Solution Found!
Can volumes. Averages are less variable than individual
Chapter , Problem 5.19(choose chapter or problem)
Can volumes. Averages are less variable than individual observations. It is reasonable to assume that the can volumes in Exercise 5.17 vary according to a Normal distribution. In that case, the mean \(x^{-}\) of an SRS of cans also has a Normal distribution.
(a) Make a sketch of the Normal curve for a single can. Add the Normal curve for the mean of an SRS of 4 cans on the same sketch.
(b) What is the probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more?
(c) What is the probability that the mean volume of an SRS of 4 cans differs from the target value by 1 ml or more?
Questions & Answers
QUESTION:
Can volumes. Averages are less variable than individual observations. It is reasonable to assume that the can volumes in Exercise 5.17 vary according to a Normal distribution. In that case, the mean \(x^{-}\) of an SRS of cans also has a Normal distribution.
(a) Make a sketch of the Normal curve for a single can. Add the Normal curve for the mean of an SRS of 4 cans on the same sketch.
(b) What is the probability that the volume of a single randomly chosen can differs from the target value by 1 ml or more?
(c) What is the probability that the mean volume of an SRS of 4 cans differs from the target value by 1 ml or more?
ANSWER:Step 1 of 3
As given in problem 5.17,
\(\begin{array}{l}
\mu=250 \\
\sigma=0.5
\end{array}\)
The mean \(\bar{x}\) of an SRS of cans also has a normal distribution. Then,
\(\begin{aligned}
\mu_{\bar{x}} & =250 \\
\sigma_{\bar{x}} & =\frac{0.5}{\sqrt{4}} \\
& =\frac{0.5}{2} \\
& =0.25
\end{aligned}\)
Therefore, \(\bar{x}\) has a \(N(250,0.25)\) distribution.