Question

the weights (in pounds) of eight vehicles and the variabilities of their braking distances (in feet) when stopping on a dry surface are shown in the table. at $alpha = 0.05$, is there enough evidence to conclude that there is a significant linear correlation between vehicle weight and variability in braking distance on a dry surface?
weight, x: 5940 5310 6500 5100 5850 4800 5700 5870
variability, y: 1.78 1.93 1.94 1.63 1.63 1.50 1.58 1.70
calculate the test statistic.
t = 1.65 (round to two decimal places as needed.)
calculate the p - value.
p - value = (round to three decimal places as needed.)

Explanation:

Step1: Determine degrees of freedom

The degrees of freedom for a correlation test with $n$ data - points is $df=n - 2$. Here, $n = 8$, so $df=8 - 2=6$.

Step2: Use t - distribution to find P - value

We know the test - statistic $t = 1.65$ and $df = 6$. Since this is a two - tailed test for correlation, we use the t - distribution. Using a t - distribution table or a statistical software (e.g., in Python: 2*(1 - stats.t.cdf(abs(1.65),6)) where stats is from scipy.stats), we find the P - value.
The P - value is the probability of getting a test - statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
$P - value=2\times(1 - P(T\leq|1.65|))$ where $T$ follows a t - distribution with $df = 6$.
$P - value = 2\times(1 - 0.921)=0.158$

Answer:

$0.158$