Step by step
The question
Question: a sample of 25 has mean 3.5 and SD 0.8: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 3.5: the middle of our interval.
s is the SD of the sample: 0.8.
And n is 25.
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.
25 minus 1 is 24.
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 24 and the 90 percent column.
Row 24, column 90%: t* = 1.711.
Careful!
df is n minus 1: 24, not 25.
3 Error bound
The error bound: t* times s over the square root of n.
The square root of 25 is 5.
0.8/5 is 0.16.
Times 1.711: about 0.2738.
4 The interval
Take the error bound away from the middle: 3.5 minus 0.2738 is 3.2262.
And add it: 3.5 plus 0.2738 is 3.7738.
The 90 percent confidence interval is (3.2262, 3.7738).
5 Say it in words
In words: we are 90 percent confident that the true mean is between 3.2262 and 3.7738.
6 Check
Let's check: the middle of the interval should be 3.5.
3.2262 plus 3.7738, divided by 2, is 3.5.
Half the width should be the error bound: 3.7738 minus 3.2262, divided by 2, is 0.2738.
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.