Question

The following table gives data from a local school district on children's ages (x) in years and reading levels(y). Age in years X: 6, 7, 8, 9, 10, 11, 12, 14, 14, 15. Reading level Y: 1.3, 2.2, 3.7, 4.1, 4.9, 5.2, 6.0, 7.1, 8.5, 9.7.
Calculate the correlation coefficient r.
Calculate the coefficient of determination r2. Find the regression coefficients and estimate the regression equation.   
Based on the estimated regression equation in (b), find the reading level of a child who is 9.5 years.

Answer 1

In the above problem

x = Age in years

y = Reading level

We construct the table as below using set of values of x and y

6	1.3	36	1.69	7.8
7	2.2	49	4.84	15.4
8	3.7	64	13.69	29.6
9	4.1	81	16.81	36.9
10	4.9	100	24.01	49
11	5.2	121	27.04	57.2
12	6	144	36	72
14	7.1	196	50.41	99.4
14	8.5	196	72.25	119
15	9.7	225	94.09	145.5
106	52.7	1212	340.83	631.8

The formula of  the correlation coefficient r is as below

  

We have to find the correlation coefficient r is

  

The correlation coefficient r is 0.9798

We have to find the coefficient of determination using the following formula

Coefficient of determination = =

  

The Coefficient of determination r2 is 0.9601

We have to find the regression coefficients b0 and b1 is as below

We have to find b1 as below

then, We have to find b0 as below

Using all above values we get b0 is as below

  

By using regression coefficients we estimate the regression equation is as below

By using the regression equation we can estimate value of that is we can find the reading level of a child who is 9.5 years

that is = is as below

  

(Round up to 1 decimal place)

The estimate value(y) that is reading level of a child who is 9.5 years is 4.4