Answer to Question #267306 in Statistics and Probability for Matilda

Question

A sociologist wants to know whether the number of children in the family is linearly dependent on the age of the mother at her wedding. He interviewed 9 housewives and the results are shown below.

Age at wedding (X)

21

15

22

22

21

25

30

18

24

No. of children (Y)

4

8

3

4

2

3

1

5

6

Find the following:

a.) Estimate the regrision equation

b.) Estimate the number of children at age 27

c.) Compute the sample correlation coefficient r and the sample coefficient of determination r² and interpret the result.

Answer 1

a)

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 21 & 4 & 84 & 441 & 16\\ & 15 & 8 & 120 & 225 & 64\\ & 22 & 3 & 66 & 484 & 9\\ & 22 & 4 & 88 & 484 & 16\\ & 21 & 2 & 42 & 441 & 4\\ & 25 & 3 & 75 & 625 & 9\\ & 30 & 1 & 30 & 900 & 1\\ & 18 & 5 & 90 & 324 & 25\\ & 24 & 6 & 144 & 576 & 36\\ Sum= & 198 & 36 & 739 & 4500 & 180\\ \end{array}$$ $$\sum_iX_i=198, \sum_iY_i=36$$

$$\sum_iX_iY_i=739,\sum_iX_i^2=4500, \sum_iY_i^2=180$$ $$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{198}{9}=22$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{36}{9}=4$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum_iX_i)^2=739-\dfrac{(198)^2}{9}$$

$$=144$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum_iY_i)^2=180-\dfrac{(36)^2}{9}$$

$$=36$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum_iX_i)(\sum_iY_i)$$

$$=739-\dfrac{198(36)}{9}=-53$$

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}$$

$$=\dfrac{-53}{144}=-0.368056$$

$$n=\bar{Y}-m\bar{X}$$

$$=4-(-0.368056)(22)$$

$$=12.097232$$

Therefore, we find that the regression equation is:

$$Y=12.097232-0.368056X$$

b)

$$Y=12.097232-0.368056(27)=2$$

The number of children at age 27 is 2.

c) Correlation coefficient:

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}=\dfrac{-53}{\sqrt{144}\sqrt{36}}$$

$$=-0.736111$$

Negative strong correlation.

$$r^2=\dfrac{(SS_{XY})^2}{SS_{XX}SS_{YY}}=\dfrac{(-53)^2}{144(36)}$$

$$=0.541860$$

54.186% of the data fit the regression model.