Is wine good for your heart? A researcher from the University of California, San Diego, collected data on average per capita wine consumption and heart disease death rate in a random sample of 19 countries for which data were available. The following table displays the data.

(a) Is there statistically significant evidence of a negative linear relationship between wine consumption and heart disease deaths in the population of countries? Carry out an appropriate significance test at the $\alpha =0.05$level.

(b) Calculate and interpret a 95% confidence interval for the slope $\beta$ of the population regression line.

Given:

$n=\text{Sample size}=19$

$c=\text{Confidence level}=95\%$

The estimate of the slope is given in the row "X Variable 1 and in the column "Coefficients" of the output:

$b\approx -22.969$

The estimate of the standard error of the slope is given in the row "X Variable 1 and in the column "standard error" of the output:

$S{E}_b\approx 3.557$

The degrees of freedom is the sample size decreased by 2 :

localid="1650543375677" $$\begin{gathered} df=n-2 \\ =19-2 \\ =17 \end{gathered}$$

The critical t-value can be found in table B in the row of d f=17 and in the column of c=95% :

$t^{*}=2.110$

The boundaries of the confidence interval then become:

localid="1650543397407" $$\begin{gathered} b-t^{*}\times S{E}_b \\ =-22.969-2.110\times 3.557 \\ =-30.4743 \end{gathered}$$

localid="1650543408380" $$\begin{gathered} b+t^{*}\times S{E}_b \\ =-22.969+2.110\times 3.557 \\ =-15.4637 \end{gathered}$$

Given:

$n=19$

$b=-22.969$

$S{E}_b=3.557$

Determine the hypothesis:

$\beta =0$

$\beta <0$

Compute the value of the test statistic:

localid="1650543432666" $$\begin{gathered} t=\frac{b-{\beta }_0}{S{E}_b} \\ =\frac{-22.969-0}{3.557} \\ \approx -6.457 \end{gathered}$$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table B containing the t-value in the row d f=n-2=19-2=17 :

$P<0.0005$

If the P-value is less than or equal to the significance level, then the null hypothesis is rejected:

$P<0.05\Rightarrow \text{Reject}{H}_0$

There is sufficient evidence to support the claim that there is a negative linear relationship between wine consumption and heart disease deaths in the population of countries.