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statistical techniques in business

Questions and Answers of Statistical Techniques in Business

True or false?(a) Adjusted R2 can possibly decrease when an explanatory variable is added to a regression model.(b) Possible effects of an influential observation include changing a correlation from
Evidence of multicollinearity exists in a multiple regression fit when(a) Strong intercorrelations occur among explanatory variables.(b) The R2-value is very large.(c) The F test of H0: β1 = · ·
Forward selection and stepwise regression are similar in the sense that, if they have the same α-level for testing a term,(a) They always select the same final regression model.(b) They always
The log transformation of the mean response in regression is useful when(a) E(y) is approximately a logarithmic function of x.(b) E(y) is approximately an exponential function of x.(c) logE(y) is
In the model E(y) = α + β1x + β2x2, the coefficientβ2(a) Is the mean change in y as x2 is increased one unit with x held constant.(b) Is a curvature coefficient that describes whether the
You invest $1000 in an account with interest compounded annually at 10%.(a) How much money do you have after x years?(b) How long does it take your savings to double in size?
Example 14.8 showed a predicted U.S. population size (in millions) x decades after 1890 of ˆy =73.175(1.130)x.(a) Show this is equivalent to 1.23% predicted growth per year. [Hint: (1.0123)10 =
A recent newspaper article quoted a planner in a Florida city as saying, “This city has been growing at the rate of 4.2% per year. That’s not slow growth by any means. It corresponds to 42%
Using the formula s/s j(n − 1)(1 − R2j) for the standard error of the estimator of βj in multiple regression, explain how precision of estimation is affected by(a) Multicollinearity.(b) The
You plan to model coital frequency in the previous month as a function of age, for a sample of subjects with ages between 20 and 90. For the ordinary bivariate model, explain what might be
Give an example of two variables that you expect to have a nonlinear relationship. Describe the pattern you expect for the relationship. Explain how to model that pattern.
A sociologist’s first reaction upon studying automated variable selection routines was that they had the danger of leading to “crass empiricism” in theory building.From a theoretical
Give an example of a response variable and a pair of explanatory variables for which an automated variable selection procedure would probably produce a model with only one explanatory variable.
For the UN2 data file at the text website, using methods of this chapter,(a) Find a good model relating x = per capita GDP to y =life expectancy. (Hint: What does a plot of the data suggest?)(b) Find
Table 14.15 shows the population size of Florida, by decade from 1830 to 2010. Analyze these data, which are the data file FloridaPop at the text website. Explain why a linear model is reasonable for
For the Mental data file at the text website and the model predicting mental impairment using life events and SES, conduct an analysis of residuals and influence diagnostics.
Analyze the Crime data set at the text website, deleting the observation for D.C., with y = violent crime rate. Use methods of this chapter. Prepare a report describing the analyses and diagnostic
Refer to the data file the class created in Exercise 1.12. Select a response variable, pose a research question, and build a model using other variables in the data set.Interpret and summarize your
Refer to the Students data file (Exercise 1.11).(a) Conduct and interpret a regression analysis using y =political ideology, selecting predictors from the variables in that file. Prepare a report
Consider the fertility and GDP data in Table 14.6, from the FertilityGDP data file.(a) Using GLM software, fit the exponential regression model, assuming fertility rate has a (i) normal, (ii) gamma
For white men in the United States, Table 14.14 presents the number of deaths per thousand individuals of a fixed age within a period of a year.(a) Plot x = age against y = death rate and against log
Consider the formula ˆy = 4(2)x.(a) Plot ˆy for integer x between 0 and 5.(b) Plot loge ˆy against x. Report the intercept and slope of this line.
