Step by step
The question
Question: a sample of 10 has mean 4.2 and SD 0.9: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 4.2: the middle of our interval.
s is the SD of the sample: 0.9.
And n is 10.
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.
10 minus 1 is 9.
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 9 and the 95 percent column.
Row 9, column 95%: t* = 2.262.
Careful!
df is n minus 1: 9, not 10.
3 Error bound
The error bound: t* times s over the square root of n.
0.9 over the square root of 10 is about 0.2846.
Times 2.262: about 0.6438.
4 The interval
Take the error bound away from the middle: 4.2 minus 0.6438 is 3.5562.
And add it: 4.2 plus 0.6438 is 4.8438.
The 95 percent confidence interval is (3.5562, 4.8438).
5 Say it in words
In words: we are 95 percent confident that the true mean is between 3.5562 and 4.8438.
6 Check
Let's check: the middle of the interval should be 4.2.
3.5562 plus 4.8438, divided by 2, is 4.2.
Half the width should be the error bound: 4.8438 minus 3.5562, divided by 2, is 0.6438.
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.