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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions

12.22 CHALL ENG E The two-sample t test and one-way ANOVA. Refer to the LDL level data in Exercise 7.61 (page 467). Find the two-sample pooled t statistic for comparing men with women.Then formulate the problem as an ANOVA and report the results of this analysis. Verify that F = t2.
12.21 Sleep deprivation and reaction times. Sleep deprivation experienced by physicians during residency training and the possible negative consequences are of concern to many in the health care community. One study of 33 resident anesthesiologists compared their changes from baseline in reaction
12.24 Restaurant ambience and consumer behavior.There have been numerous studies investigating the effects of restaurant ambience on consumer behavior. A recent study investigated the effects of musical genre on consumer spending.7 At a single high-end restaurant in England over a 3-week period,
12.23 The importance of recreational sports to college satisfaction. The National Intramural-Recreational Sports Association (NIRSA)performed a survey to look at the value of recreational sports on college campuses.6 One of the questions asked each student to rate the importance of recreational
12.22 CHALL ENG E The two-sample t test and one-way ANOVA. Refer to the LDL level data in Exercise 7.61 (page 467). Find the two-sample pooled t statistic for comparing men with women.Then formulate the problem as an ANOVA and report the results of this analysis. Verify that F = t2.
12.21 Sleep deprivation and reaction times. Sleep deprivation experienced by physicians during residency training and the possible negative consequences are of concern to many in the health care community. One study of 33 resident anesthesiologists compared their changes from baseline in reaction
12.20 Calculating the pooled standard deviation.An experiment was run to compare four groups.The sample sizes were 25, 28, 150, and 21, and the corresponding estimated standard deviations were 42, 38, 20, and 45.(a) Is it reasonable to use the assumption of equal standard deviations when we analyze
12.19 APPLET The effect of increased variation between groups. Set the pooled standard error for the One-Way ANOVA applet at a middle value. Drag the black dots so that they are approximately equal.(a) What is the F statistic? Give its P-value.(b) Drag the mean of the second group up and the mean
12.18 APPLET The effect of increased variation within groups. The One-Way ANOVA applet lets you see how the F statistic and the P-value depend on the variability of the data within groups and the differences among the means.(a) The black dots are at the means of the three groups. Move these up and
12.17 Calculating the ANOVA F test P-value, continued. For each of the following situations, find the F statistic and the degrees of freedom.Then draw a sketch of the distribution under the null hypothesis and shade in the portion corresponding to the P-value. State how you would report the
12.16 Calculating the ANOVA F test P-value. For each of the following situations, find the degrees of freedom for the F statistic and then use Table E to approximate the P-value.(a) Seven groups are being compared with 5 observations per group. The value of the F statistic is 2.31.(b) Five groups
12.15 A one-way ANOVA example. A study compared 4 groups with 8 observations per group. An F statistic of 3.33 was reported.(a) Give the degrees of freedom for this statistic and the entries from Table E that correspond to this distribution.(b) Sketch a picture of this F distribution with the
12.14 Data collection, continued. Refer to Exercise 12.10. For each situation, discuss the method of obtaining the data and how this will affect the extent to which the results can be generalized.
12.13 Data collection and the interpretation of results. Refer to Exercise 12.9. For each situation, discuss the method of obtaining the data and how this will affect the extent to which the results can be generalized.
12.12 Determining the degrees of freedom, continued. Refer to Exercise 12.10. For each situation, give the following:(a) Degrees of freedom for the model, for error, and for the total.(b) Null and alternative hypotheses.(c) Numerator and denominator degrees of freedom for the F statistic.
12.11 Determining the degrees of freedom. Refer to Exercise 12.9. For each situation, give the following:(a) Degrees of freedom for the model, for error, and for the total.(b) Null and alternative hypotheses.(c) Numerator and denominator degrees of freedom for the F statistic.
