Question

1. an amusement park records the high temperature and the number of customers they have for a random sample of 10 days. \

$$\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c} temperature (°f) & 55 & 80 & 70 & 85 & 92 & 90 & 87 & 60 & 75 & 84 \\\\ \\hline number of customers & 810 & 1409 & 1200 & 1525 & 1810 & 1802 & 1679 & 905 & 1315 & 1559 \\end{tabular}$$

a. the line of best fit describing the relationship between $x = $ temperature (°f) and $y = $ number of customers is $\hat{y} = -726.22 + 27.35x$. identify and interpret the slope of the line of best fit. b. how many customers does the model predict will be at the amusement park on a day that has a temperature of 85°? c. use the linear model to calculate and interpret the residual for a temperature of 85°.

Explanation:

Part a

Step1: Recall slope formula

The line of best fit is in the form \( \hat{y} = b_0 + b_1x \), where \( b_1 \) is the slope. Here, the equation is \( \hat{y} = -726.22 + 27.35x \), so the slope \( b_1 = 27.35 \).

Step2: Interpret the slope

In a linear regression equation \( \hat{y}=b_0 + b_1x \), the slope \( b_1 \) represents the change in the predicted value of \( y \) (number of customers) for a one - unit change in \( x \) (temperature in \( ^\circ F \)). So, for each increase of \( 1^\circ F \) in temperature, the predicted number of customers increases by 27.35.

Step1: Identify the formula and values

We use the line of best fit \( \hat{y}=-726.22 + 27.35x \). Here, \( x = 85 \) (the temperature).

Step2: Substitute \( x = 85 \) into the formula

\( \hat{y}=-726.22+27.35\times85 \)
First, calculate \( 27.35\times85 = 2324.75 \)
Then, \( \hat{y}=-726.22 + 2324.75=1598.53 \)

Step1: Recall the formula for residual

The residual \( e=y - \hat{y} \), where \( y \) is the actual value and \( \hat{y} \) is the predicted value.

Step2: Find \( y \) and \( \hat{y} \) for \( x = 85 \)

From the table, when \( x = 85 \), the actual number of customers \( y = 1525 \). From part b, when \( x = 85 \), \( \hat{y}=1598.53 \)

Step3: Calculate the residual

\( e=y-\hat{y}=1525 - 1598.53=-73.53 \)

Step4: Interpret the residual

A negative residual means that the actual number of customers (1525) is less than the number of customers predicted by the model (1598.53) when the temperature is \( 85^\circ F \). Specifically, the model over - predicted the number of customers by 73.53 (or approximately 74) when the temperature was \( 85^\circ F \).

Answer:

The slope is \( 27.35 \). Interpretation: For each increase of \( 1^\circ F \) in temperature, the predicted number of customers at the amusement park increases by 27.35.

Part b