Question: Let (X) be a random sample of size (n) from a (Nleft(mu, sigma^{2}ight)) population distribution representing the weights, in ounces, of cereal placed in cereal

Let $X$ be a random sample of size $n$ from a $N\left(\mu, \sigma^{2}ight)$ population distribution representing the weights, in ounces, of cereal placed in cereal boxes for a certain brand and type of breakfast cereal. Define $\hat{\sigma}$ as in Theorem 6.19.

(a) Show that the random variable $T=n^{1 / 2}(\bar{X}-\mu) / \hat{\sigma}$ has the $t$-distribution with $n$ - 1 degrees of freedom.

(b) Let $n=25$. What is the probability that the random interval $\left(\bar{X}-2.06 \hat{\sigma} / n^{1 / 2}, \bar{X}+2.06 \hat{\sigma} / n^{1 / 2}ight.$ ) will have an outcome that contains the value of $\mu$ ? (This random interval is an example of a confidence interval in this case for the population mean $\mu$. See Section 10.6.)

(c) Suppose that $\bar{x}=16.3$ and $s^{2}=.01$. Define a confidence interval that is designed to have a .90 probability of generating an outcome that contains the value of the population mean weight of cereal placed in the cereal boxes. Generate a confidence interval outcome for the mean weight.

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a To show that the random variable T fracn12barX muhatsigma has the tdistribution with n1 degrees of freedom we can use the fact that fracbarX musigma sqrtn follows a standard normal distribution sinc... View full answer

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