The sample size needed to estimate the mean of a universe with a specified level of precision can be calculated using the concepts presented on the preceding pages. The following example illustrates the three step procedure that can be used to make an estimate of the needed sample size.
Two marketing executives wish to estimate the average monthly sales of a particular brand of cat food in grocery stores. After some discussion, it is decided that they need an estimate which is correct within 10 percent of actual sales, and they wish to be virtually certain of their estimate. If simple random sampling is used, how large a sample of stores is needed to achieve this precision?
Use the Desired Precision Confidence Level to calculate the required standard error of the Mean:
On the basis of their previous experience (perhaps assisted by a pilot test) the executives estimate that the average monthly sales per store are 30 cans. Therefore, to be within 10 percent of average monthly store sales will require a sample result within ± 3 cans of the actual average. This means they will want a confidence interval of ± 3 cans.
Recall the concept that indicated that a sample mean is virtually certain (i.e.99.7 percent) to be within three standard errors of the universe mean. To be virtually certain that the sample result will be within ± 3 cans of the universe mean, it is necessary to set three standard errors of the mean equal to 3 cans. The required sample size (n) must then be such that the estimated standard error of the mean is
3Sx = 3 cans
Sx = 1 can
Estimate the universe Standard Deviation:
They must also use their knowledge of sales conditions to estimate the standard deviation of monthly store sales of the cat food brand. For commonly encountered distributions (e.g. positively skewed ones), a reasonable estimate of the universe standard deviation can be obtained by dividing the range of the assumed distribution of monthly store sales by four (4). For example, if the executives felt that sales per store would range from a low of 12 cans per month to a high of 60 cans per month, then the estimated universe standard deviation (s) would be ( 60 – 12)/ 4 = 12 cans per month.
Calculate the Required sample Size:
The formula presented
Sx = S / √n
Can then be used to determine the required sample size by inserting the above obtained values for Sx and S:
Sx = S / √n
1 can = 12 cans / √ n
√n = 12
n = 144
The averages sales from a sample of 144 stores should be within 3 cans of the actual universe average. That is, if x is he average. That is, if x is the average monthly sales of the stores in the sample, the executives will be almost certain that averages sales of all stores in the universe are within the range of x ± 3 cans.
Impact of sample Size Relative to Universe Size:
The above reasoning assumes that the projected sample size (n) is small relative to the universe size (N). If the indicated sample size is an appreciable proportion of the universe (more than 5 percent), this formula overestimates the sample size required. If the ratio n / N exceeds 0.05 the indicated sample size should be revised downward using the formula:
Revised sample size = n / 1 + n / N:
As an example of this refinement, suppose in the cat food example that the universe size is N = 500 stores. Then because n / N = 0.29, the required sample should be reduced to:
Revised sample size = 144 / 1 + 144 / 500 = 112.