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`

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.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators