(Solved): The following table gives the systolic blood pressure (SBP), body size (QUET), age (AGE), and smok ...

The following table gives the systolic blood pressure (SBP), body size (QUET), age (AGE), and smoking history (SMK) for a hypothetical s im the town of Angina. Data can be downloaded with this link: Download CSV Codebook: QUET stands for "quetelet index," a measure of size defined by QUET \( =100 \cdot\left(\frac{\text { weight }}{\text { height }^{2}}\right) \); SMK = 0 if a nonsmoker, SMK \( =1 \) if a current or previous smoker. (1) Determine the least-squares estimates for the regression of SBP \( (y) \) on SMK \( (x) \) : - intercept: \( b_{0}= \) - slope: \( b_{1}= \) (Report answers accurate to at least 3 decimal places.) - \( M_{\text {nonsmokers }}= \) - \( M_{\text {smokers }}= \) - Compare these values to the OLS coefficients (Hint: ¿what is the difference of the group means?) (Report answers accurate to at least 3 decimal places.) (3) Find the test statistic and \( P \)-value for the test of a zero slope, \( H_{0}: \beta_{1}=0 \) - \( t= \) - \( p= \) (Report answers accurate to at least 3 decimal places.) (4) Conduct a 2 independent samples \( t \)-test with these two groups (assume equal variance): - \( t= \) - \( p= \) - Compare these value to the hypothesis test for the slopes; note any key observations (and briefly explain). (Report answers accurate to at least 3 decimal places.) (5) Find the correlation coefficient for thse two variables, and test against \( H_{0}: \rho=0 \) - First, make a scatter plot and note any key features - \( r= \) - \( p= \) - Compare the resulting \( P \)-values across all of these tests. - Confirm the relationship between \( t \) and \( r \) is \( \frac{t^{2}}{n-2}=\frac{r^{2}}{1-r^{2}} \) (Report answers accurate to at least 3 decimal places.) Key Concept: The correlation from this context-one dichotomous variable and one scalar variable-has a special name: the point-biserial correlation. This correlation is often denoted \( r_{\mathrm{pb}} \), but one should note that it is calculated the same way as the Pearson correlation coefficient, \( r \). Additionally, the two distinct numeric values used for the dichotomous variable do not influence the value of \( r_{\mathrm{pb}} \), though only values of zero and one produce the effects notes above regarding the group means. There are additional formulas that one might use for this, and you are encouraged to explore further here (wikipediapage).