Suppose you have been following a particular airline stock for many years. You are interested in determining the average daily price of this stock in a 10-year period and you have access to the stock reports for these years. However, you do not want to average all the daily prices over 10 years because there are several thousand data points, so you decide to take a random sample of the daily prices and estimate the average. You want to be 90% confident of your results, you want the estimate to be within \$2.00 of the true average, and you believe the standard deviation of the price of this stock is about \$12.50 over this period of time. How large a sample should you take?

## Solution

Given that the margin of error $E =2$. The confidence coefficient is $1-\alpha=0.9$. Thus $\alpha = 0.1$.

The population standard deviation is $\sigma = 12.5$.

The critical value of $Z$ is $z=Z_{\alpha/2} = 1.645$.

The minimum sample size required to estimate the mean is

` $$ \begin{aligned} n &= \bigg(\frac{z* \sigma}{E}\bigg)^2\\ & = \bigg(\frac{1.645*12.5}{2}\bigg)^2\\ & =105.7041\\ &\approx 106. \end{aligned} $$ `

Thus, the sample of size $n=106$ will ensure that the $90$% confidence interval for the mean will have a margin of error $2$.