Based on 629 valid young respondents in the GSS 2018 data, how can we reveal the relationship between age and BMI? The Pearson's correlation coefficient and the simple regression are suitable methods to address this question. Descriptive statistics and covariance are given in the Table 4.

Table 4 Descriptive statistics, covariance, and valid sample size of respondent's age and BMI for young adult cases in the GSS 2018 data

Variables Covariance Mean Standard Deviation N

Age (x) 6.588 30.762 5.902 629

BMI (y) 28.145 6.847

1.Using numbers in the Table 4 above to calculate the Pearson's correlation coefficient.

2.For the entire American young adult population, do we still have a non-zero Pearson's correlation coefficient? Please fill in blanks in the following paragraph to conduct a T test of the null hypothesis of H0: =0. Let's use the =0.05 and t critical values of 1.964 for this hypothesis test.

The alternative hypothesis is H1:______________ . The T statistic of this hypothesis test is ______________ , which falls / does not fall into the rejection region. We therefore have / have no enough evidence to reject the null hypothesis.

3.Can we use age to explain variations in the BMI for American youths? Let's use "x" to represent the independent variable of "Age", and use "y" to represent the dependent variable of "BMI." Use the OLS method to obtain estimated constant (a ) and slope (b ) in the regression model of y=a+bx+e.

b =______________

a =______________

Does this regression model fit to the observed data well? Calculate and explain the coefficient of determination (R^2).

R^2 = ________

The R^2calculated above means that:

The meaning of the estimated constant (a ) is that ____________________________________________________________ . And the mean BMI for a 25-year-old person is predicted as ______________.