Question

An IQ test is designed so that the mean is 100 and the standard deviation is 24 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 8 IQ points of the true mean. Determine the required sample size using the TI-83/84. The required sample size is________

Answer 1

The required sample size is 35.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.95)/(2) = 0.025`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.025 = 0.975`, so Z = 1.96.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

Determine the required sample size.

The required sample size is n, which is found when
`M = 8`

The standard deviation of 24 means that
`\sigma = 24`. So

`M = z(\sigma)/(√(n))`

`8 = 1.96(24)/(√(n))`

`8√(n) = 1.96*24`

Simplifying both sides by 8

`√(n) = 1.96*3`

`(√(n))^2 = (1.96*3)^2`

`n = 34.6`

Rounding up:

The required sample size is 35.