1. Let X be a random sample of size n from a N(m,s2

)

population distribution representing the weights, in ounces, of cereal placed in cereal boxes for a certain brand and type of breakfast cereal. Define s^ as in Theorem 6.19.

(a) Show that the random variable T ¼ n1=2 X  m  =s^ has the t-distribution with n-1 degrees of freedom.

(b) Let n ¼ 25. What is the probability that the random interval X  2:06s^=n1=2; X þ 2:06s^=n1=2  will have an outcome that contains the value of m? (This random interval is an example of a confidence interval –

in this case for the population mean m. See Section 10.6.)

(c) Suppose that x ¼ 16.3 and s 2 ¼ .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.