Draw rough sketches of the following mathematical functions on the same set of axes, for x between 0 and 35.(a) ˆy = 6(1.02)x. (ˆy = predicted world population size in billions x years after 2000,
For United Nations data on y = world population size (billions) between 1900 and 2010, the exponential regression model with x = number of years since 1900 givesˆy = 1.4193(1.014)x.(a) Explain why
For data shown in the article “Wikipedia: Modelling Wikipedia’s growth” at en.wikipedia.org, the number of English language articles inWikipedia was well approximated from 2001 to 2008 by ˆy =
For data from 2005 to 2011 from Facebook on y =number of people (in millions) worldwide using Facebook, the prediction equation ˆy = 2.13(2.72)x fits well, where x = number of years since January 1,
The Crime2 data file at the text website illustrates how a single observation can be highly influential in determining whether the model should allow nonlinearity.(a) With all 51 observations, fit
Refer to the previous exercise.(a) Using size as a straight-line predictor, r2 = 0.695, whereas R2 = 0.704 for the quadratic model. Is the degree of nonlinearity major, or minor? Is the linear
For the Houses data file, Table 14.13 shows results of fitting a quadratic regression model with s = size as the predictor.(a) Interpret the coefficients of this equation.What shape does it have?(b)
Sketch the following mathematical functions on the same set of axes, for values of x between 0 and 4. Use these curves to describe how the coefficients of x and x2 affect their shape.(a) ˆy = 10 +
Table 14.12 shows the results of fitting two models to 54 observations on y = mental health score, x1 = degree of social interaction, and x2 = SES. The variables x1 and x2 are measured on scales of
Refer to the data from Example 14.7 on fertility rates and GDP (page 441). To allow for greater variation at higher values of mean fertility, fit a quadraticGLMwith a gamma distribution for fertility
Refer to the plot of residuals in Figure 14.13 for Exercise 14.6.(a) Explain why a more valid fit may result from assuming that income has a gamma distribution, rather than a normal distribution.(b)
For a data set for 100 adults on y = height, x1 =length of left leg, and x2 = length of right leg, the model E(y) = α + β1x1 + β2x2 is fitted. Neither H0: β1 = 0 nor H0: β2 = 0 has a P-value
Three variables have population correlationsρx1x2= 0.85, ρyx1= 0.65, and ρyx2= 0.65. For these, the partial correlations are ρyx1·x2= ρyx2·x1= 0.244. In a sample, rx1x2= 0.90, ryx1= 0.70, and
For the Houses2 data file, fit the model to y = selling price using house size, whether the house is new, and their interaction.(a) Show that the interaction term is highly significant.(b) Show that
In Exercise 14.3, backward elimination and forward selection choose the model with explanatory variables SIZE, BATHS, and NEW.(a) Fit this model with the Houses2 data set. Inspect the leverages and
For the Crime2 data file at the text website, fit the linear regression model with y = violent crime rate and x = percentage living in metropolitan areas, for all 51 observations.(a) Plot the
For the data for 21 nations in the UN2 data file at the text website that are not missing observations on literacy, Table 14.10 shows various diagnostics from fitting the multiple regression model
Figure 14.13 is a plot of the residuals versus the predicted y-values for the model discussed in Example 13.1 (page 390) relating income to education and racial–ethnic group.What does this plot
Use software with the Crime2 data file at the text website, excluding the observation for D.C. Let y = murder rate. For the five explanatory variables in that data file(excluding violent crime rate),
Refer to the previous exercise. Using software with these four predictors, find the model that would be selected using the criterion. (a) R2 adj, (b) PRESS, (c) AIC.
For the Houses2 data file at the text website, Table 14.9 shows a correlation matrix and a model fit using four predictors of selling price.With these four predictors,(a) For backward elimination,
Table 11.23 (page 347) showed results of a multiple regression using nine predictors of the quality of life in a country.(a) In backward elimination with these nine predictors, can you predict which
For Example 11.2 (page 312) on y = mental impairment, x1 = life events, and x2 = SES, the multiple regression model has output Coef. Std. Error t Sig.(Constant) 28.230 2.174 12.984 .000 LIFE .103
A recent study9 examined the role of family structure in the financial support parents provide for their children’s college education. Using data for 5070 children from 1519 families from the
Summarize advantages of using a linear mixed model to analyze repeated-measures data, compared to using standard repeated-measures ANOVA.
Explain what is meant by the term mixed model, and explain the distinction between a fixed effect and a random effect.
Explain the reason for entering random effects into a regression model. Describe a study in which it would be helpful to use this approach.
