A manufacturer of walkbehind push mowers receives refurbishe

A manufacturer of walk-behind push mowers receives refurbished small engines from two new suppliers, A and B. It is not uncommon that some of the refurbished engines need to be lightly serviced before they can be fitted into mowers. The mower manufacturer recently received 100 engines from each supplier. In the shipment from A, 13 needed further service. In the shipment from B, 10 needed further service. Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that there exists a difference in the proportions of engines from the two suppliers needing service.

Solution

We have given that A manufacturer of walk-behind push mowers receives refurbished small engines from two new suppliers, A and B. It is not uncommon that some of the refurbished engines need to be lightly serviced before they can be fitted into mowers.

Here we want to test the hypothesis that ,

H0 : P1 = P2 Vs H1: P1 P2

Since the test is two sided.

where P1 and P2 are population proportions corresponding to supplier A and supplier B.

The test statistics fir testing two proportion is,

Z = (p1^ - p2^) / sqrt(p^q^/n1 + p^q^/n2)

where p1^ = r1 /n1 = 13 / 100 = 0.13

p2^ = r2 / n2 = 10 / 100 = 0.1

Best pooled estimate p^ = (r1+r2) / (n1+n2) = ( 13 + 10) / (100+100) = 23/200 = 0.115

q^ = 1 - p^ = 1 - 0.115 = 0.885

p1^ - p2^ = 0.13 - 0.10 = 0.03

p^q^ / n1 = 0.115 * 0.885 / 100 = 0.00101775

p^q^ / n2 = 0.115 * 0.885 / 100 = 0.00101775

sqrt( p^q^ / n1 + p^q^ / n2) = sqrt( 0.00101775 + 0.00101775) = sqrt( 0.0020355) = 0.0451

Z = 0.03 / 0.0451 = 0.6649

There are two critical values because the alternative contains inequality.

alpha = 10 % = 0.1

EXCEL syntax : normsdist(z)

where z is the test statistic value.

critical values are 2.5758 and -2.5758

Since the test statistic does not fall in the critical region.

So fail to reject H0 at 1% level of significance.

Conclusion :The data provide sufficient evidence to conclude that there exists a difference in the proportions of engines from the two suppliers needing service.