Question

Suppose that 26 of 200 tires of brand A failed to last 30,000 miles whereas the corresponding figures for 200 tires of brands B, C, and D were 23, 15, and 32. Test the null hypothesis that there is no difference in the durability of the four kinds of tires at the 0.05 level of significance. Using the above data, test the null hypothesis that the failure rates of the four tire brands are 10% at the 0.05 level of significance.

Answer 1

null hypothesis:Ho: there is no difference in the durability of the four kinds of tires

Alternate hypothesis:Ha: there is a difference in the durability of the four kinds of tires

degree of freedom(df) =(rows-1)*(columns-1)=	3
for 3 df and 0.05 level , critical value       χ2=	7.815
Decision rule : reject Ho if value of test statistic X2>7.815

Applying chi square test of homogeniety:

Expected	Ei=row total*column total/grand total	A	B	C	D	Total
	failed	24.0000	24.0000	24.0000	24.0000	96.00
	passed	176.0000	176.0000	176.0000	176.0000	704.00
	total	200.00	200.00	200.00	200.00	800.00
chi square    χ2	=(Oi-Ei)2/Ei	A	B	C	D	Total
	failed	0.167	0.042	3.375	2.667	6.2500
	passed	0.023	0.0057	0.4602	0.3636	0.8523
	total	0.1894	0.0473	3.8352	3.0303	7.102
test statistic X2 =	7.102

since test statistic does not falls in rejection region (<7.815) we fail to reject null hypothesis
we do not have have sufficient evidence to conclude that there is a difference in the durability of the four kinds of tires