91Ó°ÊÓ

A research worker wishes to estimate the mean of a population using a sample large enough that the probability will be \(.95\) that the sample mean will not differ from the population mean by more than 25 percent of the standard deviation. How large a sample should he take?

The chi-square density function is the special case of a gamma density with parameters \(\alpha=\mathrm{K} / 2\) and \(\lambda=1 / 2\). Find the mean, variance and moment-generating function of a chi-square random variable.

If \(z_{1}, z_{2} \ldots . . z_{n}\) is a random sample from a standard normal distribution, then show: (i) \(\bar{z}\) has a normal distribution with mean 0 and variance \(1 / \mathrm{n}\), (ii) for \(\mathrm{n}=2, \overline{\mathrm{z}}\) and \(\mathrm{n}_{\sum \mathrm{i}=1}\left(\mathrm{z}_{\mathrm{i}}-\overline{\mathrm{z}}\right)^{2}\) are independent, and (iii) \(\sum_{i=1}\left(z_{i}-\bar{z}\right)^{2}\) has a chi-square distribution with \(n-1\) degrees of freedom.

A manufacturer of light bulbs claims that his light bulbs will burn on the average 500 hours. To maintain this average, he tests 25 bulbs each month. If the computed t value falls between \(-t_{0.05}\) and \(t_{0.05}\), he is satisfied with his claim. What conclusion should he draw from a sample that has a mean \(\overline{\mathrm{x}}=518\) hours and a standard deviation \(\mathrm{s}=40\) hours? Assume the distribution of burning times to be approximately normal.