12.10 Describing the ANOVA model, continued.For each of the following situations, identify the response variable and the populations to be compared, and give I, the ni, and N.(a) A developer of a virtual-reality (VR) teaching tool for the deaf wants to compare the effectiveness of different
12.9 Describing the ANOVA model. For each of the following situations, identify the response variable and the populations to be compared, and give I, the ni, and N.(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different
12.8 Growth of Douglas fir seedlings. An experiment was conducted to compare the growth of Douglas fir seedlings under three different levels of vegetation control (0%, 50%, and 100%). Forty seedlings were randomized to each level of control. The resulting sample means for stem volume were 50, 75,
12.7 Why no multiple comparisons? Any pooled two-sample t problem can be run as a one-way ANOVAwith I = 2. Explain why it is inappropriate to analyze the data using contrasts or multiple-comparisons procedures in this setting.
12.6 Determining the critical value of F. For each of the following situations, state how large the F statistic needs to be for rejection of the null hypothesis at the 0.05 level.(a) Compare 5 groups with 3 observations per group.(b) Compare 5 groups with 6 observations per group.(c) Compare 5
12.5 What’s wrong? For each of the following, explain what is wrong and why.(a) Within-group variation is the variation in the data due to the differences in the sample means.(b) The mean squares in an ANOVA table will add, that is, MST =MSG + MSE.(c) The pooled estimate sp is a parameter of the
12.4 Visualizing the ANOVA model. For each of the following situations, draw a picture of the ANOVAmodel similar to Figure 12.6 (page 645).Use numerical values for the μi. To sketch the Normal curves, you may want to review the 68–95–99.7 rule on page 59.(a) μ1 = 15, μ2 = 16, μ3 = 21, and
12.3 Computing the pooled standard deviation. An experiment was run to compare three groups. The sample sizes were 25, 22, and 19, and the corresponding estimated standard deviations were 22, 20, and 18.(a) Is it reasonable to use the assumption of equal standard deviations when we analyze these
12.2 What’s wrong? For each of the following, explain what is wrong and why.(a) In rejecting the null hypothesis, one can conclude that all the means are different from each other.(b) A one-way ANOVA can be used only when there are fewer than five means to be compared.(c) A two-way ANOVA is used
12.1 What’s wrong? For each of the following, explain what is wrong and why.(a) ANOVA tests the null hypothesis that the sample means are all equal.(b) A strong case for causation is best made in an observational study.(c) You use one-way ANOVA when the response variable has only two possible
Finding a multiple regression model on the Internet. Search the Internet to find an example of the use of multiple regression. Give the setting of the example, describe the data, give the model, and summarize the results. Explain why the use of multiple regression in this setting was appropriate or
The final multiple regression model of Taste.Use the three explanatory variables Acetic, H2S, and Lactic in a multiple regression to predict Taste. Write a short summary of your results, including an examination of the residuals. Based on all of the regression analyses you have carried out on these
Another multiple regression model of Taste.Carry out a multiple regression using H2S and Lactic to predict Taste. Comparing the results of this analysis with the simple linear regressions using each of these explanatory variables alone, it is evident that a better result is obtained by using both
Multiple regression model of Taste. Carry out a multiple regression using Acetic and H2S to predict Taste. Summarize the results of your analysis. Compare the statistical significance of Acetic in this model with its significance in the model with Acetic alone as a predictor(Exercise 11.53). Which
Comparing the simple linear regression models. Compare the results of the regressions performed in the three previous exercises.Construct a table with values of the F statistic, its P-value, R2, and the estimate s of the standard deviation for each model. Report the three regression equations. Why
The final simple linear regression model of Taste. Repeat the analysis of Exercise 11.53 using Taste as the response variable and Lactic as the explanatory variable.
Another simple linear regression model of Taste. Repeat the analysis of Exercise 11.53 using Taste as the response variable and H2S as the explanatory variable.
