To determine how the amount of time, [tex]\(X\)[/tex], a customer spends browsing in the store affects their spending, [tex]\(\hat{Y}\)[/tex], we have the given regression equation:
[tex]\[\hat{Y} = 40 + 0.43X\][/tex]
Here is a step-by-step explanation of how we analyze this equation:
### Step 1: Understand the components of the equation
1. Intercept ([tex]\(40\)[/tex]):
- The intercept in the regression equation is 40. This means that if a customer spends zero minutes in the store ([tex]\(X = 0\)[/tex]), the predicted amount they will spend is [tex]$40.
2. Slope (\(0.43\)):
- The slope is 0.43. This value indicates that for each additional minute a customer spends in the store, the amount they are predicted to spend increases by $[/tex]0.43.
### Step 2: Predicting the amount spent for a specific browsing time
To predict the amount spent by a customer who spends a certain amount of time browsing, we can substitute [tex]\(X\)[/tex] with the given browsing time into the regression equation.
Let's calculate the predicted amount a customer will spend if they browse for 10 minutes:
1. Substitute [tex]\(X = 10\)[/tex] into the regression equation:
[tex]\[
\hat{Y} = 40 + 0.43 \times 10
\][/tex]
2. Perform the multiplication:
[tex]\[
0.43 \times 10 = 4.3
\][/tex]
3. Add the result to the intercept:
[tex]\[
\hat{Y} = 40 + 4.3 = 44.3
\][/tex]
So, if a customer spends 10 minutes browsing in the store, the predicted amount they will spend is [tex]$44.3.
### Summary
Given the regression equation \(\hat{Y} = 40 + 0.43X\), the components are:
- Intercept: 40
- Slope: 0.43
For a customer who spends \(X = 10\) minutes browsing, the predicted spending amount \( \hat{Y} \) is calculated as follows:
\[
\hat{Y} = 40 + 0.43 \times 10 = 44.3
\]
Thus, if a customer spends 10 minutes in the store, they are expected to spend $[/tex]44.3.
