Question

1. a software company kept a record of their annual budget for advertising and their profit for each of the last eight years. these data are shown in the table below:

annual advertising budget (in thousands, $) (x)	profit (in millions, $) (y)
13	2.4
14	3.2
16	4.6
19	5.7
24	6.9
24	7.9
28	9.3

a. write the linear regression equation for this set of data, rounding all values to the nearest hundredth.

b. state, to the nearest hundredth, the correlation coefficient of these linear data.

c. state what this correlation coefficient indicates about the linear fit of the data.

Explanation:

Step1: Calculate necessary sums

First, compute sums for regression:
$n=8$
$\sum x = 10+13+14+16+19+24+24+28 = 148$
$\sum y = 2.2+2.4+3.2+4.6+5.7+6.9+7.9+9.3 = 42.2$
$\sum xy = (10×2.2)+(13×2.4)+(14×3.2)+(16×4.6)+(19×5.7)+(24×6.9)+(24×7.9)+(28×9.3) = 862.1$
$\sum x^2 = 10^2+13^2+14^2+16^2+19^2+24^2+24^2+28^2 = 2910$

Step2: Find slope ($b$)

Use slope formula:
$$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$
$$b = \frac{8×862.1 - 148×42.2}{8×2910 - 148^2} = \frac{6896.8 - 6245.6}{23280 - 21904} = \frac{651.2}{1376} ≈ 0.37$$

Step3: Find y-intercept ($a$)

Use intercept formula:
$$a = \bar{y} - b\bar{x} = \frac{\sum y}{n} - b\frac{\sum x}{n}$$
$$a = \frac{42.2}{8} - 0.37×\frac{148}{8} = 5.275 - 0.37×18.5 ≈ 5.275 - 6.845 = -1.57$$
*Correction: Using precise slope value ~0.3715, $a ≈ 5.275 - 0.3715×18.5 ≈ -1.52$
Form regression equation: $\hat{y} = 0.37x - 1.52$

Step4: Calculate correlation coefficient ($r$)

Use correlation formula:
$$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$
$\sum y^2 = 2.2^2+2.4^2+3.2^2+4.6^2+5.7^2+6.9^2+7.9^2+9.3^2 = 263.92$
$$r = \frac{651.2}{\sqrt{1376×(8×263.92 - 42.2^2)}} = \frac{651.2}{\sqrt{1376×(2111.36 - 1780.84)}} = \frac{651.2}{\sqrt{1376×330.52}} ≈ 0.99$$

Step5: Interpret correlation coefficient

A $r$ value close to +1 indicates a very strong positive linear association, so the linear model fits the data extremely well.

Answer:

a. $\hat{y} = 0.37x - 1.52$
b. $0.99$
c. There is a very strong positive linear relationship between advertising budget and profit, meaning the linear fit is excellent.