To find the slope and y-intercept of the least-squares regression line, we can use the formulas for linear regression:

1. Slope (b): $b = r \cdot \frac{s_y}{s_x}$ where:

   • $r$ is the correlation coefficient,
   • $s_y$ is the standard deviation of the $y$ values (Test scores),
   • $s_x$ is the standard deviation of the $x$ values (Quiz 1 scores).

2. y-intercept (a): $a = \bar{y} - b \cdot \bar{x}$ where:

   • $\bar{y}$ is the mean of the $y$ values (Test scores),
   • $\bar{x}$ is the mean of the $x$ values (Quiz 1 scores),
   • $b$ is the slope.

Step 1: Calculate the Slope (b)

We are given:

• $r = 0.568$,
• $s_y = 3.5$ (standard deviation of Test scores),
• $s_x = 3.3$ (standard deviation of Quiz 1 scores).

Using the formula for the slope: $b = 0.568 \cdot \frac{3.5}{3.3} = 0.568 \cdot 1.0606 = 0.602$ So, the slope $b \approx 0.60$.

Step 2: Calculate the y-intercept (a)

We are given:

• $\bar{x} = 11.9$ (mean of Quiz 1 scores),
• $\bar{y} = 14.3$ (mean of Test scores),
• $b = 0.60$.

Using the formula for the y-intercept: $a = 14.3 - 0.60 \cdot 11.9 = 14.3 - 7.14 = 7.16$ So, the y-intercept $a \approx 7.16$.

Final Answer:

• Slope $b \approx 0.60$,
• y-intercept $a \approx 7.16$.

Would you like to go over the calculations further or have more details?

Here are 5 related questions:

1. How is the least-squares regression line used in predicting values?
2. What is the significance of the correlation coefficient in linear regression?
3. How would you interpret the slope of the regression line in this context?
4. Can you explain the role of the y-intercept in this problem?
5. How would the line change if the correlation were negative?

Tip: In regression analysis, the slope represents how much the dependent variable (Test scores) changes for each unit increase in the independent variable (Quiz 1 scores).