63. More on insurance An insurance company claims that in the entire population of homeowners, the mean annual loss from fire is $\mu = \$250$ and the standard deviation of the loss is $\sigma = \$1000$. The distribution of losses is strongly right-skewed: many policies have $\$0$ loss, but a few have large losses. An auditor examines a random sample of 10,000 of the company's policies. If the company's claim is correct, what's the probability that the average loss from fire in the sample is no greater than $\$275$? Show your work.
64. Bad carpet The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies a random sample of 200 square yards of the material, records the number of flaws found in each square yard, and calculates $\bar{x}$, the mean number of flaws per square yard inspected. Find the probability that the mean number of flaws exceeds 1.8 per square yard. Show your work.