Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean.

Mean IQ of Attorneys See the preceding exercise, in which we can assume that for the IQ scores. Attorneys are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of attorneys, given that we want 98% confidence that the sample mean is within 3 IQ points of the population mean. Does the sample size appear to be practical?

The IQ test ofAttorneys has a standard deviation of $\sigma =15$. The level of confidence is 98%, and the sample mean lies within 3 IQ points of the true mean.

The sample size n can be determined by using the following formula.

$n={\left[\frac{z_{\frac{\alpha }{2}}\times \sigma }{E}\right]}^2 ...\left(1\right)$

Here, E is the margin of error.

The $z_{\frac{\alpha }{2}}$is a z-score that separates an area of $\frac{\alpha }{2}$in the right tail of the standard normal distribution.

The confidence level of 98% corresponds to $\alpha =0.02 \text{and} \frac{\alpha }{2}=0.01$.

The value$z_{\frac{\alpha }{2}}$has the cumulative area to its left as $1-\frac{\alpha }{2}$.

Mathematically,

$$\begin{gathered} P\left(z<z_{\frac{\alpha }{2}}\right)=1-\frac{\alpha }{2} \\ =0.99 \end{gathered}$$

From the standard normal table, the area of 0.99 is observed corresponding intersection of the row value 2.3 and column value 0.03, which implies $z_{\frac{\alpha }{2}}$is 2.33.

The sample size is calculated by substituting the values of $z_{\frac{\alpha }{2}}$,$\sigma$, and E in equation (1).

$$\begin{gathered} \text{n}={\left[\frac{z_{\frac{\alpha }{2}}\times \sigma }{\text{E}}\right]}^2 \\ ={\left[\frac{2.33\times 15}{3}\right]}^2 \\ =135.722 \\ =136 \left(\text{rounded} \text{up}\right) \end{gathered}$$

Thus, with 136 samples values, you can be 98% confident that your sample mean lies within 3 IQ points of the true mean.

The sample size required to estimate the meanIQ test ofAttorneys is 136. Yes, the sample size does appear to be very practical.