This paper presents estimation for a sample size with known population standard deviation of $400 and a margin of error of $2

Abstract

This paper presents estimation for a sample size with known population standard deviation of $400 and a margin of error of ± $25 from the mean. As with any statistical procedures that infer to the population and true mean, this estimation assumes that the population from which the sample is to be drawn takes a z-distribution with a characteristic akin to N (0, 1). The implication of this assumption is that the sample drawn is also expected to have a normal distribution characteristic, except for any error that might result from random sampling. Given the above information and assumption, z-tables are utilized to achieve the objective of the analysis by helping in getting the z-values for the limits. The estimation is done at a confidence coefficient of 98%.

Estimating the Sample size

Generally, we would estimate a sample size through the following formula:

n = ((z x s)/D)) 2

Where n is the sample size being estimated, s represents the known standard deviation and z represents the percentile falling under the 1-α/2 in a standard normal distribution. The division by two is important because there is an upper limit and a lower limit.

0.98/2 = 0.490 and the area under this value in the z-table is 2.32 as shown in the sketch below. The sketch is not drawn to scale though it is intended to present a notion of a standard normal distribution.

Figure SEQ Figure * ARABIC 1: Normal curve for z-values at 98% Confidence Interval

We then multiply the z-value by the standard deviation and get the following:

2.32 x 400 = 928

However, we had a margin of error of 25 hence we divide the product above by the margin of error and get:

928/25 = 37.12

n = 37.12 x 37.12 = 1, 377.8944

= 1,378

Implications of reducing Sample Size

Implications of a small sample size or reducing the estimated sample are manifold. First with respect to the standard error, a small sample size increases the possibility of a large standard error since the standard error is a function of the sample size. What this implies is that if a sample that is needed should give a proper representation of the target population, it should be as large as possible to represent the population with more precision. Hinton (2004) gives an important point to discourage small sample sizes, especially in non-experimental designs where the target population is large. Since we already have boundaries for estimate, the problem with reducing sample size is that it will force us to increase the margin of error and this also have in impact in reducing the intended precision. A small sample size will definitely have an impact on the generalizability of the results of the study (Hinton 2004). Therefore, the sample size should only be reduced if the company is ready to withstand problems of accuracy, precision and generalizability.

Reference:

Hinton P. “Business Statistics Explained” Routledge 2004: 200 – 266.