Question

A study is conducted to determine the relationship between a driver's age and the number of accidents that he or she has had over a 1-year period.

The data from this study is in the table:

Age 16 24 18 17 23 27 32

Accidents 3 2 5 2 0 1 1

The correlation coefficient for this problem is

There is/ is not evidence of significant correlation at the 5% significance level

The best estimate for the number of accidents a 20-year old driver has in 1 year is

Answer 1

Observation table -

Sr. No.	Age(x)	Accidents(y)	(x-x_bar)	(y-y_bar)	(x-x_bar)(y-y_bar)	(x-x_bar)^2	(y-y_bar)^2
1	16	3	-6.4286	1	-6.4286	41.3269	1
2	24	2	1.5714	0	0	2.4693	0
3	18	5	-4.4286	3	-13.2858	19.6125	9
4	17	2	-5.4286	0	0	29.4697	0
5	23	0	0.5714	-2	-1.1428	0.3265	4
6	27	1	4.5714	-1	-4.5714	20.8977	1
7	32	1	9.5714	-1	-9.5714	91.6117	1
Total	157	14	-	-	-35	205.7143	16

Formula for correlation coefficient is -

Calculations -

Hence, the correlation coefficient between age and accidents is -0.6101.

Now, we have to test that correlation coefficient is significant or not. We use t-test for significance of correlation coefficient.

Null hypothesis - H₀ : Correlation coefficient is not significant, i.e. = 0.

Alternative hypothesis - H₁ : Correlation coefficient is significant. i.e. 0.

Test statistic -

Test criterion -

Reject H₀ if t t_/2,n-2 or  -t -t_/2,n-2 .

Calculations -

r = -0.6101, n = 7

Critical value -

t_/2,n-2 = t_0.05/2,7-2 = t_0.025,5 = 2.5706

Conclusion -

-t (-1.1003) > -t_/2,n-2 (-2.5706), hence H₀ is accepted at 5% level of significance.

Result -

There is no significant correlation between age & accidents.

Now, we have to find out the best estimate for the number of accidents a 20 year old has in 1 year -

Let, linear regression equation be -

Y = a + bX

Where,

a = - b

Calculations -

a = - b = 2 - (22.4286)(-0.1701) = 2 + 3.8159 = 5.8159

linear regression equation is -

Y = a + bX = 5.8159 - 0.1701X

the best estimate for the number of accidents a 20 year old has in 1 year =

Y = 5.8159 - 0.1701X  = 5.8159 - 0.1701(20) = 5.8159 - 3.402 = 2.4139 2

Hence, the best estimate for the number of accidents a 20 year old has in 1 year is 2.

Note : t-table provided below.