in Table D.9 of the Data Bank concerning age (years) and canine length (mm). (a) Obtain the least squares fit of canine length to the predictor age. (b) Test H0: β1 = 0 versus H1 : β1 ≠ 0 with α = .05. (c) Obtain a 90% confidence interval for the canine length
To extend known results on the social hierarchy in monkeys, researchers scanned n = 14 healthy persons using positron emission tomography to image dopamine type 2/3 in the brain. The social status of each person was assessed using the Bartlett Simplified Measure of Social Status. Letting x =
Identify the values of the parameters nu β0, and β1 in the statistical modelY = 4 + 2x + e where e is a normal random variable with mean 0 and standard deviation 5.
Refer to Exercise 11.25.(a) What proportion of the y variability is explained by the linear regression on x?(b) Find the sample correlation coefficient.Exercise 11.25We all typically go to the shortest line in the grocery store. Data were collected on the number of carts ahead in line and the total
Refer to Exercise 11.28.(a) What proportion of y variability is explained by the linear regression on x?(b) Find the sample correlation coefficient.Exercise 11.28An engineer found that by adding small amounts of a compound to rechargeable batteries during manufacture, she could extend their
Scientists6, looking for ways to reduce the amount of organic aerosols (OA) in the atmosphere, measured the nighttime rate of production of NO3 and also the rate of production of particular organic nitrates which comprise about 30% of the organic aerosol present. The summary statistics for the
(b) Show that SSE = (1 - r2)Syy.
Concerns that were raised for the environment near a government facility led to a study of plants. Since leaf area is difficult to measure, the leaf area ( cm2) was fit to x = Leaf length × Leaf width using a least squares approach. For data collected one year, the fitted regression line is = .2
Identify the values of the parameters β0, βl, and cr in the statistical modelY = 6 - 3x + ewhere e is a normal random variable with mean 0 and standard deviation 3.
Last week's total number of hours worked by a student, y, depends on the number of days, x, he reported to work last week. Suppose the data from nine students provided(a) Plot the scatter diagram. (b) Calculate ,, Sxx ,Syy and Sxy (c) Determine the equation of the least squares fitted line and draw
Refer to Exercise 11.50.(a) Find the residuals.(b) Calculate the SSE by (i) summing the squares of the residuals and also (ii) using the formula SSE = Syy - S2xy/Sxx.(c) Estimate the error variance.Exercise 11.50Last week's total number of hours worked by a student, y, depends on the number of
Refer to Exercise 11.50.(a) Construct a 95% confidence interval for the slope of the regression line.(b) Obtain a 90% confidence interval for the expected y value corresponding to x = 4 days.Exercise 11.50Last week's total number of hours worked by a student, y, depends on the number of days, x, he
An experiment is conducted to determine how the strength y of plastic fiber depends on the size x of the droplets of a mixing polymer in suspension. Data of (x, y) values, obtained from 15 runs of the experiment, have yielded the following summary statistics.(a) Obtain the equation of the least
Refer to Exercise 11.53.(a) Obtain the decomposition of the total y variability into two parts: one explained by linear relation and one not explained.(b) What proportion of the y variability is explained by the straight line regression?(c) Calculate the sample correlation coefficient between x and
A recent graduate moving to a new job collected a sample of monthly rent (dollars) and size (square feet) of 2-bedroom apartments in one area of a midwest city.(a) Plot the scatter diagram and find the least squares fit of a straight line. (b) Do these data substantiate the claim that the monthly
Refer to Exercise 11.55.(a) Calculate the sample correlation coefficient.(b) What proportion of the y variability is explained by the fitted regression line?Exercise 11.55A recent graduate moving to a new job collected a sample of monthly rent (dollars) and size (square feet) of 2-bedroom
A top Internet site features used cars for sale. The age, x, measured in years and the price, y, expressed in thousands of dollars was recorded from a sample of listings for a popular midsize foreign car.(a) Plot the scatter diagram. (b) Determine the equation of the least squares regression line
Refer to Exercise 11.57.(a) From the fitted regression line, determine the predicted value for the average selling price of a 5-year-old car and construct a 95% confidence interval.(b) Determine the predicted value for a 5-year-old car to be listed on the site next week. Construct a 90% prediction
Again referring to Exercise 11.57, find the sample correlation coefficient between age and selling price. What proportion of the variability in price is explained by the fitted straight line? Comment on the adequacy of the straight line fit.Exercise 11.57A top Internet site features used cars for
Under the linear regression model: (a) Determine the mean and standard deviation of Y, for x = 4, when β0 = 1, β1 = 3, and σ =2. (b) Repeat part (a) with x = 2.
