Question

If a random sample of 53 students was asked for the number of semester hours they are taking this semester. The sample standard deviation was found to be s = 4.7 semester hours. How many more students should be included in the sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean  for all students at this college?

Answer 1

94 more students should be included in the sample.

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`.

So it 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.

How many students we need to sample to be 99% sure that the sample mean x is within 1 semester hour of the population mean?

We need to survey n students.

n is found when M = 1.

We have that
`\sigma = 4.7`

So

`M = z*(\sigma)/(√(n))`

`1 = 2.575*(4.7)/(√(n))`

`√(n) = 2.575*4.7`

`(√(n))^(2) = (2.575*4.7)^(2)`

`n = 146.47`

Rounding up

147 students need to be surveyed.

How many more students should be included...?

53 have already been surveyed

147 - 53 = 94

94 more students should be included in the sample.