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 slope \( b \) and the intercept \( a \) of the least - squares regression line, we use the following formulas:

Step 1: Calculate the slope \( b \)

The formula for the slope of the least - squares regression line is \( b=r\times\frac{s_y}{s_x} \), where \( r \) is the linear correlation coefficient, \( s_y \) is the standard deviation of the response variable (number of ice bags sold), and \( s_x \) is the standard deviation of the explanatory variable (temperature).

We are given that \( r = 0.9974\), \( s_y=26.8113 \), and \( s_x = 11.5221 \).

$$
LATEXBLOCK0
$$

Step 2: Calculate the intercept \( a \)

The formula for the intercept of the least - squares regression line is \( a=\bar{y}-b\bar{x} \), where \( \bar{y} \) is the mean of the response variable, \( \bar{x} \) is the mean of the explanatory variable, and \( b \) is the slope we just calculated.

We know that \( \bar{y}=69.8873 \), \( \bar{x} = 84.1359 \), and \( b\approx2.32 \)

$$
LATEXBLOCK1
$$

Step 3: Predict the number of ice bags sold (\(\hat{y}\)) when \( x = 87.00^{\circ}\text{F}\)

The equation of the least - squares regression line is \( \hat{y}=a + bx \)

Substitute \( a=- 125.32 \), \( b = 2.32 \), and \( x = 87.00 \) into the equation:

$$
LATEXBLOCK2
$$

Final Answers

• The slope \( b=\boldsymbol{2.32} \)
• The intercept \( a=\boldsymbol{- 125.32} \)
• The predicted number of ice bags sold when the temperature is \( 87.00^{\circ}\text{F} \) is \( \boldsymbol{77} \)

Answer:

To determine the slope \( b \) and the intercept \( a \) of the least - squares regression line, we use the following formulas:

Step 1: Calculate the slope \( b \)

The formula for the slope of the least - squares regression line is \( b=r\times\frac{s_y}{s_x} \), where \( r \) is the linear correlation coefficient, \( s_y \) is the standard deviation of the response variable (number of ice bags sold), and \( s_x \) is the standard deviation of the explanatory variable (temperature).

We are given that \( r = 0.9974\), \( s_y=26.8113 \), and \( s_x = 11.5221 \).

$$
LATEXBLOCK0
$$

Step 2: Calculate the intercept \( a \)

The formula for the intercept of the least - squares regression line is \( a=\bar{y}-b\bar{x} \), where \( \bar{y} \) is the mean of the response variable, \( \bar{x} \) is the mean of the explanatory variable, and \( b \) is the slope we just calculated.

We know that \( \bar{y}=69.8873 \), \( \bar{x} = 84.1359 \), and \( b\approx2.32 \)

$$
LATEXBLOCK1
$$

Step 3: Predict the number of ice bags sold (\(\hat{y}\)) when \( x = 87.00^{\circ}\text{F}\)

The equation of the least - squares regression line is \( \hat{y}=a + bx \)

Substitute \( a=- 125.32 \), \( b = 2.32 \), and \( x = 87.00 \) into the equation:

$$
LATEXBLOCK2
$$

Final Answers

• The slope \( b=\boldsymbol{2.32} \)
• The intercept \( a=\boldsymbol{- 125.32} \)
• The predicted number of ice bags sold when the temperature is \( 87.00^{\circ}\text{F} \) is \( \boldsymbol{77} \)