Question

A researcher is interested in determining the average number of years employees of a company stay with the company. If past information shows a standard deviation of 7 months, what size sample should be selected so that at 95% confidence the margin of error will be 2 months or less?

Answer 1

Concepts and reason

The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and the estimation process on the sample parameter is repeated.

The margin of error is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter.

The minimum sample size is used to estimate the true population mean with the required margin of error and confidence level.

Fundamentals

The formula for calculating the sample size for proportion is,

$n = {\left( {\frac{{{z_{\alpha /2}} \times \sigma }}{E}} \right)^2}$

Here,

$\sigma =$ Population standard deviation

$E =$ Margin of error

${z_{\alpha /2}} =$ Critical value

The Excel formula for calculating the critical values is,

$\pm {z_{\alpha /2}} = NORMSINV\left( {\frac{\alpha }{2}} \right)$

Let $\alpha$ be the level of significance = 0.05

At 95% confidence level, the critical values are,

$\begin{array}{c}\\ \pm {z_{0.05/2}} = NORMSINV\left( {\frac{{0.05}}{2}} \right)\\\\ = \pm 1.96\\\end{array}$

Let $E$ be the margin of error = 2.

Let $\sigma$ be the population standard deviation = 7.

The sample size is,

$\begin{array}{c}\\n = {\left( {\frac{{{z_{\alpha /2}} \times \sigma }}{E}} \right)^2}\\\\ = {\left( {\frac{{1.96 \times 7}}{2}} \right)^2}\\\\ = 47.058\\\\ \approx 48\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{Round up to next integer}}} \right)\\\end{array}$

Ans:

The value of sample size is, 48.