Question 7.4: The manager of a small supermarket chain wants to estimate t......
The manager of a small supermarket chain wants to estimate the proportion of time stock clerks spend making price changes on previously marked merchandise. The manager wants a 98 percent confidence that the resulting estimate will be within 5 percent of the true value. What sample size should she use?
e = .05 z = 2.33 (see p. 307) \hat{p} is unknown
When no sample estimate of p is available, a preliminary estimate of sample size can be obtained using \hat{p} = .50. After 20 or so observations, a new estimate of \hat{p} can be obtained from those observations and a revised value of n computed using the new \hat{p}. It would be prudent to recompute the value of n at two or three points during the study to obtain a better indication of the necessary sample size. Thus, the initial estimate of n is
n\,=\,\left(\frac{2.33}{.05}\right)^{2}\,.50(1-.50)\,=\,542.89,\ \text{or}\;543\;{\mathrm{observations}}Suppose that, in the first 20 observations, stock clerks were found to be changing prices twice, making \hat{p} = 2/ 20 = .10. The revised estimate of n at that point would be
n\;=\;\left(\frac{2.33}{.05}\right)^{2}.10(1-.10)\;=\;195.44,\,\mathrm{or}\;196\;\mathrm{~observations}Suppose a second check is made after a total of 100 observations, and \hat{p} = .11 at this point (including the initial 20 observations). Recomputing n yields
n~=\left({\frac{2.33}{.05}}\right)^{2}.11(.89)~=~212.60,\,\mathrm{or}\,\,213\,\mathrm{~observations}Note: As before, if the resulting value of n is noninteger, round up.
Perhaps the manager might make one more check to settle on a final value for n. If the computed value of n is less than the number of observations already taken, sampling would be terminated at that point.