c. 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?
To find if there is a statistically significant linear relationship between number of friends and grade point average, use N-2 for degrees of freedom at the .05 significance level for a two-tailed test. Use a table for critical or cutoff values for a Pearson r. Compare the value from the table to the test statistic stated in the problem. If the test statistic exceeds the critical value from the table (either direction), the null will be rejected. There will be a linear relationship in the population and the test will be statistically significant. If the test statistic does not exceed the critical value from the table, then the null will not be rejected and you cannot conclude a linear relationship in the population.
I hope this brief explanation will get you started.
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
To determine whether the correlation between the number of friends and GPA for 50 adolescents is significant at the .05 level for a two-tailed test, we need to conduct a hypothesis test.
The null hypothesis (H0) for this test states that there is no correlation between the number of friends and GPA in the population. The alternative hypothesis (Ha) states that there is a correlation between the number of friends and GPA in the population.
The significance level, also known as alpha (α), determines the level of significance we are willing to accept. In this case, the significance level is .05, which means we are willing to accept a 5% chance of rejecting the null hypothesis when it is true (Type I error).
For a correlation test, we can use the t-test to determine the significance. The test statistic, denoted as t, is computed as:
t = r * sqrt((n - 2) / (1 - r^2))
Where r is the correlation coefficient and n is the sample size.
In this case, the correlation coefficient (r) is 0.37, and the sample size (n) is 50. Substituting these values into the formula, we can calculate the test statistic (t).
t = 0.37 * sqrt((50 - 2) / (1 - 0.37^2))
Now, we need to compare the absolute value of the test statistic (|t|) to the critical value from the t-distribution at the given significance level and degrees of freedom (df = n - 2). Since this is a two-tailed test, we need to consider the critical value for both the upper and lower tails.
To find the critical value from the t-distribution table, we need to find the degrees of freedom (df = n - 2). In this case, df = 50 - 2 = 48. For a desired significance level of .05 in a two-tailed test, the critical value is ±2.010.
If |t| > 2.010, we reject the null hypothesis at the .05 significance level, which means the correlation is significant. Otherwise, if |t| ≤ 2.010, we fail to reject the null hypothesis, and the correlation is not significant.
Compare the calculated |t| value with the critical value to determine the significance of the correlation.