Question

question 8 of 15
suppose a gas station monitors how many bags of ice they sell along with the maximum daily temperature for 100 days. a gas station data analyst plots the data in a scatter plot with temperature, in °f, on the horizontal axis and the number of bags of ice sold on the vertical axis. she calculates a linear correlation coefficient of r = 0.9974. the mean temperature is 84.1359 °f with a standard deviation of 11.5221 °f. the mean number of ice bags sold is 69.8873 with a standard deviation of 26.8113.
determine the slope, b, and the intercept, a, of the least-squares regression line for this dataset, precise to two decimal places.

b = 2.32
correct answer
a = 125.31
incorrect answer
predict how many bags of ice are sold, ŷ, to the nearest whole number, when the temperature is 87.00°f.

ŷ = 77
correct answer

Explanation:

To determine the intercept \( a \) of the least - squares regression line, we use the formula for the least - squares regression line \( \hat{y}=a + bx \), where \( a=\bar{y}-b\bar{x} \).

Step 1: Recall the formula for the intercept

The formula for the intercept \( a \) of the least - squares regression line is \( a=\bar{y}-b\bar{x} \), where:

• \( \bar{y} \) is the mean of the response variable (number of ice bags sold), \( \bar{y} = 69.8873\)
• \( b \) is the slope of the regression line, \( b = 2.32\)
• \( \bar{x} \) is the mean of the explanatory variable (temperature), \( \bar{x}=84.1359\)

Step 2: Substitute the values into the formula

Substitute \( \bar{y}=69.8873 \), \( b = 2.32\) and \( \bar{x}=84.1359 \) into the formula \( a=\bar{y}-b\bar{x} \):

$$
LATEXBLOCK0
$$

Rounding to two decimal places, \( a\approx - 125.31\)

Step 3: Predict the number of ice bags sold

To predict the number of ice bags sold \( \hat{y} \) when \( x = 87.00^{\circ}\text{F} \), we use the regression equation \( \hat{y}=a + bx \). We know that \( a=-125.31 \), \( b = 2.32\) and \( x = 87.00 \)

$$
LATEXBLOCK1
$$

Rounding to the nearest whole number, \( \hat{y}=77 \)

For the intercept \( a \):

Step 1: Identify formula for \( a \)

The formula is \( a=\bar{y}-b\bar{x} \).

Step 2: Substitute values

Substitute \( \bar{y}=69.8873 \), \( b = 2.32 \), \( \bar{x}=84.1359 \) into \( a=\bar{y}-b\bar{x} \).
\( a = 69.8873-2.32\times84.1359=69.8873 - 195.195288=-125.307988\approx - 125.31\)

Step 1: Recall the regression equation

The regression equation is \( \hat{y}=a + bx \), with \( a=-125.31 \), \( b = 2.32 \) and \( x = 87.00 \)

Step 2: Substitute values into the equation

\( \hat{y}=-125.31+2.32\times87.00=-125.31 + 201.84 = 76.53\approx77\)

Answer:

(for intercept \( a \)):
\( a=-125.31 \)

For the prediction of \( \hat{y} \) when \( x = 87.00 \):