Answer to Question #151159 in Statistics and Probability for John David Manabat

Question

The publisher wants to determine the relationship between the number of copies of books sold and the expenditure made. A sample of 10 books studied shows the following data.

Copies sold
(in thousands) –x
5
13
24
35
58
73
98
100
121
189

Advertising Expense (thousands of pesos)-y
5
9
5
7
8
6
12
21
38
30


1. Determine the regression model.
2. How many copies will be sold if expenditure is P40k ?
3. If there are 220k copies, how much will be the advertising expense?
4. Test the regression coefficient.

Answer 1

Solution

$$\sum x = 716 \space \sum y=141 \space n=10$$

$$\sum x^2 = 80654 \space \sum y^2 =3209 \sum xy =14953$$

1. Regression Model

$$\hat y= \hat\alpha +\hat\beta x$$

$$\hat\beta = {n \sum xy - \sum x \sum y \over n \sum x^2 - {( \sum x)} ^2}$$

$$={10(14953)-(716*141) \over 10(80654)-{(716)}^2}$$ $$=0.1653$$

$$\alpha = {\sum y - \beta \sum x \over n}$$

$$={141 - 0.1653(716) \over 10} =2.2645$$

2. When expenditure is p40k

y=40

$$40=2.2645 +0.1653x$$

$$x= {40-2.2645 \over 0.1653} = 228.28$$

$\approx 228.28k \space copies$

3. If there are 220k copies.

x=220

$$y=2.2645 + 0.1653(220) = 38.63$$

=p38.63k

4. Test regression coefficient

$H_0 : \beta =0 \space vs \space H_1: \beta \not = 0$

$$t={ \beta - 0 \over {S \over \sqrt {SS_x}}}$$

$$S= \sqrt {SS_y - \beta SS_x \over n-2}$$

$$SS_y = {n \sum y^2 - {( \sum y)} ^2 \over n}=1220.9$$

$$SS_x = {n \sum x^2 - {( \sum x)} ^2 \over n} = 29388.4$$

$$S_{xy} = {n \sum xy - \sum x \sum y \over n} =4857.4$$

$$S= \sqrt {1220.9 - 0.1653 (4857.4) \over 8} = 7.228$$

$$t= {0.1653 - 0 \over {7.228 \over 29388.4}} = 3.921$$

T-critical

$t_{0.025,8}=2.306$

Since t-stat > t-critical, we reject H₀

Therefore the regression coefficient is significant