Chapter 7.2, Problem 9E

Refer to Example 7.2. The amount of fill dispensed by a bottling machine is normally distributed with σ = 1 ounce. If n = 9 bottles are randomly selected from the output of the machine, we found that the probability that the sample mean will be within .3 ounce of the true mean is .6318. Suppose that $\bar{Y}$ is to be computed using a sample of size n.

1. a If n = 16, what is $P\left(\left|\bar{Y}-\mu \right|\le .3\right)$?
2. b Find $P\left(\left|\bar{Y}-\mu \right|\le .3\right)$ when $\bar{Y}$ is to be computed using samples of sizes n = 25. n = 36. n = 49, and n = 64.
3. c What pattern do you observe among the values for $P\left(\left|\bar{Y}-\mu \right|\le .3\right)$ that you observed for the various values of n?
4. d Do the results that you obtained in part (b) seem to be consistent with the result obtained in Example 7.3?

   7.2 A bottling machine can be regulated so that it discharges an average of µ ounces per bottle. It has been observed that the amount of fill dispensed by the machine is normally distributed with σ = 1.0 ounce. A sample of n = 9 filled bottles is randomly selected from the output of the machine on a given day (all bottled with the same machine setting), and the ounces of fill are measured for each. Find the probability that the sample mean will be within .3 ounce of the true mean µ for the chosen machine setting.