b. The correlation between the number correct on a math test and the time it takes to complete the test is -.45. Test whether this correlation is significant for 80 children at the .05 level of significance. Choose either a one-or two-tailed test and justify your choice.
To find if there is a statistically significant linear relationship between the number correct on a math test and the time it takes to complete the test, use N-2 for degrees of freedom at the .05 significance level for a one-tailed test or 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 (negative direction for a one-tailed in this case or either direction for a two-tailed), 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.
-4.45
To test whether the correlation between the number correct on a math test and the time it takes to complete the test is significant, we need to use a hypothesis test.
First, let's define the null and alternative hypotheses:
- Null Hypothesis (H₀): The correlation between the number correct on the math test and the time it takes to complete the test is not significant (ρ = 0).
- Alternative Hypothesis (H₁): The correlation between the number correct on the math test and the time it takes to complete the test is significant (ρ ≠ 0).
Next, we need to determine whether to perform a one-tailed or two-tailed test. This decision is based on the research question and the context.
In this case, we are only interested in determining if there is any significant correlation (positive or negative) between the number correct on the math test and the time it takes to complete the test. Therefore, we should choose a two-tailed test because we want to detect any type of significant relationship, regardless of the direction.
Now, we can perform the hypothesis test using a statistical test, such as the Pearson's correlation coefficient (r) or the t-test. However, since we only have the value of the correlation (-0.45) and the sample size (80 children), we'll use the t-test.
To calculate the t-value, we need to transform the correlation coefficient using Fisher's transformation. The formula is as follows:
t = (r √(N-2))/(√(1-r²))
Given that r = -0.45 and N = 80, we can plug these values into the equation:
t = (-0.45 √(80-2))/(√(1-(-0.45)²))
Calculating this further, we find:
t ≈ -4.37
To determine the critical t-value for a two-tailed test at a significance level of 0.05, we need the degrees of freedom (df). In this case, df = N - 2 = 80 - 2 = 78. Using a t-table or a statistical software, we find that the critical t-value is approximately ±1.99.
Since our calculated t-value (-4.37) is outside the range of the critical t-value (-1.99 to 1.99), we can reject the null hypothesis.
Therefore, we can conclude that the correlation between the number correct on the math test and the time it takes to complete the test is significant for 80 children at the 0.05 level of significance.