Question

compute the correlation coefficient for your bivariate data. test the significance of your correlation coefficient. use a significance level of 0.05. you may use either the p - value method or the critical value method. if using the p - value method: be sure to compute the appropriate test statistic for linear correlation and regression. you will use a two - tailed t - test. if using the critical value method: be sure to check against the table of critical values for your significance level and degrees of freedom. the correlation coefficient is significant if it is more extreme than the corresponding critical value. make a regression equation for your variables. if you determined your correlation coefficient is significant, then use your regression equation to make a prediction. pick some value of your explanatory value and compute the corresponding response variable. if you determined your correlation coefficient is not significant, then identify the sample mean for each variable. scatter chart (with age on x - axis and gender on y - axis, data points around y = 1 from x = 20 to x = 80)

Explanation:

To solve this problem, we'll follow the steps for analyzing bivariate data (age and gender) using statistics. We'll assume gender is coded as a binary variable (e.g., 1 for one gender, 0 for another, but here most points are at 1).

Step 1: Compute the Correlation Coefficient

The formula for the Pearson correlation coefficient \( r \) is:

$$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

• Let \( x = \text{Age} \), \( y = \text{Gender} \) (coded as 1 for most points, so \( \sum y \approx n \), \( \sum y^2 \approx n \), \( \sum xy \approx \sum x \)).
• For a scatter plot with most \( y = 1 \), the correlation will be weak (since age varies but gender is mostly constant). Let’s assume sample size \( n \), \( \sum x = S_x \), \( \sum y = S_y = n \), \( \sum xy = S_x \), \( \sum x^2 = S_{x^2} \), \( \sum y^2 = n \).

Substituting into \( r \):

$$ r = \frac{n(S_x) - S_x(n)}{\sqrt{[nS_{x^2} - S_x^2][n(n) - n^2]}} = \frac{0}{\sqrt{[nS_{x^2} - S_x^2][0]}} $$

The denominator has a term \( n^2 - n^2 = 0 \), so \( r \approx 0 \) (no linear relationship).

Step 2: Test Significance of \( r \)

Using the critical value method:

• Degrees of freedom \( df = n - 2 \).
• For \( \alpha = 0.05 \) (two - tailed), the critical value \( r_{\text{crit}} \) (from correlation tables) depends on \( n \). Since \( r \approx 0 \), and \( |r| < |r_{\text{crit}}| \), the correlation is not significant.

Step 3: Regression Equation

The regression equation is \( \hat{y} = b_0 + b_1x \), where:

$$ b_1 = r \cdot \frac{s_y}{s_x}, \quad b_0 = \bar{y} - b_1\bar{x} $$

Since \( r \approx 0 \), \( b_1 \approx 0 \), so \( \hat{y} \approx \bar{y} \).

Step 4: Prediction (or Sample Means)

Since the correlation is not significant, we report the sample means:

• \( \bar{x} = \frac{\sum x}{n} \) (mean age), \( \bar{y} = \frac{\sum y}{n} \approx 1 \) (mean gender, since most points are 1).

Final Answer

• Correlation coefficient: \( \boldsymbol{r \approx 0} \) (no linear relationship).
• Significance: Not significant ( \( |r| < r_{\text{crit}} \) ).
• Regression equation: \( \hat{y} \approx \bar{y} \approx 1 \) (or sample mean of gender).
• Prediction (if not significant): Sample mean of age \( \bar{x} \), sample mean of gender \( \bar{y} \approx 1 \).

Answer:

To solve this problem, we'll follow the steps for analyzing bivariate data (age and gender) using statistics. We'll assume gender is coded as a binary variable (e.g., 1 for one gender, 0 for another, but here most points are at 1).

Step 1: Compute the Correlation Coefficient

The formula for the Pearson correlation coefficient \( r \) is:

$$ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

• Let \( x = \text{Age} \), \( y = \text{Gender} \) (coded as 1 for most points, so \( \sum y \approx n \), \( \sum y^2 \approx n \), \( \sum xy \approx \sum x \)).
• For a scatter plot with most \( y = 1 \), the correlation will be weak (since age varies but gender is mostly constant). Let’s assume sample size \( n \), \( \sum x = S_x \), \( \sum y = S_y = n \), \( \sum xy = S_x \), \( \sum x^2 = S_{x^2} \), \( \sum y^2 = n \).

Substituting into \( r \):

$$ r = \frac{n(S_x) - S_x(n)}{\sqrt{[nS_{x^2} - S_x^2][n(n) - n^2]}} = \frac{0}{\sqrt{[nS_{x^2} - S_x^2][0]}} $$

The denominator has a term \( n^2 - n^2 = 0 \), so \( r \approx 0 \) (no linear relationship).

Step 2: Test Significance of \( r \)

Using the critical value method:

• Degrees of freedom \( df = n - 2 \).
• For \( \alpha = 0.05 \) (two - tailed), the critical value \( r_{\text{crit}} \) (from correlation tables) depends on \( n \). Since \( r \approx 0 \), and \( |r| < |r_{\text{crit}}| \), the correlation is not significant.

Step 3: Regression Equation

The regression equation is \( \hat{y} = b_0 + b_1x \), where:

$$ b_1 = r \cdot \frac{s_y}{s_x}, \quad b_0 = \bar{y} - b_1\bar{x} $$

Since \( r \approx 0 \), \( b_1 \approx 0 \), so \( \hat{y} \approx \bar{y} \).

Step 4: Prediction (or Sample Means)

Since the correlation is not significant, we report the sample means:

• \( \bar{x} = \frac{\sum x}{n} \) (mean age), \( \bar{y} = \frac{\sum y}{n} \approx 1 \) (mean gender, since most points are 1).

Final Answer

• Correlation coefficient: \( \boldsymbol{r \approx 0} \) (no linear relationship).
• Significance: Not significant ( \( |r| < r_{\text{crit}} \) ).
• Regression equation: \( \hat{y} \approx \bar{y} \approx 1 \) (or sample mean of gender).
• Prediction (if not significant): Sample mean of age \( \bar{x} \), sample mean of gender \( \bar{y} \approx 1 \).