Question

In a survey of 215 adult Americans, it was found that the average amount of sleep per night is 7.8 hours with a standard deviation of 1.2 hours. How many adult Americans would need to be surveyed to estimate the mean amount of sleep per night within 0.12 hour with 95% confidence?

Answer 1

We would need at least 385 adult Americans.

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 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.

How many adult Americans would need to be surveyed to estimate the mean amount of sleep per night within 0.12 hour with 95% confidence?

You would need at least n adults, in which n is found when M = 0.12,
`\sigma = 1.2`. So

`M = z*(\sigma)/(√(n))`

`0.12 = 1.96*(1.2)/(√(n))`

`0.12√(n) = 1.96*1.2`

`√(n) = (1.96*1.2)/(0.12)`

`√(n) = 19.6`

`(√(n))^(2) = (19.6)^(2)`

`n = 384.16`

Rounding up, 385

We would need at least 385 adult Americans.