Hi Matthew,
Is your x actually x-bar, a sample mean, or is it just a single observation? If just a single observation, I'm not sure how to compute a confidence interval from that. I suspect, since you were given s, that it is a sample mean. GIven that you don't know population standard deviation, we use a t-confidence interval here. The formula for that is:
CI= x-bar +/- t*SE where:
x-bar=sample mean
t*=t critical value, obtained from t-table or software
SE=standard error, which we also must calculate with formula below:
SE=s/sqrt(n)
Here,
s2=13, so
s=sqrt(13)
n=36
SE=[sqrt(13)]/[sqrt(36]
SE=sqrt(13)/6
SE=0.60
Now, back to our confidence interval formula:
x-bar=31
SE=0.60
Now, we need to compute t*. We can get this from a table, but first we need to know degrees of freedom and confidence level.
The formula:
df=n-1
df=35
Now, for significance level, you were given:
a=0.01
For a two-sided t-interval, which we must assume this is--we weren't told otherwise--we have to divide this by 2 and get 0.005. When we check the t-table--many are available online but Wyzant reviews posts with links so I can't link here--they list the reverse of this 0.995.
On the t-table, look on the left side for degrees of freedom--35 in this case, and along the top for the confidence level--0.995 here. My t-table did not have 35; the closest it has is 30, so we'll use that. You want closest without going over in this case. From table:
df=30, C. Level 0.995;
t*= 2.75
So, plugging back in:
x-bar=31
t*=2.75
SE=0.60
CI=31 +/- (2.75*0.60)
CI=31 +/- 1.65
CI=(29.35, 32.65)
I hope this helps.
