Hi Bailey,
We do not know population standard deviation, so we must use a t-confidence interval. Before we can do that, though, we need the sample mean (x-bar). To compute that, add up the values you were given and divide by 6. This gives:
x-bar=53.10
Now, we also need sample standard deviation (s), and I recommend using a calculator or software for that. It would involve a lot of subtraction and squaring if done manually. From calculator:
s=8.48
Now, to the formula for the t-confidence interval:
CI=x-bar +/- t*SE
Let's break this down:
x-bar=sample mean
t*=t-critical value, discussed below
SE=standard error, described below
Now, since we are working with a sample, we cannot just go with standard deviation. We must compute standard error, which is:
SE=s/sqrt(n) where:
s=sample standard deviation
n=sample size
Here,
s=8.48
n=6
SE=8.48/sqrt(6)
SE=3.46
Now, we have standard error but still need t*, the t-critical value. To find this, we need our confidence level (95%) and a metric called degrees of freedom, calculated by:
df=n-1 where n=sample size
n=6
df=6-1
df=5
So, go to t-table, search for 95% confidence in the column and 5 degrees of freedom in the row. This gives:
t*=2.571
Now, we can substitute these values back into our original formula:
CI=x-bar +/- t*SE
x-bar=53.10
SE=3.46
t*=2.571
CI=53.10 +/- (3.46*2.571)
CI=53.10 +/- 8.90
CI=(44.20, 62.00)
I hope this helps.
