Question

ed below are annual data for various years. the data are the numbers of cars sold (thousands) and the numbers of ts scored in the super bowl. construct a scatterplot, find the value of the linear correlation coefficient r, and find p - value using α = 0.05. is there sufficient evidence to conclude that there is a linear correlation between those variables? would it be reasonable to expect a correlation?

car sales	8173	8214	8515	8995	8635	8535	8273	8145
super bowl points	60	70	44	76	43	55	54	53

the linear correlation coefficient is r = 0.194. (round to three decimal places as needed.)
the test statistic is t = \square. (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for test statistic t

The formula for the test statistic \( t \) for the linear correlation coefficient \( r \) is \( t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}} \), where \( n \) is the number of pairs of data. Here, \( n = 8 \), \( r = 0.194 \).

Step2: Substitute the values into the formula

First, calculate \( n - 2 = 8 - 2 = 6 \). Then, calculate \( \sqrt{n - 2}=\sqrt{6}\approx2.449 \). Next, calculate \( 1 - r^2 = 1 - (0.194)^2 = 1 - 0.037636 = 0.962364 \), and \( \sqrt{1 - r^2}=\sqrt{0.962364}\approx0.981 \). Now, substitute into the \( t \) formula: \( t=\frac{0.194\times2.449}{0.981} \). Calculate the numerator: \( 0.194\times2.449\approx0.475 \). Then, \( t=\frac{0.475}{0.981}\approx0.48 \).

Answer:

\( 0.48 \)