Question

Problem 1 The past output of a machine indicates that each unit it produces will be of a given quality with the below frequency. Assuming you purchase a new machine had have it produce 500 units, given the below data can we ascribe the difference in output to chance? New Machine (500 total) Frequency with older machines 40% 30% 20% 10% Quality Top High Medium Low 234 81 68

Answer 1

To answer this question we would be using chi square test of goodness of fit. We would be assuming 5% level of significance.

The following table is obtained (fo-fe)2/fe (234-200)2/200 5.78 (117-150)2/150-7.26 (81-100)2/100 3.61 (68-50)/50-6.48 Categories Top High Medium LoW Observed Expected 500 0.4-200 500*0.3-150 500 0.2 100 500 0.1 50 234 81 68 Sum 500 23.13

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho : p1-0.4, p2 0.3, p3-0.2, p4 0.1

Ha​: Some of the population proportions differ from the values stated in the null hypothesis i.e. difference in output is not due to chance.

This corresponds to a Chi-Square test for Goodness of Fit.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df=4−1=3, so then the rejection region for this test is R={χ2:χ2>7.815}.

(3) Test Statistics

The Chi-Squared statistic is computed as follows:

5.78+7.26 +3.61 6.48 23.13 Ei

The p value is 0.00004

The x (chi-square) distribution with v-3 degrees of freedom 0.30 Пр-0 0.25 0.20 ac) 0.15 0.10 0.05 0.00 23.130 If X is a random variable having a 0.0000 distribution with v 3 degrees of freedom, then p -Prx2 23.130]

(4) Decision about the null hypothesis

Since it is observed that χ2=23.13>χc2​=7.815, it is then concluded that the null hypothesis is rejected. Since p value is so small, we would even reject it at 1% level of significance.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the α=0.05 or 0.01, or even 0.0001 significance level. Hence the difference in output is not due to chance.

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