Step by step
The question
Question: a sample of 9 has mean 20 and SD 6: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 20: the middle of our interval.
s is the SD of the sample: 6.
And n is 9.
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.
9 minus 1 is 8.
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 8 and the 90 percent column.
Row 8, column 90%: t* = 1.860.
Careful!
df is n minus 1: 8, not 9.
3 Error bound
The error bound: t* times s over the square root of n.
The square root of 9 is 3.
6/3 is 2.
Times 1.86: 3.72.
4 The interval
Take the error bound away from the middle: 20 minus 3.72 is 16.28.
And add it: 20 plus 3.72 is 23.72.
The 90 percent confidence interval is (16.28, 23.72).
5 Say it in words
In words: we are 90 percent confident that the true mean is between 16.28 and 23.72.
6 Check
Let's check: the middle of the interval should be 20.
16.28 plus 23.72, divided by 2, is 20.
Half the width should be the error bound: 23.72 minus 16.28, divided by 2, is 3.72.
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.