(Solved): How much influence does the media have on one's decision to undergo cosmetic surgery? This was the ...

How much influence does the media have on one's decision to undergo cosmetic surgery? This was the question of interest in a study done by a journal dedicated to body image research. In the study, 170 college students answered questions about their impressions of reality TV shows featuring cosmetic surgery. The data for the study (simulated based on statistics reported in the journal articles) are given below. Multiple regression was used to model desire to have a cosmetic surgery \( (y) \) as a function of gender \( \left(x_{1}\right) \), self-esteem \( \left(x_{2}\right) \), body satisfaction \( \left(x_{3}\right) \), and impression of reality TV \( \left(x_{4}\right) \). Complete parts a through \( \mathbf{f} \). a. Fit the first-order model, \( E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4} \), to the data. Give the least squares prediction equation. \[ E(y)=1+(\quad) x_{1}+(\quad) x_{2}+(\quad) x_{4} \] (Round to two decimal places as needed.) b. Interpret the \( \beta \) estimates in the words of the problem. Interpret the \( \beta_{0} \) estimate in the model. Choose the correct answer below. A. The mean desire is \( \beta_{0} \) units higher for males than females, when all other variables are being held constant. B. The mean desire will increase by \( \beta_{0} \) for each additional increase of 1 unit of self-esteem, when all other variables are held constant. C. The mean desire will decrease by \( \beta_{0} \) for each additional increase of 1 unit of impression of reality TV, when all other variables are held constant. D. It is the estimate of the \( y \)-intercept. Interpret the \( \beta_{1} \) estimate in the model. Choose the correct answer below. A. The mean desire will increase by \( \beta_{1} \) for each additional increase of 1 unit of body satisfaction, when all other variables are held constant. B. The mean desire will increase by \( \beta_{1} \) for each additional increase of 1 unit of self-esteem, when all other variables are held constant. C. It is the estimate of the \( y \)-intercept. D. The mean desire is \( \beta_{1} \) units higher for males than females, when all other variables are being held constant. Interpret the \( \beta_{2} \) estimate in the model. Choose the correct answer below. A. The mean desire will increase by \( \beta_{2} \) for each additional increase of 1 unit of impression of reality TV, when all other variables are held constant. B. The mean desire is \( \beta_{2} \) units higher for males than females, when the variable self-esteem is held constant. C. The mean desire will increase by \( \beta_{2} \) for each additional increase of 1 unit of self-esteem, when all other variables are held constant. D. It is the estimate of the \( y \)-intercept. Interpret the \( \beta_{3} \) estimate in the model. Choose the correct answer below. A. It is the estimate of the \( y \)-intercept. B. The mean desire will increase by \( \beta_{3} \) for each additional increase of 1 unit of impression of reality TV, when all other variables are held constant. C. The mean desire is \( \beta_{3} \) units higher for males than females, when the variable body satisfaction is held constant. D. The mean desire will increase by \( \beta_{3} \) for each additional increase of 1 unit of body satisfaction, when all other variables are held constant. Interpret the \( \beta_{4} \) estimate in the model. Choose the correct answer below. A. It is the estimate of the \( y \)-intercept. B. The mean desire will increase by \( \beta_{4} \) for each additional increase of 1 unit of impression of reality TV, when all other variables are held constant. C. The mean desire is \( \beta_{4} \) units higher for males than females, when the variable impression of reality TV is held constant. D. The mean desire will decrease by \( \beta_{4} \) for each additional increase of 1 unit of body satisfaction, when all other variables are held constant. c. Test the adequacy of the model, using \( \alpha=0.05 \). Determine the null and alternative hypotheses. Choose the correct answer below. A. \( \mathrm{H}_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0, \mathrm{H}_{\mathrm{a}}: \) At least one \( \beta_{\mathrm{i}} \neq 0, \mathrm{i}=1,2,3,4 \) B. \( H_{0} \) : At least one \( \beta_{i} \neq 0, i=1,2,3,4, H_{a}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0 \) C. \( \mathrm{H}_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0, \mathrm{H}_{\mathrm{a}}: \) Exactly one \( \beta_{\mathrm{i}} \neq 0, \mathrm{i}=1,2,3,4 \) D. \( \mathrm{H}_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0, \mathrm{H}_{\mathrm{a}}: \beta_{1} \neq \beta_{2} \neq \beta_{3} \neq \beta_{4} \neq 0 \) The test statistic is \( \mathrm{F}= \) (Round to two decimal places as needed.) The p-value is (Round to three decimal places as needed.) Since the \( \mathrm{p} \)-value is (1) \( \quad \alpha=0.05 \), (2) \( \quad \mathrm{H}_{0} \). There (3) sufficient evidence to indicate at least one of the independent variables is useful in the prediction of the interest in having cosmetic surgery. d. Which statistic, \( \mathrm{R}^{2} \) or \( \mathrm{R}_{\mathrm{a}}^{2} \) is the preferred measure of model fit? Interpret the value of this statistic. The value of (4) \( \quad=\quad \% \) is the preferred measure of model fit, because it accounts for both the (5) and the number of (6) in the model. (Round to one decimal place as needed.) e. Conduct a test to determine whether desire to have cosmetic surgery decreases linearly as level of body satisfaction increases. Use \( \alpha=0.05 \). Determine the null and alternative hypotheses. \( \mathrm{H}_{0}:(7) \) (8) \( ) \) \( \mathrm{H}_{\mathrm{a}}:(9) \) (10) 0 The test statistic is (Round to two decimal places as needed.) The p-value is (Round to three decimal places as needed.) Since the p-value is (11) \[ \alpha=0.05, \] \( \mathrm{H}_{0} \). There (13) sufficient evidence to indicate that cosmetic surgery decreases linearly as level of body satisfaction increases. f. Find a \( 95 \% \) confidence interval for \( \beta_{4} \). Interpret the result. The confidence interval is I 1. (Round to three decimal places as needed.) Interpret the result. Choose the correct answer below. A. The true value of the coefficient of a subject's belief that reality television shows featuring cosmetic surgery are realistic has a \( 95 \% \) probability of being within the confidence interval. B. The true value of the coefficient of a subject's belief that reality television shows featuring cosmetic surgery are realistic is within the upper and lower bound of the confidence interval. C. The confidence interval contains \( 95 \% \) of the possible values of the coefficient of a subject's belief that reality television shows featuring cosmetic surgery are realistic. D. One can be \( 95 \% \) confident that the increase in mean desire for cosmetic surgery is between the lower and upper bounds of the confidence interval for each unit increase in impression of reality TV, holding all other variables constant. 1: Data Table