We are given summary data for two variables: extroversion ($x$) and volunteering ($y$). We need to calculate the line of best fit predicting volunteering from extroversion and then test for statistical significance. The given data includes: - $\bar{X} = 12.58$ - $s_x = 4.65$ - $\bar{Y} = 7.44$ - $s_y = 2.12$ - $r = 0.34$ - $n = 67$ - $SS_M = 19.79$ - $SS_E = 215.77$

1. Problem Description

We are given summary data for two variables: extroversion ($x$) and volunteering ($y$). We need to calculate the line of best fit predicting volunteering from extroversion and then test for statistical significance. The given data includes:

- $\bar{X} = 12.58$

- $s_x = 4.65$

- $\bar{Y} = 7.44$

- $s_y = 2.12$

- $r = 0.34$

- $n = 67$

- $SS_M = 19.79$

- $SS_E = 215.77$

2. Solution Steps

Step 1: Calculate the slope ($b$) of the regression line.

The formula for the slope $b$ is:

$b = r \cdot \frac{s_y}{s_x}$

Plugging in the values, we get:

$b = 0.34 \cdot \frac{2.12}{4.65} = 0.34 \cdot 0.4559 \approx 0.1550$

Step 2: Calculate the y-intercept ($a$) of the regression line.

The formula for the y-intercept $a$ is:

$a = \bar{Y} - b \cdot \bar{X}$

Plugging in the values, we get:

$a = 7.44 - 0.1550 \cdot 12.58 = 7.44 - 1.9499 \approx 5.49$

Step 3: Write the equation of the regression line.

The regression line is given by:

$\hat{Y} = a + bx$

$\hat{Y} = 5.49 + 0.1550x$

Step 4: Test for statistical significance using an F-test.

We are given $SS_M = 19.79$ (Sum of Squares Model) and $SS_E = 215.77$ (Sum of Squares Error).

We need to find the Mean Square Model ($MS_M$) and Mean Square Error ($MS_E$).

$MS_M = \frac{SS_M}{df_M}$

$MS_E = \frac{SS_E}{df_E}$

Here, $df_M = 1$ (degrees of freedom for the model) and $df_E = n - 2 = 67 - 2 = 65$ (degrees of freedom for the error).

$MS_M = \frac{19.79}{1} = 19.79$

$MS_E = \frac{215.77}{65} \approx 3.32$

Now, we can calculate the F-statistic:

$F = \frac{MS_M}{MS_E} = \frac{19.79}{3.32} \approx 5.96$

To determine statistical significance, we need to compare this F-statistic to a critical F-value. We need to look up the F-critical value for $df_M = 1$ and $df_E = 65$ at a significance level (alpha) of 0.

0

5. Using an F-distribution table or calculator, we find that the critical F-value is approximately 3.

9

9.

Since our calculated F-statistic (5.96) is greater than the critical F-value (3.99), we reject the null hypothesis. This indicates that there is a statistically significant relationship between extroversion and volunteering.

3. Final Answer

The equation of the line of best fit is $\hat{Y} = 5.49 + 0.1550x$.

There is a statistically significant relationship between extroversion and volunteering (F = 5.96, p < 0.05).