13.33.* Using the graphical representation in Figure 13.10, explain why yi= yi+ b( ¯x − ¯xi), where b is the estimated slope. So, when b > 0, yi is adjusted upward if ¯x > ¯xi and adjusted
13.32.* Suppose we use a centered variable for the covariate and express the interaction model when the categorical factor has two categories as E(y) = α + β1(x − μx) + β2z + β3(x − μx) ×
Summarize the differences in purpose of a one-way analysis of variance and an analysis of covariance.
In the United States, the mean annual income for blacks (μ1) is smaller than for whites (μ2), the mean number of years of education is smaller for blacks than for whites, and annual income is
In the model E(y) = α + β1x + β2z, where z = 1 for females and z = 0 formales,(a) The categorical factor has two categories.(b) One line has slope β1 and the other has slope β2.(c) β2 is the
For a regression model fitted to annual income(thousands of dollars) using predictors age and marital status, Table 13.23 shows the sample mean incomes and the adjusted means. How could the adjusted
Draw a scatterplot with sets of points representing two groups such that H0: equal means would be rejected in a one-wayANOVAbut not in an analysis of covariance.
Let y = death rate and x = mean age of residents, measured for each county in Louisiana and in Florida.Sketch a hypothetical scatterplot, identifying points for each state, when the mean death rate
In analyzing GSS data relating y = frequency of having sex in the past year to frequency of going to bars, DeMaris (2004, p. 62) noted that the slope for unmarried subjects is more than double the
You have two groups, and you want to compare their regressions of y on x, to test the hypothesis that the true slopes are identical for the two groups. Explain how to do this using regression
For the Crime2 data file at the text website, let z be a dummy variable for whether a state is in the South, with z = 1 for AL, AR, FL, GA, KY, LA, MD, MS, NC, OK, SC, TN, TX,VA, WV.(a) Not including
Analyze the Houses2 data file at the text website by modeling selling price in terms of size of house and whether it is new. (a) Fit the model allowing interaction, and test whether the interaction
You plan a study of factors associated with fertility(a woman’s number of children) in a Latin American city.Of particular interest is whether migrants fromother cities or migrants from rural areas
Table 13.21 shows output for GSS data with y =index of attitudes toward premarital, extramarital, and homosexual sex, for which higher scores represent more permissive attitudes. The categorical
For the 2014 GSS, Table 13.20 shows estimates(with se values in parentheses) for four regression models for y = political party identification in the United States, scored from 1 = strong Democrat to
An article8 on predicting attitudes toward homosexuality modeled a response variable with a four-point scale in which homosexual relations were scaled from 1 =always wrong to 4 = never wrong, with x1
Refer to the OECD data file at the text website, shown in Table 3.13 (page 58). Pose a research question about how the human development index and whether a nation is in Europe relate to carbon
Refer to the data file your class created in Exercise 1.12. For variables chosen by your instructor, use regression analysis as the basis of descriptive and inferential statistical analyses.
Refer to the Students data file (Exercise 1.11).Using software, prepare a report presenting graphical, descriptive, and inferential analyses with(a) y = political ideology and the predictors
Refer to the regression modeling of the familyclustered data in Table 13.13.Add to the Family data file the data for family 9, who had (y, x1, x2) values (0, 2, 0) and (1, 2, 1). Fit the linear mixed
Exercise 13.1 reported the regression equation relating y = education to race (z = 1 for whites) and to father’s education (x) of E(y) = 3 + 0.8x − 0.6z. The means ¯y = 11 for nonwhites, ¯y =
Table 13.1 did not report the observations for 10 Asian Americans. Their (x, y) values were Subject 1 2 3 4 5 6 7 8 9 10 Education 16 14 12 18 13 12 16 16 14 10 Income 70 42 24 56 32 38 58 82 36
Refer to the previous exercise. The means of percentage registered for the three categories are ¯x1 = 76.2,¯x2 = 49.5, and ¯x3 = 39.7.The overall mean ¯x = 60.4.(a) Find the adjusted mean of the
The software outputs in Table 13.19 show results of fitting two models to data from a study of the relationship between y = percentage of adults voting, percentage of adults registered to vote, and
Using software, replicate all the analyses shown in Sections 13.1 and 13.2 using the Income data file at the text website.
