To find the linear regression line for the given data, we need to determine the line of best fit, which can be represented by the equation:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept. Given the data points, we have the following steps:
### Step 1: Listing the data points
We have the following data points:
- [tex]\( x = [3, 7, 13, 16, 17] \)[/tex] (Number of TV commercials)
- [tex]\( y = [2, 3, 5, 8, 9] \)[/tex] (Car sales in hundreds)
### Step 2: Perform Linear Regression
Using linear regression formulas, we find:
- [tex]\( slope (m) \)[/tex]
- [tex]\( intercept (b) \)[/tex]
- [tex]\( r \)[/tex]-value
- [tex]\( p \)[/tex]-value
- Standard error
### Step 3: Calculation Results
After performing the calculations, we find the following numerical results:
- The slope [tex]\( m \)[/tex] is approximately [tex]\( 0.49 \)[/tex]
- The y-intercept [tex]\( b \)[/tex] is approximately [tex]\( -0.06 \)[/tex]
- The correlation coefficient (r-value) is approximately [tex]\( 0.961942 \)[/tex]
- The p-value is approximately [tex]\( 0.008861 \)[/tex]
- The standard error is approximately [tex]\( 0.079964 \)[/tex]
### Step 4: Formulate the equation
Using the slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] rounded to two decimal places, we can write the equation of the regression line as:
[tex]\[ y = 0.49x - 0.06 \][/tex]
### Conclusion
So, the linear regression line for the given data is:
[tex]\[ y = 0.49x - 0.06 \][/tex]
This equation indicates that for every additional TV commercial aired, the car sales increase by [tex]\( 0.49 \)[/tex] hundred units, and if no TV commercials are aired, the starting car sales can be approximated as [tex]\(-0.06\)[/tex] hundreds.
