Step by step
The question
Question: a sample of 100 has mean 250, and the SD of all the numbers is 40: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 250: the middle of our interval.
σ, the SD of all the numbers, is known: 40.
And n, the sample size, is 100.
The interval goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.
1 Find z*
95 percent of the bell is in the middle.
That leaves 5 percent: 2.5 percent in each end.
z* marks the cut between the middle and the ends.
Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.
Now we look for 0.9750 inside the table.
0.9750 is in the table: row 1.9, column .06.
So z = 1.96.
2 Error bound
The error bound: z* times σ over the square root of n.
The square root of 100 is 10.
40/10 is 4.
Times 1.96: 7.84.
Careful!
We divide by the square root of 100, not by 100.
3 The interval
Take the error bound away from the middle: 250 minus 7.84 is 242.16.
And add it: 250 plus 7.84 is 257.84.
The 95 percent confidence interval is (242.16, 257.84).
4 Say it in words
In words: we are 95 percent confident that the true mean is between 242.16 and 257.84.
5 Check
Let's check: the middle of the interval should be 250.
242.16 plus 257.84, divided by 2, is 250.
Half the width should be the error bound: 257.84 minus 242.16, divided by 2, is 7.84.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.