Question

A survey of 22 employed workers found that the correlation between the number of years of post-secondary education and current annual income in dollars is 0.51. The researchers hypothesize a positive relationship between number of years of post-secondary education and annual income. What can the researchers conclude with an α of 0.05 ? a) Obtain/compute the appropriate values to make a decision about H 0

​ . Critical Value = Test Statistic = Decision: b) Compute the corresponding effect size(s) and indicate magnitude(s If not appropriate, input and/or select "na" below. Effect Size = ; Magnitude: c) Make an interpretation based on the results. There is a significant positive relationship between years of post-secondary education and current annual income. There is a significant negative relationship between years of post-secondary education and current annual income. There is no significant relationship between years of post-secondary education and current annual income.

Answer 1

Based on the results, we can conclude that there is a significant positive relationship between the number of years of post-secondary education and current annual income. This means that as the number of years of education increases, the annual income also tends to increase. However, we cannot make any causal inferences based on this correlation study.

Explanation:

To make a decision about the null hypothesis (H0), we need to perform a hypothesis test using the correlation coefficient and the sample size. The null hypothesis is that there is no correlation between the number of years of post-secondary education and current annual income, which can be written as:

H0: ρ = 0

The alternative hypothesis is that there is a positive correlation between the two variables, which can be written as:

Ha: ρ > 0

We can use a one-tailed test with a significance level (α) of 0.05.

a) To obtain/compute the appropriate values to make a decision about H0, we need to calculate the test statistic and compare it to the critical value from the t-distribution. The test statistic for testing the null hypothesis of no correlation is given by:

t = r * sqrt(n - 2) / sqrt(1 - r^2)

where r is the sample correlation coefficient, n is the sample size, and sqrt is the square root function. Substituting the given values, we get:

t = 0.51 * sqrt(22 - 2) / sqrt(1 - 0.51^2)

t ≈ 2.24

The critical value for a one-tailed test with 20 degrees of freedom (22-2) and a significance level of 0.05 is:

tcrit = 1.725

Since the test statistic (t) is greater than the critical value (tcrit), we reject the null hypothesis and conclude that there is a significant positive relationship between the number of years of post-secondary education and current annual income.

b) To compute the effect size, we can use Cohen's d, which measures the standardized difference between two means. However, since this is a correlation study, we can use the correlation coefficient (r) as the effect size. The magnitude of the effect size can be interpreted using the following guidelines:

Small effect size: r = 0.10 - 0.29

Medium effect size: r = 0.30 - 0.49

Large effect size: r ≥ 0.50

In this case, the effect size is r = 0.51, which indicates a large positive relationship between the two variables.

c) Based on the results, we can conclude that there is a significant positive relationship between the number of years of post-secondary education and current annual income. This means that as the number of years of education increases, the annual income also tends to increase. However, we cannot make any causal inferences based on this correlation study.