The calculations involved in a regression analysis become increasingly tedious with larger data sets. Access to a computer proves to be of considerable advantage. We repeat here a computer-based analysis of linear regression using the data of Example 4 and the MINITAB package.The sequence of steps
Consider the data on wolves in Table D.9 of the Data Bank concerning body length (cm) and weight (lb). Calculate the correlation coefficient r and r2 for (a) All wolves. (b) Male wolves. (c) Female wolves. (d) Comment on the differences in your answers. Make a multiple scatter diagram (see Chapter
Consider the data on all of the wolves in Table D.9 of the Data Bank concerning body length (cm) and weight (lb). Using MINITAB or some other software program: (a) Plot weight versus body length. (b) Obtain the least squares fit of weight to the predictor variable body length. (c) Test H0 : β1 = 0
Refer to Exercise 11.62 and a least squares fit using the data on all of the wolves in Table D.9 of the Data Bank concerning body length (cm) and weight (lb). There is one obvious outlier, row 18 with body length 123 and weight 106, indicated in the MINITAB output. Drop this observation. (a) Obtain
Many college students obtain college degree credits by demonstrating their proficiency on exams developed as part of the College Level Examination Program (CLEP). Based on their scores on the College Qualification Test (CQT), it would be helpful if students could predict their scores on a
Use MINITAB or some other software program to regress the marine growth on freshwater growth for the fish growth data in Table D.7 of the Data Bank. Do separate regression analyses for: (a) All fish. (b) Males. (c) Females. Your analysis should include (i) a scatter diagram, (ii) a fitted line,
Use MINITAB or some other software to obtain the scatter diagram, correlation coefficient, and the regression line of the final time to run 1.5 miles on the initial times given in Table D.5 of the Data Bank.
The data on the maximum height (feet) and top speed (mph) of the 12 highest roller coasters, displayed in the chapter opener, are(a) Use MINITAB or some other software program to determine the proportion of variation in speed due to regression on height. (b) What top speed is predicted for a new
Under the linear regression model: (a) Determine the mean and standard deviation of Y, for x = 1, when β0 = 3, β1 = -2, and σ = 3. (b) Repeat part (a) with x = 2.
Graph the straight line for the means of the linear regression model Y = β0 + β1x + e having β0 = -3, β1, = 4.
Graph the straight line for the means of a linear regression model Y = β0 + β1x + e having β0 = 7 and β1, = 2.
Developers have built a small robotic vehicle that can travel over rough terrain. They recorded the time y, in minutes, that it takes to travel a fixed distance over various but similar terrains. For a fixed run, the robot's motor is set at a nominal speed x, in feet per second, but this varied
Consider the data on all of the wolves in Table D.9 of the Data Bank concerning age (years) andcanine length (mm).(a) Obtain the least squares fit of the straight line regression model Y = β0 + β1x + e to Predict canine length from age.(b) Obtain the least squares fit of the multiple regression
Consider the response variable miles per gallon on highways and the two predictor variables x1 = engine volume (liter) and x2 = size of battery (volt). Using a recent government Fuel Economy Guide, and the data on hybrid-electric cars and SUVs, we obtain the regression analysis given in Table 9.(a)
Referring to Example 5, Chapter 11, and Table D.l3 of the Data Bank, we now use two predictors, x1 = girth (cm) and x2 = length to predict y = weight (lb) of a polar bear. The output from a regression analysis is given in Table 10.(a) How many polar bears were included in the analysis?(b) Identify
With reference to Exercise 12.11:(a) Test H0 : p1, =0 versus H1 : βl ≠ 0 with a = .05.(b) Test H0: 32 = 0 versus H1 : β2 ≠ 0 with a = .05.(c) Estimate the expected y value corresponding to x1, = 3.2 and x2 = 200.(d) Construct a 90% confidence interval for the intercept β0.
