Finding the sample size of students with the confidence level

Question

Here is my question: How large a sample of students would be needed in order to estimate the population prediction mean within ±2 with 85% confidence? Given that the sample of wight students predictions was 72 83 78 65 69 77 81 71 that I have calculated the sample standard deviation(s) = 6.2335, the sample mean = 74.5 and the sample size is 8.

In my own trial, 85% refers to z=1.440. The question requires my calculation base on t(20,0.075) but I cannot find the 0.075 in the table. So I doubt whether my directions aren't correct.

Thanks a lot for help!

Answer 1

Here is a simulation that serves as a 'reality check' on my answer in Comments that the necessary sample size is around 20 or 22.

In R, the procedure t.test with parameter conf.lev=.85) will use data x show an 85% confidence interval for the population mean $\mu.$.

Suppose, for planning purposes, we use the sample mean and SD from your pilot experiment with eight observations to be the population mean and SD. Then we can simulate vector x of $n=22$ observations from $\mathsf{Norm}(\mu=74.5, \sigma=6.23).$ And use t.test to find the resulting 85% CI for $\mu.$ Finally, we can find the half-length of the CI to be the margin of error $E$ and see how close it is to $E = 2.$

set.seed(320)
x = rnorm(22, 74.5, 6.23)
CI = t.test(x, conf.lev = .85)$conf.int;  CI
[1] 73.21693 77.44468
attr(,"conf.level")
[1] 0.85

me = diff(CI)/2;  me
[1] 2.113874

The result is a little larger than we hoped. But this is only for one sample. If we do this for 10,000 samples and average the results, we can get an idea whether $n = 22$ is, on average, sufficient to have a margin of error of the desired size.

set.seed(2021)
me = replicate(10000, 
                diff(t.test(rnorm(21,14.5,6.24),
                 conf.lev=.85)$conf.int)/2)
mean(me)
[1] 1.969126

hdr = "Simulated Margins of Error for n = 21"
hist(me, prob=T, col="skyblue2", main=hdr)

So margins of error are often as small as 1.5 or as large as 2.3, but they are mostly near 2.0, as desired.