18. \(Y=\) height, \(X_{1}=\) length of left leg, and \(X_{2}=\) length of right leg are measured for a sample of 100 adults. The model \(E(Y)=\alpha+\beta_{1} X_{1}+\beta_{2} X_{2}\) is fitted to the data, and neither \(H_{0}: \beta_{1}=0\) nor \(H_{0}: \beta_{2}=0\) is rejected.

1. Does this imply that length of leg is not a good predictor of height? Why?
2. Does this imply that \(H_{0}: \beta_{1}=\beta_{2}=0\) would not have a small \(P\) -value? Why?
3. Suppose \(r_{Y X_{1}}=.901, r_{Y X_{2}}=.902\), and \(r_{X_{1} X_{2}}=.999\). What model would you expect to select, using forward selection and the predictors \(X_{1}\) and \(X_{2}\) ? Why?

Before you try to answer 18.a, look at the suggested correlations in 18.c. What correlation would you expect between length of right and left legs in a general population sample of adults?

SPSS Book. Use gssnet.sav . Ch. 23 Data Analysis #3. a-g only Remember "stepwise" is the variation of forward selection where variables are entered from the strongest to weakest, only allowing in those with p< .05, but removing any variables if p > .10 at subsequent steps. Also note that you should have a 3 step solution. Part G. suggests a 4 step solution, and that is incorrect. For d, e, and f calculate the predicted value using the equation written out for each step. Note how prediction is refined by each variable added.

3. Use stepwise linear regression to obtain a model to predict years of education from the following independent variables: father's education (variable paeduc), spouse's education (speduc), hours of television watched per day (tvhours), age (age), income in dollars (rincdol), and hours worked (hrs 1). Use the default criteria for variable entry and removal.

1. By including the variable speduc, you are restricting your analysis to what kind of people?
2. What is the first independent variable to enter the model? How is it selected?
3. What proportion of the variability in the dependent variable is explained by the independent variable? Can you reject the null hypothesis that the population value for the regression coefficient is 0 ?
4. Based on the model, what do you predict for years of education for a 50 -year-old person whose father has 12 years of education, whose spouse has 14 years of education, who watches 3 hours of television a day, has an income of $30,000, worked 42 hours last week, and is male?
5. What variable enters the model at the second step? What is the change in \(R^{2}\) when the variable enters? Write the regression equation and obtain the predicted years of education for the person described in question 3 d. How has your prediction changed?
6. What variable enters at the third step? How much does \(R^{2}\) change? Write the regression equation and obtain the predicted years of education for the person described in question 3 d.
7. What is the last variable to enter the model? How much does $R^{2}$ change when it is entered? What is the predicted value for years of education for the person described in question 3 d ? How much has the predicted value for the person changed from question 3 d to question 3 g ?
8. Why does variable selection stop?
9. What is the partial correlation between education and father's education, "controlling" for spouse's education and respondent's income in dollars?
10. Based on the size of the partial regression coefficients, is it reasonable to conclude that a person's income is least strongly related to education? Why or why not?
11. Consider the spouse's education variable. How has its coefficient changed at each step of model building? Why has this happened?

Ch. 23 Data Analysis #7. You will have to select full-time workers only. Under Data , select Select Cases , use "if condition satisfied," with workstat = 1. The question states "Develop a regression equation…" using hours worked last week as the dependent variable. To do this, select a half-dozen or so independent variables to build a regression model, starting with a correlation matrix (see the multiple regression lab handout if necessary). There are a few opportunities to create scales in the gssnet datafile- try creating at least one scale by running correlations, Cronbach’s alpha, and summing the items together. The question suggests taking the square root of hours worked- you would do this under transform > compute square root is under the Arithmetic functions.

Develop a regression equation to predict hours worked last week for full-time employees only (variable wrkstat equals 1). Summarize your findings. Be sure to check for violations of the regression assumptions. If the assumptions appear to be violated, try taking the square root of the number of hours worked as the dependent variable. Rerun the model and check the diagnostics again. Do you see any improvement?

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