Referring to Exercise 12.12, we have added one more predictor x3 = skull width (cm). The output for a regression analysis is shown in Table 11.(a) Test H0 : Î²3 = 0 versus H0 : Î²3 # 0 with a = .05.(b) Test H0: Î²2 = 0 versus H0: Î²2 ‰ 0 with a = .05.(c) Estimate the expected value
An environmental scientist identified a point source for E. Coli at the edge of a stream. She then measured y = E. Coli, in colony forming units per 100 ml of water, at different distances, in feet, downstream from the point source. Suppose she obtains the following pairs of (x, y ).(a) Transform
Obtain a linearizing transformation in each case(a)(b) y = eaxb
A genetic experiment is undertaken to study the competition between two types of female flies (Drosophihz melanogaster) in cages with one male genotype acting as a substrate. The independent variable x is the time spent in cages, and the dependent variable y is the ratio of the numbers of type 1 to
Some researchers3 illustrate a multiple linear regression equation to predict the yearly output of an oil field. We use their latest 20 years of data and change the units of three of their variables toy = loge (yearly output) where output is measured in 100,000 barrelsx1 = loge (number of startup
Refer to Exercise 12.18. The estimated standard errors ofare .0606 and .0632, respectively.(a) Obtain a 90% confidence interval for Î²1(b) Test H0:Î²2 = .5 versus H1: Î²2 ‰ .5 with a = .05.Exercise 12.18Some researchers3 illustrate a multiple linear regression equation to predict the
Refer to the data of Exercise 12.1.(a) Consider the reciprocal transformation y €² = 1 / y and plot the scatter diagram of y€² versus x.(b) Fit a straight line regression to the transformed data.(c) Calculate r2 and comment on the adequacy of the fit.In Exercise 12.1.
A second-degree polynomialis fitted to a response y, and the following predicted values and residuals are obtained.Do the assumptions appear to be violated?
The following predicted values and residuals are obtained in an experiment conducted to determine the degree to which the yield of an important chemical in the manufacture of penicillin is dependent on sugar concentration (the time order of the experiments is given in parentheses).(a) Plot the
An experimenter obtains the following residuals after fitting a quadratic expression in x.Do the basic assumptions appear to be violated?
An interested student used the method of least squares to fit the straight line = 850.7 + 320.3 x to gross domestic product, y, in real dollars. The results for 26 recent years, x = 1,2,..., 26, appear here. Which assumption(s) for a linear regression model appear to be seriously violated by the
Consider the data on male wolves in Table D.9 of the Data Bank concerning age (years) and canine length (mm).(a) Obtain the least squares fit of canine length to the predictor age.(b) Obtain the least squares fit of canine length to a quadratic function of the predictor age. The MINITAB commands
The resident population of the United States grew from 1910 to 2010 but the growth was not linear. The response variable is y = population in millions and, to simplify the calculations, the predictor variable is x = year - 1900.(a) Fit a quadratic regression model y = Î²0 + Î²1x + Î²2x2 + e
Listed below are the price quotations on an Internet site for a midsize foreign used car along with their age and odometer mileage.Perform a multiple regression analysis of these data. In particular(a) Determine the equation for predicting the price from age and mileage. Interpret the meaning of
Refer to the data of speed x and stopping distance y given in Table 1.The MINITAB commands for fitting a straight line regression to y€² = ˆšy and x are (a) Obtain the computer output and identify the equation of the fitted line and the value of r2 (see Example 1).(b) Give a 95% confidence
A forester seeking information on basic tree dimensions obtains the following measurements of the diameters 4.5 feet above the ground and the heights of 12 sugar maple trees (courtesy of A. Ek). The forester wishes to determine if the diameter measurements can be used to predict the tree height.(a)
Recorded here are the scores x1 and x2 in two midterm examinations, the GPA x3, and the final examination score y for 20 students in a statistics class.(a) Ignoring the data of GPA and the first midterm score, fit a simple linear regression of y on x2. Compute r2. (b) Fit a multiple linear
Refer to Exercise 11.64.(a) Fit a quadratic model Y = Î²0 + Î²1x + Î²2x2 + e to the data for CLEP scores y and CQT scores x.(b) Use the fitted regression to predict the expected CLEP score when x = 160.(c) Compute r2 for fitting a line and R2 for fitting a quadratic
Write the design matrix X for fitting a multiple regression model to the data of Exercise 12.26.In Exercise 12.26
Write the design matrix X for fitting a quadratic- regression model using the data of Exercise 12.25.In Exercise 12.25.
