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Refer to Exercise 6.41. Let Y1, Y2, . . . , Yn be
Chapter 6, Problem 43E(choose chapter or problem)
Refer to Exercise 6.41. Let \(Y_{1}, Y_{2}, \ldots \ldots Y_{n}\) be independent, normal random variables, each with mean \(\mu\) and variance \(\sigma^{2}\).
a. Find the density function of \(\bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}\).
b. If \(\sigma^{2}=16\) and \(n=25\), what is the probability that the sample mean, \(\bar{Y}\), takes on a value that is within one unit of the population mean, \(\mu\)? That is, find \(P(|\bar{Y}-\mu| \leq 1)\).
c. If \(\sigma^{2}=16\), find \(P(|\bar{Y}-\mu| \leq 1)\) if \(n=36,n=64\), and \(n=81\). Interpret the results of your calculations.
Questions & Answers
QUESTION:
Refer to Exercise 6.41. Let \(Y_{1}, Y_{2}, \ldots \ldots Y_{n}\) be independent, normal random variables, each with mean \(\mu\) and variance \(\sigma^{2}\).
a. Find the density function of \(\bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}\).
b. If \(\sigma^{2}=16\) and \(n=25\), what is the probability that the sample mean, \(\bar{Y}\), takes on a value that is within one unit of the population mean, \(\mu\)? That is, find \(P(|\bar{Y}-\mu| \leq 1)\).
c. If \(\sigma^{2}=16\), find \(P(|\bar{Y}-\mu| \leq 1)\) if \(n=36,n=64\), and \(n=81\). Interpret the results of your calculations.
ANSWER:Step 1 of 4
Let \(Y_{1}, Y_{2}, \ldots \ldots Y_{n}\) be independent random variables with mean \(\mu\) and variance \(\sigma^{2}\).