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 the failure rates of the four tire brands are 10% at the 0.05 level of significance.

Answer 1

Null and alternative hypothesis are          
Ho : The failure rates of all brands are same          
Ha : The failure rates of all brands are not same         

Degrees of freedom = df = number of categories - 1 = 4 - 1 = 3                      
Degrees of Freedom = 3                      
                      
α = 0.05   that is 5% significance level                  
                      
We calculate critical χ2 using Excel function CHISQ.INV.RT(α, df)                      
Critical Chi Square χ2 = CHISQ.INV.RT(0.05, 3)                      
Critical Chi Square χ2 = 7.815                      
Decision Rule :                      
Reject Ho if chi-square test statistic > 7.815                      
From the above table, calculate value of χ2 test statistic ,            
Chi Square Value = χ2 = 0.0535                     
                      
0.0535 < 7.815                      
that is Chi Square calculate value < critical Chi Square value                      
Hence, as per the decision rule, we Do Not Reject Ho                      
                      
p-value approach                      
We calculate p-value using Excel function CHISQ.DIST                      
p-value = CHISQ.DIST.RT(0.0535, 3)                      
p-value = 0.9968                      
0.9968 > 0.05                      
that is p-value > α                      
Hence, we Do Not Reject Ho                     
                      
                      
Conclusion :                      
There does not exist sufficient statistical evidence to conclude that the failure rates of all brands are different from 10%                      
that is, failure rates of all brands are same and equal to 10%