Step by step
The question
Question: a sample of 16 has mean 50 and SD 8: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 50: the middle of our interval.
s is the SD of the sample: 8.
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*
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.
Its columns are named by the middle part: the 95 percent column, which leaves 2.5 percent in each end.
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.
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.
8/4 is 2.
Times 2.131: 4.262.
4 The interval
Take the error bound away from the middle: 50 minus 4.262 is 45.738.
And add it: 50 plus 4.262 is 54.262.
The 95 percent confidence interval is (45.738, 54.262).
5 Say it in words
In words: we are 95 percent confident that the true mean is between 45.738 and 54.262.
6 Check
Let's check: the middle of the interval should be 50.
45.738 plus 54.262, divided by 2, is 50.
Half the width should be the error bound: 54.262 minus 45.738, divided by 2, is 4.262.
So it is right!
7 Pick the choice
Our answer is (45.738, 54.262).
That is choice C.
8 Why not the others
A is (46.08, 53.92).
Wrong: used z = 1.96: with a sample SD and n = 16 use t = 2.131.
B is (32.952, 67.048).
Wrong: used the SD 8 without dividing by √16.
D is (48, 52).
Wrong: used 1 SE.
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.