Step by step
The question
Question: a sample of 8 has mean 250 and SD 40: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 250: the middle of our interval.
s is the SD of the sample: 40.
And n is 8.
1 Why t
We do not know σ, the SD of all the numbers.
We only have s, the SD of the sample.
So we use t instead of z.
The t curve is a bit wider than the bell.
It makes the interval a bit wider, to be safe.
The t-table has a row for each df, the degrees of freedom: n minus 1.
8 minus 1 is 7.
2 Find t*
90 percent of the bell is in the middle.
That leaves 10 percent: 5 percent in each end.
t* marks the cut between the middle and the ends.
In the t-table we take row 7 and the 90 percent column.
Row 7, column 90%: t* = 1.895.
Careful!
df is n minus 1: 7, not 8.
3 Error bound
The error bound: t* times s over the square root of n.
40 over the square root of 8 is about 14.1421.
Times 1.895: about 26.7993.
4 The interval
Take the error bound away from the middle: 250 minus 26.7993 is 223.2007.
And add it: 250 plus 26.7993 is 276.7993.
The 90 percent confidence interval is (223.2007, 276.7993).
5 Say it in words
In words: we are 90 percent confident that the true mean is between 223.2007 and 276.7993.
6 Check
Let's check: the middle of the interval should be 250.
223.2007 plus 276.7993, divided by 2, is 250.
Half the width should be the error bound: 276.7993 minus 223.2007, divided by 2, is 26.7993.
So it is right!
Let's look back.
First: why t.
Then: find t*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.