Step by step
The question
Question: a sample of 25 has mean 72 and SD 10: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 72: the middle of our interval.
s is the SD of the sample: 10.
And n is 25.
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.
25 minus 1 is 24.
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 24.
Its columns are named by the middle part: the 95 percent column, which leaves 2.5 percent in each end.
Row 24, column 95%: t* = 2.064.
Careful!
df is n minus 1: 24, not 25.
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 25 is 5.
10/5 is 2.
Times 2.064: 4.128.
4 The interval
Take the error bound away from the middle: 72 minus 4.128 is 67.872.
And add it: 72 plus 4.128 is 76.128.
The 95 percent confidence interval is (67.872, 76.128).
5 Say it in words
In words: we are 95 percent confident that the true mean is between 67.872 and 76.128.
6 Check
Let's check: the middle of the interval should be 72.
67.872 plus 76.128, divided by 2, is 72.
Half the width should be the error bound: 76.128 minus 67.872, divided by 2, is 4.128.
So it is right!
7 Pick the choice
Our answer is (67.872, 76.128).
That is choice D.
8 Why not the others
A is (68.08, 75.92).
Wrong: used z = 1.96: with a sample SD and n = 25 use t = 2.064.
B is (70, 74).
Wrong: used 1 SE.
C is (51.36, 92.64).
Wrong: used the SD 10 without dividing by √25.
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.