Step by step
The question
Question: a sample of 16 has mean 8.4 and SD 1.2: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 8.4: the middle of our interval.
s is the SD of the sample: 1.2.
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*
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 15 and the 95 percent column.
Row 15, column 95%: t* = 2.131.
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.
1.2/4 is 0.3.
Times 2.131: 0.6393.
4 The interval
Take the error bound away from the middle: 8.4 minus 0.6393 is 7.7607.
And add it: 8.4 plus 0.6393 is 9.0393.
The 95 percent confidence interval is (7.7607, 9.0393).
5 Say it in words
In words: we are 95 percent confident that the true mean is between 7.7607 and 9.0393.
6 Check
Let's check: the middle of the interval should be 8.4.
7.7607 plus 9.0393, divided by 2, is 8.4.
Half the width should be the error bound: 9.0393 minus 7.7607, divided by 2, is 0.6393.
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.