Question

A study of peach trees found that the average number of peaches per tree was 725. The standard deviation of the population is 70 peaches per tree. A scientist wishes to find the 95% confidence interval for the mean number of peaches per tree. How many trees does she need to sample to obtain an average accurate to within 10 peaches per tree

Answer 1

She needs to sample 189 trees.

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.

The standard deviation of the population is 70 peaches per tree.

This means that
`\sigma = 70`

How many trees does she need to sample to obtain an average accurate to within 10 peaches per tree?

She needs to sample n trees.

n is found when M = 10. So

`M = z(\sigma)/(√(n))`

`10 = 1.96(70)/(√(n))`

`10√(n) = 1.96*70`

Dividing both sides by 10:

`√(n) = 1.96*7`

`(√(n))^2 = (1.96*7)^2`

`n = 188.2`

Rounding up:

She needs to sample 189 trees.