Simple linear regression model of Taste.Perform a simple linear regression analysis using Taste as the response variable and Acetic as the explanatory variable. Be sure to examine the residuals carefully. Summarize your results.Include a plot of the data with the least-squares regression line. Plot
Pairwise scatterplots of the explanatory variables. Make a scatterplot for each pair of variables in the CHEESE data set (you will have six plots). Describe the relationships. Calculate the correlation for each pair of variables and report the P-value for the test of zero population correlation in
Describing the explanatory variables. For each of the four variables in the CHEESE data set, find the mean, median, standard deviation, and interquartile range. Display each distribution by means of a stemplot and use a Normal quantile plot to assess Normality of the data. Summarize your findings.
Interpretation of coefficients in log PCB regressions. Use the results of your analysis of the log PCB data in Exercise 11.48 to write an explanation of how regression coefficients, standard errors of regression coefficients, and tests of significance for explanatory variables can change depending
CH ALLENGE Predicting total TEQ using transformed variables. Use the log data set that you created in Exercise 11.46 to find a good multiple regression model for predicting the log of TEQ.Use only log PCB variables for this analysis. Write a report summarizing your results and comparing them with
CH ALLENGE Even more on predicting total amount of PCB using transformed variables. Use the log data set that you created in Exercise 11.46 to find a good multiple regression model for predicting the log of PCB. Use only log PCB variables for this analysis. Write a report summarizing your results.
CH ALLENGE Predicting total amount of PCB using transformed variables, continued. Refer to the previous exercise.(a) Use numerical and graphical summaries to describe the relationships between each pair of log variables.(b) Compare these summaries with the summaries that you produced in Exercise
CHALLENGE Predicting total amount of PCB using transformed variables. Because distributions of variables such as PCB, the PCB congeners, and TEQ tend to be skewed, researchers frequently analyze the logarithms of the measured variables. Create a data set that has the logs of each of the variables
CHALLENGE Multiple regression model for total TEQ, continued. The information summarized in TEQ is used to assess and manage risks from these chemicals. For example, theWorld Health Organization (WHO) has established the tolerable daily intake (TDI) as 1 to 4 TEQs per kilogram of body weight per
Multiple regression model for total TEQ.Dioxins and furans are other classes of chemicals that can cause undesirable health effects similar to those caused by PCB. The three types of chemicals are combined using toxic equivalent scores (TEQs), which attempt to measure the health effects on a common
More on predicting the total amount of PCB.Run a regression to predict PCB using the variables PCB52, PCB118, and PCB138. Note that this is similar to the analysis that you did in Exercise 11.41, with the change that PCB180 is not included as an explanatory variable.(a) Summarize the results.(b) In
Adjusting analysis for potential outliers. The examination of the residuals in part (c) of the previous exercise suggests that there may be two outliers, one with a high residual and one with a low residual.(a) Because of safety issues, we are more concerned about underestimating PCB in a specimen
Predicting the total amount of PCB. Use the four congeners, PCB52, PCB118, PCB138, and PCB180, in a multiple regression to predict PCB.(a) Write the statistical model for this analysis.Include all assumptions.(b) Run the regression and summarize the results.(c) Examine the residuals. Do they appear
Relationship among PCB congeners. Production of polychlorinated biphenyls (PCBs) was banned in the United States in 1977, but because of their widespread use, these compounds are found in many species of fish. As a result, 38 states have issued advisories about limiting consumption of certain
CHALL ENGE Predicting bone resorption using transformed variables. Refer to the previous exercise. Rerun using logs.The following eleven exercises use the PCB data set described in the Data Appendix.
CHALL ENG E Predicting bone resorption. Refer to Exercises 11.34 to 11.36. Answer these questions with the roles of VO+ and VO− reversed;that is, run models to predict VO−, with VO+ as an explanatory variable.
CHALL ENGE Predicting bone formation using transformed variables. Because the distributions of VO+, VO−, OC, and TRAP tend to be skewed, it is common to work with logarithms rather than the measured values. Using the questions in the previous three exercises as a guide, analyze the log data.
