Step by step
The question
Question: a sample of 16 has mean 60 and SD 8: find the 80% confidence interval for the true mean.
x̄, the sample mean, is 60: the middle of our interval.
s is the SD of the sample: 8.
And n is 16.
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.
16 minus 1 is 15.
2 Find t*
80 percent of the bell is in the middle.
That leaves 20 percent: 10 percent in each end.
t* marks the cut between the middle and the ends.
In the t-table we take row 15 and the 80 percent column.
Row 15, column 80%: t* = 1.341.
Careful!
df is n minus 1: 15, not 16.
3 Error bound
The error bound: t* times s over the square root of n.
The square root of 16 is 4.
8/4 is 2.
Times 1.341: 2.682.
4 The interval
Take the error bound away from the middle: 60 minus 2.682 is 57.318.
And add it: 60 plus 2.682 is 62.682.
The 80 percent confidence interval is (57.318, 62.682).
5 Say it in words
In words: we are 80 percent confident that the true mean is between 57.318 and 62.682.
6 Check
Let's check: the middle of the interval should be 60.
57.318 plus 62.682, divided by 2, is 60.
Half the width should be the error bound: 62.682 minus 57.318, divided by 2, is 2.682.
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.