Chapter 8, part C.

V. Determining Sample Size Suppose that we wish to choose a sample size that will guarantee us a specified degree of accuracy. Recall: We can call the latter part of this interval the error (E). Once we specify a degree of confidence (1-), you can find the appropriate z/2.

The procedure. Given our desired level of error (E), the standard deviation and z/2, you can find the required sample size. This n will provide a precision statement with a 1- probability that the sampling error will be E or less.

An Example. A firm is concerned with the time it takes to train an employee. In a sample of 15 workers, you construct a 95% confidence interval of 53.87 ± 3.78 days. Your manager feels that this error is too large and wants a .95 probability that the value of x-bar will provide a sampling error of ± 2 days or less.

The Solution. Your boss wants E=2 days, 1-  =.95, which dictates a z.025=1.96. From an earlier study, you know that s=6.82 days. You should sample 45 employees to lower the sampling error to 2 days.

VI. Interval Estimates of a Population Proportion If you recall from our coverage of sampling distributions, one of the only modifications that need to be made when dealing with proportions is that the standard deviation of the sampling distribution differs. If p is unknown, we can use the sample value, p-bar.

A. Interval Estimate Use the following formula: or,

An Example Scheer Industries gives an exam at the end of employee training and p is the proportion of all workers that pass. In a sample of 45, 36 passed (p-bar=.80). You need to construct a 95% confidence interval. First, calculate Now: .80 ± 1.96(.060) or .80 ± .12

B. Determining Sample Size Much the same way as before, E is the sampling error specified by the user. With proportions, we would expect E.10. Although this looks easy enough, if we don’t know p, we need to use a “planning value” as a starting point.

Solution The text gives several methods of choosing a planning value for p, but the most common method is to use sample proportions from a previous sample of same or similar units. If your boss desires E=.10 at 95% confidence, z.025=1.96 and we can use p-bar=.80 as our planning value for the unknown p.

The sample size Given our information, we should sample 62 workers. Had there been no other information, use p=.5 and see how many workers you should sample. With a more conservative estimate of p, you should find that it’s necessary to sample more workers. Does this make sense?