More on predicting bone formation. Now consider a regression model for predicting VO+using OC, TRAP, and VO−.(a) Write out the statistical model for this analysis including all assumptions.(b) Run the multiple regression to predict VO+using OC, TRAP, and VO−. Summarize the results.(c) Make a
Predicting bone formation. Let’s use regression methods to predict VO+, the measure of bone formation.(a) Since OC is a biomarker of bone formation, we start with a simple linear regression using OC as the explanatory variable. Run the regression and summarize the results. Be sure to include an
Bone formation and resorption. Consider the following four variables: VO+, a measure of bone formation; VO−, a measure of bone resorption;OC, a biomarker of bone formation; and TRAP, a biomarker of bone resorption.(a) Using numerical and graphical summaries, describe the distribution of each of
Selecting from among several models. Refer to the results from the previous exercise.(a) Make a table giving the estimated regression coefficients, standard errors, t statistics, and P-values.(b) Describe how the coefficients and P-values change for the four models.(c) Based on the table of
Building a multiple linear regression model.Let’s now build a model to predict the lifesatisfaction score, LSI.(a) Consider a simple linear regression using GINI as the explanatory variable. Run the regression and summarize the results. Be sure to check assumptions.(b) Now consider a model using
Predicting a nation’s “average happiness”score. Consider the following five variables for each nation: LSI, life-satisfaction score, an index of happiness; GINI, a measure of inequality in the distribution of income; CORRUPT, a measure of corruption in government; LIFE, the average life
Predicting GPA of seventh-graders. Refer to the educational data for 78 seventh-grade students given in Table 1.9 (page 29). We view GPA as the response variable. IQ, gender, and self-concept are the explanatory variables.(a) Find the correlation between GPA and each of the explanatory variables.
Multiple linear regression model. Now consider a regression model using all three explanatory variables.(a) Write out the statistical model for this analysis, making sure to specify all assumptions.(b) Run the multiple regression model and specify the fitted regression equation.(c) Generate a 95%
Looking at the simple linear regressions.Now let’s look at the relationship between each explanatory variable and the total score.(a) Generate scatterplots for each explanatory variable and the total score. Do these relationships all look linear?(b) Compute the correlation between each
Annual ranking of world universities. Let’s consider developing a model to predict total score based on the peer review score (PEER), faculty-tostudent ratio (FtoS), and citations-to-faculty ratio(CtoF).(a) Using numerical and graphical summaries, describe the distribution of each explanatory
Interpretation of coefficients in a multiple regression. Recall that the relationship between an explanatory variable and a response variable can depend on what other explanatory variables are included in the model.(a) Use a simple linear regression to predict assets using the number of accounts.
Transforming the variables. Sometimes we attempt to model curved relationships by transforming variables. Take the logarithm of assets and the logarithm of the number of accounts. Does the relationship between the logs appear to be approximately linear? Analyze the data and provide a summary of
Curvilinear relationship versus a couple of outliers. To one person, the plot of assets versus the number of accounts indicates that the relationship is curved. Another person might see this as a linear relationship with two outliers.Identify the two outliers and rerun the linear regression and the
Adjusting for correlated explanatory variables.In the multiple regression you performed in the previous exercise, the P-value for the number of accounts was 0.8531, while the P-value for the square was 0.0070. Unless we have a strong theoretical reason for considering a model with a quadratic term
Online stock trading. Online stock trading has increased dramatically during the past several years. An article discussing this new method of investing provided data on the major Internet stock brokerages who provide this service.9 Below are some data for the top 10 Internet brokerages.The
Even more on nutrition labels for foods.Refer to the previous two exercises. When the researchers planned these studies, they expected both unfavorable nutrients and favorable nutrients to be positively associated with the overall product nutrition score. They also expected the unfavorable
