Question

31. A manufacturing company has 2 different instruments they use to measure the Rockwell hardness of an object. They believe that one of the instruments may not be working properly, and giving readings that are not completely accurate. To test this, they do the following. They take a large sheet of metal, and cut it into 60 different pieces, and randomly divide the pile in two. They believe it is safe to assume that the hardness of the metal is the same for any particular piece cut from the same sheet of metal. They measure the Rockwell hardness of 30 randomly selected pieces of metal using instrument 1 and find a sample mean hardness of 45.8 with a sample standard deviation of 1.2. They measure the hardness for the other 30 pieces of metal using instrument 2 and find a sample mean hardness of 46.2 with a sample standard deviation of 1.1. (a) Make and interpret a 95% confidence interval for the difference in the mean hardness readings using instrument 1 and instrument 2. 10 (b) Based on this interval alone, would you reject or fail to reject the null hypothesis that the mean hardness readings are equal for the two measurement instruments? Make sure to justify your answer with a reason. (c) Complete a hypothesis test to decide if you think that the two measurement instruments result in significantly different readings in the hardness of an object. Use a significance level of 5%

Answer 1

c)

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

(3) Test Statistics Since it is assumed that the population variances are equal, the t-statistic is computed as follows: +2-2 45.8-46.2 (30-11.23-11 12-1.346 (30-1)1.2+(30-1)1.1 1 30+30-2 (4) Decision about the null hypothesis Since it is observed th.346t rejected 2.002, it is then concluded that the null hypothesis is not Using the P-value approach: The p-value is p-0.1836, and since p-0.1836 2 0.05, it is concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.05 significance level.