Question

A market surveyor wishes to know how many energy drinks teenagers drink each week. They want to construct a 95% confidence interval with an error of no more than 0.06. A consultant has informed them that a previous study found the mean to be 7.3 energy drinks per week and found the standard deviation to be 1.3. What is the minimum sample size required to create the specified confidence interval? Round your answer up to the next integer.

Answer 1

The minimum sample size required to create the specified confidence interval is 1804.

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`.

So it 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.

Minimum sample size for a margin of error of 0.06:

This sample size is n.

n is found when
`M = 0.06, \sigma = 1.3`

So

`M = z*(\sigma)/(√(n))`

`0.06 = 1.96*(1.3)/(√(n))`

`0.06√(n) = 1.96*1.3`

`√(n) = (1.96*1.3)/(0.06)`

`(√(n))^(2) = ((1.96*1.3)/(0.06))^(2)`

`n = 1803.41`

Rounding up to the next integer

The minimum sample size required to create the specified confidence interval is 1804.