Question

When comparing 4 different brands failure rates for the same product, what would be the appropriate model to use for a confidence interval and hypothesis test? My data looks like this,

Brand 1: N=281, p-hat= .95805 pq=.04195

Brand 2: N=118 p-hat= .90799 pq=.09201

Brand 3: N=61 p-hat= .86767 pq= .13233

Brand 4: N=63 p-hat= .89112 pq= .10888

These are simple pass/fail trials. I am not sure what direction to go from here, I know how to compare 2 proportions I think but not sure on 4, just need some advice to get on track.

Answer 1

The chi-square test for homogeneity can be used here to determine whether the several populations are equal or homogeneous in some charateristic which is also reffered to as chi-square test for independence.

The test can be performed in following steps,

The Chi-Square test of independence is performed in following steps,

Step 1: The hypothesis is defined as,

Null hypothesis, Ho:All the population proportion are equal

Alternative hypothesis, Ha There is one population is different.

Step 2: Define the significance level for the test,

Step 3: The Chi-Square test statistic is obtained as follow,

The observe values are,

Observed values
	Pass	Fail	Total
Brand 1	269	12	281
Brand 2	107	11	118
Brand 3	53	8	61
Brand 4	56	7	63
Total	485	38	523

Step 4: The expected values are obtained using the formula,

The expected values are,

Expected values
	Pass	Fail	Total
Brand 1	260.5832	20.41683	281
Brand 2	109.4264	8.573614	118
Brand 3	56.56788	4.432122	61
Brand 4	58.42256	4.577438	63
Total	485	38	523

Step 5: Now the Chi-Square Value is obtained using the formula,

Step 4: The P-value for Chi-Square statistic is obtained using the chi square distribution table,

Step 5:

The null hypothesis is rejected. It can be stated now, there is a statisticallys significant difference between the population proportions.