For the previous exercise, Table 13.18 shows results of fitting the model allowing interaction.(a) Report the lines relating the predicted selling price to the size for homes that are (i) new, (ii)
For the Houses data file at the text website, Table 13.17 shows results of modeling y = selling price (in dollars)in terms of size of home (in square feet) and whether the home is new (1 = yes; 0 =
Consider the results in the previous exercise.(a) Marital status has three estimates. Dividing the coefficient of the divorced dummy variable by its standard error yields a t statistic. What
Based on a national survey, Table 13.16 shows results of a prediction equation for y = alcohol consumption, measured as the number of alcoholic drinks the subject drank during the past month.(a) For
For 2014 data, the GSS website yields the prediction equation ˆy = 9.59+0.166x1+0.347x2 for y = highest year of school completed, x1 = sex (1 = male, 2 = female), and x2 = highest year of mother’s
A regression analysis for the 100th Congress predicted the proportion of each representative’s votes on abortion issues that took the “pro-choice” position.7 The prediction equation wasˆy =
Table 3.9 on page 53 showed data for several nations on y = C02 emissions (in metric tons per capita) and x =per capitaGDP(in thousands of dollars). Let z =whether the nation is in Europe (1 = yes, 0
The regression equation relating y = education(number of years completed) to race (z = 1 for whites, z = 0 for nonwhites) in a certain country isE(y) = 11+2z.The regression equation relating
12.44.* This exercise motivates the formula for the between-groups variance estimate in one-way ANOVA.Suppose the sample sizes all equal n and the population means all equal μ. The sampling
Use the ANOVA applet at www.artofstat.com/webapps.html to illustrate how between-groups and within-groups variability affect the result of the ANOVA F test. Print results of two scenarios that result
Interaction terms are needed in a two-way ANOVA model when(a) Each pair of variables is associated.(b) Both explanatory variables have significant effects in the model without interaction terms.(c)
For four means, a multiple comparison method provides 95% confidence intervals for the differences between the six pairs. Then(a) For each confidence interval, there is a 0.95 chance that it contains
One-way ANOVA provides relatively more evidence that H0: μ1 = · · · = μg is false(a) The smaller the between-groups variation and the larger the within-groups variation.(b) The smaller the
Analysis of variance and regression are similar in the sense that(a) They both assume a quantitative response variable.(b) They both have F tests for testing that the response variable is
True or false? Suppose that for subjects aged under 50, there is little difference in mean annual medical expenses for smokers and nonsmokers, but for subjects aged over 50 there is a large
Refer to Exercise 12.20.The students were also asked about their attitudes toward abortion. Each received a score according to how many from a list of eight possible reasons for abortion she would
The 25 women faculty in the humanities division of a college have a mean salary of $76,000, and the five women in the science division have a mean salary of$90,000. The 20 men in the humanities
Construct a numerical example of means for a twoway classification under the following conditions:(a) Main effects are present only for the row variable.(b) Main effects are present for each
Table 7.29 (page 212) summarized a study that reported the mean number of dates in the past three months.For men, the mean was 9.7 for the more attractive and 9.9 for the less attractive. For women,
For a two-way classification of means by factors A and B, at each level of B the means are equal for the levels of A. Does this imply that the overall means are equal at the various levels of A,
(a) Explain carefully the difference between a probability of Type I error of 0.05 for a single comparison of two means and a multiple comparison error rate of 0.05 for comparing all pairs of
A study6 compared verbal memory of men and women for abstract words and for concrete words. It found a gender main effect in favor of women. It also reported,“There was no sex × word-type
A study5 described an experiment that randomly assigned participants to receive $3 to spend on themselves(self-interest), or to receive $3 to donate to a nonprofit charity (imposed charity), or to
Goto the GSS website sda.berkeley.edu/GSS.(a) Analyze the change over time (GSS variable YEAR)in the mean of political ideology (POLVIEWS) by political party identification (PARTYID). Compare
For y=number of times used public transportation in previous week and x = number of cars in family (which takes value 0, 1, or 2 for the given sample), explain the difference between conducting a
Refer to the Students data file (Exercise 1.11 on page 9), with response variable the number of weekly hours engaged in sports and other physical exercise. Using software, conduct an analysis of

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