# Solved (Free): The correlation between number of friends and grade point average (GPA) for 50 adolescents is .37

#### ByDr. Raju Chaudhari

Mar 15, 2021

The correlation between number of friends and grade point average (GPA) for 50 adolescents is .37. Is this significant at the .05 level for a two-tailed test?

#### Solution

Given that $n = 50$, $r=0.37$, $\alpha =0.05$.

State the hypothesis testing problem

The hypothesis testing problem is $H_0: \rho = 0$ against $H_a: \rho \neq 0$.

Define the test statistic

The test statistic for testing above hypothesis is
 \begin{aligned} t&=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}\\ &=\frac{0.37\sqrt{50 -2}}{\sqrt{1-0.37^2}}\\ &=2.759 \end{aligned}

The test statistic $t$ follows Students' $t$ distribution with $n-2=50-2 =48$ degrees of freedom.

The level of significance is $\alpha = 0.05$.

Determine the critical values
For the specified value of $\alpha$ determine the critical region.

 \begin{aligned} P(t t_{\alpha/2,n-2}) = \alpha. \end{aligned}

The critical values are $t_{\alpha/2,n-2}=-2.011$ and $t_{1-\alpha/2,n-2}=2.011$.

Decision

As the observed value of test statistic $t$ falls inside the critical region, we reject the null hypothesis.

We conclude that the correlation between number of friends and grade point average (GPA) is not significant at 0.05 level of significance.