More on nutrition labels for foods. The product used in the previous exercise was described by the researchers as a poor-nutrition product.The label information for this product had high values for unfavorable nutrients such as fat and low values for favorable nutrients such as fiber. The
Nutrition labels for foods. Labels providing nutrition facts give consumers information about the nutritional value of food products that they buy. A study of these labels collected data from 152 consumers who were sent information about a frozen chicken dinner. Each subject was asked to give an
CHALL ENGE Enjoyment of physical exercise. Although the benefits of physical exercise are well known, most people do not exercise and many who start exercise programs drop out after a short time. A study designed to determine factors associated with exercise enjoyment collected data from 282 female
Demand for non-biotech cereals. A study designed to determine how willing consumers are to pay a premium for non-biotech breakfast cereals (cereals that do not include gene-altered ingredients) included both U.S. and U.K. subjects.6 The response variable was a measure of how much extra they would
CH ALLENGE More on predicting substance abuse.Refer to the previous exercise. The researchers also studied cigarette use, alcohol use, and cocaine use. Here is a summary of the results for the individual regression coefficients:b t P GPA −0.340 2.16 < 0.05 Cigarette Popularity 0.338 2.24 < 0.05
Predicting substance abuse. What factors predict substance abuse among high school students? One study designed to answer this question collected data from 89 high school seniors in a suburban Florida high school.5 One of the response variables was marijuana use, which was rated on a four-point
Understanding the tests of significance. Using a new software package, you ran a multiple regression. The output reported an F statistic with P < 0.05, but none of the t tests for the individual coefficients were significant (P > 0.05). Does this mean that there is something wrong with the
Childhood obsesity. The prevalence of childhood obesity in industrialized nations is constantly rising. Since between 30% and 60% of obese children maintain their obesity into adulthood, there is great interest in better understanding the reasons for this rising trend. In one study, researchers
More on constructing the ANOVA table. A multiple regression analysis of 73 cases was performed with 5 explanatory variables. Suppose that SSM = 14.1 and SSE = 100.5.(a) Find the value of the F statistic for testing the null hypothesis that the coefficients of all of the explanatory variables are
Constructing the ANOVA table. Seven explanatory variables are used to predict a response variable using a multiple regression.There are 140 observations.(a) Write the statistical model that is the foundation for this analysis. Also include a description of all assumptions.(b) Outline the analysis
What’s wrong? In each of the following situations, explain what is wrong and why.(a) One of the assumptions for multiple regression is that the distribution of each explanatory variable is Normal.(b) The smaller the P-value for the ANOVA F test, the greater the explanatory power of the model.(c)
What’s wrong? In each of the following situations, explain what is wrong and why.(a) In a multiple regression with a sample size of 40 and 4 explanatory variables, the test statistic for the null hypothesis H0: b2 = 0 is a t statistic that follows the t(35) distribution when the null hypothesis
More on significance tests for regression coefficients. For each of the settings in the previous exercise, test the null hypotheses that the coefficient of x1 is zero versus the two-sided alternative.
95% confidence intervals for regression coefficients. In each of the following settings, give a 95% confidence interval for the coefficient of x1.(a) n = 30, ˆy = 10.6 + 10.8x1 + 7.9x2, SEb1= 2.4.(b) n = 53, ˆy = 10.6 + 10.8x1 + 7.9x2, SEb1= 2.4.(c) n = 30, ˆy = 10.6 + 10.8x1 + 7.9x2 + 5.2x3,
Residual plots for the CSDATA analysis. The CSDATA data set can be found in the Data Appendix. Using a statistical package, fit the linear model with HSM and HSE as predictors and obtain the residuals and predicted values. Plot the residuals versus the predicted values, HSM, and HSE. Are the
Pairwise relationships among variables in the CSDATA data set.The CSDATA data set can be found in the Data Appendix. Using a statistical package, generate the pairwise correlations and scatterplots discussed previously. Comment on any unusual patterns or observations.
