Question

An IQ test is designed so that the mean is 100 and the standard deviation is 21 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 ​99% confidence that the sample mean is within 6 IQ points of the true mean. Assume that σ=21 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.

Answer 1

The required sample size is 82. This number of people could be sampled in an afternoon, for example, at a commercial center in a city, which means that it is a reasonable sample size for a real world calculation.

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 - 0.99)/(2) = 0.005`

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 - 0.005 = 0.995`, so Z = 2.575.

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.

Assume that σ=21 and determine the required sample size.

A sample size of n is needed.

n is found when M = 6. So

`M = z(\sigma)/(√(n))`

`6 = 2.575(21)/(√(n))`

`6√(n) = 2.575*21`

Dividing both sides by 3

`√(n)= 2.575*3.5`

`(√(n))^2 = (2.575*3.5)^2`

`n = 81.2`

Rounding up:

The required sample size is 82. This number of people could be sampled in an afternoon, for example, at a commercial center in a city, which means that it is a reasonable sample size for a real world calculation.