. A professor is interested in knowing the correlation, if any, of students’ scores on the first and second exams given in his class.
Test 1 Test 2
92 86
90 87
89 85
82 90
88 79
92 90
100 92
91 84
Is there a significant correlation? Explain its statistical significance, if any.
With respect to the situation in the above question , what other test could the professor use that would yield valid results by looking for differences. What would be the statistical result of the test?
Sample size, n: 8
Degrees of freedom: 6
Correlation Results:
Correlation coefficient, r: 0.3201344
Critical r: ±0.706734
P-value (two-tailed): 0.4395
Fail to Reject the Null Hypothesis
Sample does not provide enough evidence to support a linear correlation
Regression Results:
Y= b0 + b1x:
Y Intercept, b0: 62.71449
Slope, b1: 0.2642045
Total Variation: 119.875
Explained Variation: 12.28551
Unexplained Variation: 107.5895
Standard Error: 4.23457
Coefficient of Det, R^2: 0.102486
Conclusion: There is no significant correlation
To determine if there is a significant correlation between students' scores on the first and second exams, the professor can use a statistical measure called the Pearson correlation coefficient (also known as Pearson's r). This coefficient measures the linear relationship between two variables, in this case, the scores on the first and second exams.
To calculate the Pearson correlation coefficient, follow these steps:
1. Calculate the mean (average) of each set of scores (test 1 and test 2).
2. Calculate the difference between each score and its respective mean for both tests.
3. Multiply the differences for each pair of scores.
4. Calculate the sum of all the differences for each test.
5. Calculate the standard deviation for both sets of scores.
6. Divide the sum of the differences by the product of the standard deviations to calculate the covariance.
7. Finally, divide the covariance by the product of the standard deviations to obtain the Pearson correlation coefficient.
After calculating the Pearson correlation coefficient, its statistical significance can be determined through hypothesis testing. The most common hypothesis test for Pearson's r is the t-test.
The null hypothesis (H0) states that there is no correlation between the two variables (scores on the first and second exams), while the alternative hypothesis (Ha) states that there is a correlation.
The statistical result of the test will provide a p-value, which represents the probability of observing the correlation coefficient (or a more extreme one) if the null hypothesis is true. If this p-value is below a predetermined significance level (usually 0.05), the null hypothesis is rejected, indicating that there is a significant correlation between the two sets of scores. If the p-value is greater than the significance level, the null hypothesis cannot be rejected, suggesting that there is no significant correlation.
Regarding the second part of your question, if the professor is interested in looking for differences between the two groups (first and second exams), they can use a statistical test called the independent samples t-test. This test determines whether the means of two independent groups differ significantly. The result of the t-test will provide a p-value, allowing the professor to ascertain if there is a significant difference between the two sets of scores.