Step by step
The question
Question: a sample of 4 has mean 140 and SD 18: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 140: the middle of our interval.
s is the SD of the sample: 18.
And n is 4.
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.
4 minus 1 is 3.
2 Find t*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
t* marks the cut between the middle and the ends.
In the t-table we take row 3 and the 95 percent column.
Row 3, column 95%: t* = 3.182.
Careful!
df is n minus 1: 3, not 4.
3 Error bound
The error bound: t* times s over the square root of n.
The square root of 4 is 2.
18/2 is 9.
Times 3.182: 28.638.
4 The interval
Take the error bound away from the middle: 140 minus 28.638 is 111.362.
And add it: 140 plus 28.638 is 168.638.
The 95 percent confidence interval is (111.362, 168.638).
5 Say it in words
In words: we are 95 percent confident that the true mean is between 111.362 and 168.638.
6 Check
Let's check: the middle of the interval should be 140.
111.362 plus 168.638, divided by 2, is 140.
Half the width should be the error bound: 168.638 minus 111.362, divided by 2, is 28.638.
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.