1. Do these data provide sufficient evidence for a real relationship in the population? Test at the .05 level, two tails. Use Pearson Correlation

X Y
8 0
6 1
4 4
3 6
4 6
5 1

2. To test the effectiveness of a new studying strategy, a psychologist randomly divides a sample of 8 students into two groups, with n=4 in each group. The students in one group receive training in the new studying strategy. Then all students are given 30 minutes to study a chapter from a history textbook before they take a quiz on the chapter. The quiz scores for the two groups are as follows:

Training:6,13,12,9
No Training : 1,6,6,3

a. Perform a t-test to answer the research question: Is the new studying strategy effective? (In other words, whether there is difference in student scores between the two groups.) Use an alpha of .05.
b. Convert these data into a form suitable for the point-biserial correlation. (use X=1 for training, X=0 for no training, and the quiz scores for Y). Calculate the point-biserial correlation, r, for these data
c. Test if the point-biserial correlation, r, is different from zero using an alpha of .05.
d. Calculate the squared correlation, r2. What does this number mean
e. What statement can be made regarding the results of the independent sample t-test comparing two group means and the results t-test of whether the point-biserial correlation is different from zero?

1. To determine if there is a real relationship between the X and Y variables, we can conduct a Pearson correlation test. The test will help us determine if there is a significant correlation between the two variables. Here's how you can perform the test:

Step 1: Calculate the correlation coefficient (r) using the formula:
r = sum((X - mean(X)) * (Y - mean(Y))) / sqrt(sum((X - mean(X))^2) * sum((Y - mean(Y))^2))

Step 2: Calculate the degrees of freedom (df) using the formula:
df = number of pairs - 2

Step 3: Compare the obtained correlation coefficient (r) with the critical value from the t-distribution table. For a two-tailed test at the 0.05 significance level, the critical value is approximately ±2.306.

Step 4: If the absolute value of the correlation coefficient (|r|) is greater than the critical value, then we can conclude that there is a significant relationship in the population. Otherwise, we fail to reject the null hypothesis of no correlation.

2. Now, let's address the questions about the t-tests and point-biserial correlation:

a. To test the effectiveness of the new studying strategy, we can conduct an independent samples t-test. The research question is if there is a difference in student scores between the two groups. Here's how you can perform the t-test:

Step 1: Calculate the mean and standard deviation for each group.

Step 2: Calculate the pooled standard deviation using the formula:
sp = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 + n2 - 2))

Step 3: Calculate the t-statistic using the formula:
t = (mean1 - mean2) / (sp * sqrt(1/n1 + 1/n2))

Step 4: Determine the degrees of freedom using the formula:
df = n1 + n2 - 2

Step 5: Compare the obtained t-statistic with the critical value from the t-distribution table. For a two-tailed test at the 0.05 significance level with 6 degrees of freedom, the critical value is approximately ±2.446. Alternatively, you can use statistical software to calculate the p-value directly.

Step 6: If the t-statistic falls outside the critical region or the p-value is less than 0.05, then we can conclude that there is a significant difference in student scores between the two groups. Otherwise, we fail to reject the null hypothesis of no difference.

b. To convert the data into a suitable format for point-biserial correlation, assign X=1 for the training group and X=0 for the no training group. Then, calculate the point-biserial correlation (r) using the formula:
r = (mean(X1) - mean(X0)) / sqrt((n1/n) * (n0/n) * (1 - mean(X1))^2 * mean(X0)^2)

c. To test if the point-biserial correlation (r) is different from zero, we can use a t-test. Calculate the t-statistic using the formula:
t = r * sqrt(n - 2) / sqrt(1 - r^2)

Compare the obtained t-statistic with the critical value from the t-distribution table for (n-2) degrees of freedom. For a two-tailed test at the 0.05 significance level with 6 degrees of freedom, the critical value is approximately ±2.447.

d. Calculate the squared correlation (r^2) using the formula:
r^2 = (mean(X1) - mean(X0))^2 / (mean(X1) * (1 - mean(X1)))

The squared correlation represents the proportion of variance in the dependent variable (quiz scores) that can be explained by the independent variable (training).

e. The results of the independent sample t-test comparing two group means and the t-test for the point-biserial correlation are related but measure different aspects of the relationship. The t-test for the group means determines if there is a significant difference in scores between the two groups, while the t-test for the point-biserial correlation determines if the correlation between the binary variable (training) and scores is different from zero. Both tests provide complementary information about the relationship between the variables.