Step by step
The question
Question: a sample of 16 has mean 15 and SD 4: find the 90% confidence interval for the true mean.
x̄, the sample mean, is 15: the middle of our interval.
s is the SD of the sample: 4.
And n is 16.
There are four choices.
First we solve it, the normal way.
Then we pick.
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, more so for small samples: a small sample makes us less sure.
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*
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 15.
Its columns are named by the middle part: the 90 percent column, which leaves 5 percent in each end.
Row 15, column 90%: t* = 1.753.
Careful!
df is n minus 1: 15, not 16.
3 Error bound
The error bound: t* times s over the square root of n.
s over the square root of n is the SD of sample means, from the sample means lesson.
t* of those SDs on each side holds the middle part we want.
The square root of 16 is 4.
4/4 is 1.
Times 1.753: 1.753.
4 The interval
Take the error bound away from the middle: 15 minus 1.753 is 13.247.
And add it: 15 plus 1.753 is 16.753.
The 90 percent confidence interval is (13.247, 16.753).
5 Say it in words
In words: we are 90 percent confident that the true mean is between 13.247 and 16.753.
6 Check
Let's check: the middle of the interval should be 15.
13.247 plus 16.753, divided by 2, is 15.
Half the width should be the error bound: 16.753 minus 13.247, divided by 2, is 1.753.
So it is right!
7 Pick the choice
Our answer is (13.247, 16.753).
That is choice D.
8 Why not the others
A is (13.355, 16.645).
Wrong: used z = 1.645: with n = 16 use t = 1.753.
B is (13.04, 16.96).
Wrong: used 1.96: that is z for 95%.
C is (7.988, 22.012).
Wrong: used the SD 4 without dividing by √16.
Let's look back.
First: why t.
Then: find t*.
Then: error bound.
Then: the interval.
Then: say it in words.
Then: check.
Then: pick the choice.
Last: why not the others.