Question

You are going to estimate how long a person holds down a button on their calculator by randomly selecting people and asking them to do something on your calculator (which has a special timer to record how long the enter button is pressed). You practiced on the people at work and found a standard deviation of about 0.37 seconds. You want to get a 94% confidence interval that is only 0.06 in width. How many people do you need to have in your study

Answer 1

You need to have 538 people in your study.

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 - 0.94)/(2) = 0.03`

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 - 0.03 = 0.97`, so Z = 1.88.

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.

You practiced on the people at work and found a standard deviation of about 0.37 seconds.

This means that
`\sigma = 0.37`

You want to get a 94% confidence interval that is only 0.06 in width.

This means that
`M = (0.06)/(2) = 0.03`

How many people do you need to have in your study?

This is n for which M = 0.03. So

`M = z(\sigma)/(√(n))`

`0.03 = 1.88(0.37)/(√(n))`

`0.03√(n) = 1.88*0.37`

`√(n) = (1.88*0.37)/(0.03)`

`(√(n))^2 = ((1.88*0.37)/(0.03))^2`

`n = 537.6`

Rounding up:

You need to have 538 people in your study.