The problem asks us to test for a significant relationship between the level of education and levels of civic engagement using a four-step hypothesis testing procedure. We are given the following descriptive statistics from a sample of 100 people: $\bar{X} = 4.02$ $s_x = 1.15$ $\bar{Y} = 15.92$ $s_y = 5.01$ $SS_x = 130.93$ $SS_y = 2484.91$ $SP = 159.39$

1. Problem Description

The problem asks us to test for a significant relationship between the level of education and levels of civic engagement using a four-step hypothesis testing procedure. We are given the following descriptive statistics from a sample of 100 people:

$\bar{X} = 4.02$

$s_x = 1.15$

$\bar{Y} = 15.92$

$s_y = 5.01$

$SS_x = 130.93$

$SS_y = 2484.91$

$SP = 159.39$

2. Solution Steps

Step 1: State the null and alternative hypotheses.

Null hypothesis ($H_0$): There is no relationship between level of education and levels of civic engagement. In other words, the correlation coefficient $\rho = 0$.

Alternative hypothesis ($H_1$): There is a relationship between level of education and levels of civic engagement. In other words, the correlation coefficient $\rho \neq 0$.

Step 2: Set the criteria for a decision.

We need to find the degrees of freedom $df$ and choose a significance level $\alpha$. Since the problem does not give the significance level, we will assume $\alpha = 0.05$. The degrees of freedom for testing the significance of a correlation coefficient is $df = n - 2 = 100 - 2 = 98$.

Using a t-table or calculator, we find the critical t-values for a two-tailed test with $\alpha = 0.05$ and $df = 98$ to be approximately $t_{crit} = \pm 1.984$. We will reject the null hypothesis if our calculated t-value is greater than 1.984 or less than -1.

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Step 3: Compute the test statistic.

First, we calculate the Pearson correlation coefficient, $r$:

$r = \frac{SP}{\sqrt{SS_x \cdot SS_y}}$

$r = \frac{159.39}{\sqrt{130.93 \cdot 2484.91}}$

$r = \frac{159.39}{\sqrt{325340.0553}}$

$r = \frac{159.39}{570.385885}$

$r \approx 0.2794$

Next, we calculate the t-statistic:

$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

$t = \frac{0.2794\sqrt{100-2}}{\sqrt{1 - (0.2794)^2}}$

$t = \frac{0.2794\sqrt{98}}{\sqrt{1 - 0.07806436}}$

$t = \frac{0.2794 \cdot 9.899}{\sqrt{0.92193564}}$

$t = \frac{2.7657}{\sqrt{0.92193564}}$

$t = \frac{2.7657}{0.96017479}$

$t \approx 2.8804$

Step 4: Make the decision.

Our calculated t-value is $t \approx 2.8804$. Since $2.8804 > 1.984$, we reject the null hypothesis.

3. Final Answer

There is a significant relationship between the level of education and levels of civic engagement.