Sample Size for Mean Find the sample size required to estimate the mean IQ of professional musicians. Assume that we want 98% confidence that the mean from the sample is within three IQ points of the true population mean. Also assume that $\sigma$= 15.

A confidence level of 98%is supposed.

The sample mean should be within 3 IQ points of the true population mean.

The population standard deviation is equal to 15.

The formula used to determine the sample size for estimating the mean IQ of professional musicians is given as follows:

$n={\left[\frac{z_{\frac{\alpha }{2}}\sigma }{E}\right]}^2$

Here, $\sigma$is given to be equal to 15.

Since the confidence level is equal to 98%, the value of the level of significance will be equal to 0.02.

Thus, the corresponding value of $z_{\frac{\alpha }{2}}$is equal to 2.3263.

It is given that the sample mean value should lie within 3 IQ points of the true population mean.

Thus, the margin of error (E) is equal to 3.

Now, the value of the sample size is computed below:

$$\begin{gathered} n={\left[\frac{z_{\frac{\alpha }{2}}\sigma }{E}\right]}^2 \\ ={\left[\frac{\left(2.3263\right)\left(15\right)}{3}\right]}^2 \\ =135.29 \\ \approx 136 \end{gathered}$$

Thus, the sample size for estimating the mean IQ of professional musicians is equal to 136.