ANOVA table for multiple regression. Use the following information to perform the ANOVA F test and compute R2.Degrees Source of freedom Sum of squares Model 175 Error 60 Total 65 1015
Significance tests for regression coefficients. Recall Exercise 11.1(page 610). Due to missing values for some students, only 86 students were used in the multiple regression analysis. The following table contains the estimated coefficients and standard errors:Variable Estimate SE Intercept
Understanding the fitted regression line. The fitted regression equation for a multiple regression isˆy = −1.4 + 2.6x1 − 2.3x2(a) If x1 = 4 and x2 = 2, what is the predicted value of y?(b) For the answer to part (a) to be valid, is it necessary that the values x1 = 4 and x2 = 2 correspond to a
Describing a multiple regression. As part of a recent study titled“Predicting Success for Actuarial Students in Undergraduate Math graduates were obtained.2 The researchers were interested in describing how students’ overall math grade point averages are explained by SAT Math and SAT Verbal
CHALLENGE Inference over different ranges of X.Think about what would happen if you analyzed a subset of a set of data by analyzing only data for a restricted range of values of the explanatory variable. What results would you expect to change? Examine your ideas by analyzing the fuel efficiency
CHALL ENGE Resting metabolic rate and exercise, continued. Refer to the previous exercise.It is tempting to conclude that there is a strong linear relationship for the women but no relationship for the men. Let’s look at this issue a little more carefully.(a) Find the confidence interval for the
Resting metabolic rate and exercise. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. The table below gives data on the lean body mass and resting metabolic rate for 12 women and 7 men who are subjects in a study of dieting.
Personality traits and scores on the GRE.A study reported correlations between several personality traits and scores on the Graduate Record Examination (GRE) for a sample of 342 test takers.22 Here is a table of the correlations:GRE score Personality trait Analytical Quantitative Verbal
Food neophobia. Food neophobia is a personality trait associated with avoiding unfamiliar foods.In one study of 564 children who were 2 to 6 years of age, food neophobia and the frequency of consumption of different types of food were measured.21 Here is a summary of the correlations:Type of food
CHALLENGE Index of biotic integrity. Refer to the data on the index of biotic integrity and area in Exercise 10.14 (page 596) and the additional data on percent watershed area that was forest in Exercise 10.15. Find the correlations among these three variables, perform the test of statistical
CHALLENGE Creating a new explanatory variable.Refer to the previous two exercises.(a) Create a new variable that is the product of length and width. Make a plot and run the regression using this new variable. Summarize the results.(b) Write a short report summarizing and comparing the different
C HALLENGE Transforming the perch data. Refer to the previous exercise.(a) Try to find a better model using a transformation of length. One possibility is to use the square. Make a plot and perform the regression analysis. Summarize the results.(b) Do the same for width.
Length, width, and weight of perch. Here are data for 12 perch caught in a lake in Finland:20 Weight Length Width Weight Length Width(grams) (cm) (cm) (grams) (cm) (cm)5.9 8.8 1.4 300.0 28.7 5.1 100.0 19.2 3.3 300.0 30.1 4.6 110.0 22.5 3.6 685.0 39.0 6.9 120.0 23.5 3.5 650.0 41.4 6.0 150.0 24.0 3.6
CHALL ENGE Matching standardized scores. Refer to the previous two exercises. An alternative to the least-squares method is based on matching standardized scores. Specifically, we set( ˆy − y)sy= (x − x)sx and solve for y. Let’s use the notation y = a0 + a1x for this line. The slope is a1 =
CHALL ENGE SAT versus ACT, continued. Refer to the previous exercise. Find the predicted value of ACT for each observation in the data set.(a) What is the mean of these predicted values?Compare it with the mean of the ACT scores.(b) Compare the standard deviation of the predicted values with the
SAT versus ACT. The SAT and the ACT are the two major standardized tests that colleges use to evaluate candidates. Most students take just one of these tests. However, some students take both.Table 10.9 gives the scores of 60 students who did this. How can we relate the two tests?(a) Plot the data
Verifying the effect of bank size. Refer to the bank wages data given in Table 10.8 and described in Exercise 10.37 (page 601). The data also include a variable “Size,” which classifies the bank as large or small. Obtain the residuals from the regression used to predict wages from LOS, and plot
Significance test of the correlation. A study reported a correlation r = 0.5 based on a sample size of n = 20; another reported the same correlation based on a sample size of n = 10.For each, perform the test of the null hypothesis that ρ = 0. Describe the results and explain why the conclusions

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