Problem 1 To test whether it can be concluded that there is a difference in the hardness of gray cast iron resulting from two production sources, ten independent measurements of hardness are taken from the iron produced by each of the two sources. Assuming both measurement populations are normally distributed with known standard deviations $\sigma_{X}=\sigma_{Y}=0.8$, set up the appropriate hypothesis test to check a difference in means at the $\alpha=0.05$ level. If the sample means yield $\bar{x}=160.1$ and $\bar{y}=160.8$, what conclusion is drawn ?