Questions: The following table shows the relationship between weight and calories burned per minute for five people. Weight (in pounds) Calories burned per minute ---------------------------------------------- 112 7.25 129 9.15 150 9.85 174 10.25 Mean 182 11.75 Standard Deviation 149.4 9.65 29.51 1.64 Weight is the explanatory variable and has a mean of 149.4 and a standard deviation of 29.51. Calories burned per minute is the response variable and has a mean of 9.65 and a standard deviation of 1.64. The correlation was found to be 0.944. Select the correct slope and y-intercept for the least squares line (answer choices are rounded to the hundredths places).

Solution Steps

To find the slope and y-intercept of the least squares line, we use the formulas for the slope (\(b\)) and y-intercept (\(a\)) of the regression line \(y = a + bx\). The slope is calculated using the formula \(b = r \times \frac{\text{std\_dev\_y}}{\text{std\_dev\_x}}\), where \(r\) is the correlation coefficient, \(\text{std\_dev\_y}\) is the standard deviation of the response variable, and \(\text{std\_dev\_x}\) is the standard deviation of the explanatory variable. The y-intercept is calculated using the formula \(a = \text{mean\_y} - b \times \text{mean\_x}\), where \(\text{mean\_y}\) and \(\text{mean\_x}\) are the means of the response and explanatory variables, respectively.

To find the slope (\(b\)) of the least squares line, we use the formula: \[ b = r \times \frac{\text{std\_dev\_y}}{\text{std\_dev\_x}} \] Substituting the given values: \[ b = 0.944 \times \frac{1.64}{29.51} \approx 0.05246 \]

The y-intercept (\(a\)) is calculated using the formula: \[ a = \text{mean\_y} - b \times \text{mean\_x} \] Substituting the calculated slope and given means: \[ a = 9.65 - 0.05246 \times 149.4 \approx 1.812 \]

Final Answer