Question

an automobile company is running a new television commercial in five cities with approximately the same population. the following table shows the number of times the commercial is run on tv in each city and the number of car sales (in hundreds). find the linear regression line for the data given in the table. round any intermediate calculations to no less than six decimal places, and round the coefficients to two decimal places.
number of tv commercials, x: 3 5 11 16 18
car sales, y (in hundreds): 3 2 9 7 8
answer

Explanation:

Step1: Calculate sums

Let \(n = 5\).
\(\sum_{i = 1}^{n}x_{i}=3 + 5+11+16+18=53\)
\(\sum_{i = 1}^{n}y_{i}=3 + 2+9+7+8=29\)
\(\sum_{i = 1}^{n}x_{i}^{2}=3^{2}+5^{2}+11^{2}+16^{2}+18^{2}=9 + 25+121+256+324 = 735\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=3\times3+5\times2+11\times9+16\times7+18\times8=9 + 10+99+112+144 = 374\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
Substitute the values:
\[

$$\begin{align*} m&=\frac{5\times374 - 53\times29}{5\times735-53^{2}}\\ &=\frac{1870-1537}{3675 - 2809}\\ &=\frac{333}{866}\\ &\approx0.38 \end{align*}$$

\]

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\)
\[

$$\begin{align*} b&=\frac{29-0.38\times53}{5}\\ &=\frac{29 - 20.14}{5}\\ &=\frac{8.86}{5}\\ & = 1.77 \end{align*}$$

\]

Answer:

\(y = 0.38x+1.77\)