Refer to the physical conditioning data given in Table D.5 of the Data Bank. Use MINITAB or some other package to fit a regression of the final number of situps on the initial number of situps and the gender of the student.
Refer to the physical fitness data in Table D.5 of the Data Bank. Use both the data on the pretest run time and gender for predicting the posttest run time. Obtain the least squares fit and plot the residuals versus fitted value.
An experiment was conducted for the purpose of studying the effect of temperature on the life-length of an electrical insulation. Specimens of the insulation were tested under fixed temperatures, and their times to failure recorded.(a) Fit a straight line regression to the transformed data(b) Is
In an experiment (courtesy of W. Burkholder) involving stored-product beetles {Trogoderma glabrum) and their sex-attractant pheromone, the pheromone is placed in a pit-trap in the centers of identical square arenas. Marked beetles are then released along the diagonals of each square at various
A student fit the regression model Y = Î²0 + Î²1x1 + Î²2x2 + e to data from the fifty states and Washington, D.C., so n = 51. The response y = median income in thousands of dollars and the two predictor variables are x1 = median monthly housing costs for home owners
Consider the multiple linear regression model Y = β0 + β1x1 + β2x2 + e where β0 = - 1> β1 = -2, β2 = 3, and the normal random variable e has standard deviation 3. What is the mean of the response Y when x1 = 3 and x2 = -2?
In Exercise 12.6, the residual sum of squares (SSE) is 531.6 and the SS due to regression is 4337.7. (a) Estimate the error standard deviation a. State the degrees of freedom. (b) Find R2 and interpret the result.
Refer again to Exercise 12.6 and assume that the assumptions about the model prevail. The estimated standard errors ofare 00161, and .1862, respectively. (a) Determine a 95% confidence interval for 82- (b) TestH0: Î²1 = .0125 versus H1: Î²1 > 0125, with a = .05.
Given below are the frequencies observed from 320 tosses of a die. Do these data cast doubt on the fairness of the die?
Alternative expressions for Pearson's x2.(a) By expanding the square on the right-hand side ofShow that the x2 statistic can also be expressed as (b)Express the x2 statistic as
Among a sample of 800 adult males, 414 said they usually open all of their mail. Among 900 adult females, 532 said they usually open all of their mail.4 (a) Construct a two-way table based on these frequencies. (b) Formulate the null hypothesis. (c) Conduct a/2 test of your null hypothesis. Use a =
Chapter 3, Table 1, gives the counts of underclassmen and upperclassmen according to hours worked.(a) Formulate the null hypothesis of no difference in working hours. (b) Conduct ax2 test of your null hypothesis. Use a = .05. (c) Comment on the form of any departure from the null hypothesis
Nausea from air sickness affects some travelers. In a comparative study of the effectiveness of two brands of motion sickness pills, brand A pills were given to 45 persons randomly selected from a group of 90 air travelers, while the other 45 persons were given brand B pills. The following results
Refer to the hardness of mattresses data in the beginning of the chapter. Confirm that these data establish a difference in the proportions who did not have lower back pain using:(a) The x2 test with level a = .01.(b) The Z test with level a = .01. Calculate the P-value.

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