To analyze the data in question #1 using the Pearson correlation coefficient, we need to calculate the correlation between the scores in Chemistry and Calculus for the ten students. The Pearson correlation coefficient, also known as Pearson's r, measures the strength and direction of the linear relationship between two variables.
Here are the steps to calculate the Pearson correlation coefficient:
1. Calculate the mean (average) of the Chemistry and Calculus scores separately.
2. Subtract the mean of each set of scores from each individual score in that set. This will give us the deviation from the mean for each score in both Chemistry and Calculus.
3. Multiply each pair of deviations together.
4. Sum up all the products from step 3.
5. Calculate the standard deviation of the Chemistry and Calculus scores separately.
6. Multiply the standard deviation of Chemistry scores by the standard deviation of Calculus scores.
7. Divide the sum from step 4 by the product from step 6.
8. This will give us the Pearson correlation coefficient, which ranges from -1 to +1.
To determine whether the correlation is significant at α = 0.05 using a t-test, we need to state the null and alternative hypotheses and perform the calculations.
Null Hypothesis (H0): There is no significant correlation between the Chemistry and Calculus scores for the population.
Alternative Hypothesis (Ha): There is a significant correlation between the Chemistry and Calculus scores for the population.
Here are the steps to perform the t-test:
1. Calculate the degrees of freedom (df) using the formula df = n - 2, where n is the number of data points.
2. Calculate the critical t-value using the significance level α and the degrees of freedom.
3. Calculate the t-value using the Pearson correlation coefficient and the degrees of freedom.
4. Compare the calculated t-value with the critical t-value to determine if the correlation is significant.
If the calculated t-value is greater than the critical t-value, we reject the null hypothesis and conclude that there is a significant correlation between the Chemistry and Calculus scores. If the calculated t-value is less than or equal to the critical t-value, we fail to reject the null hypothesis and conclude that there is no significant correlation.
Please note that I would need the actual values of the Chemistry and Calculus scores to perform the calculations and provide you with the specific results.