Step by step
The question
Question: a sample of 25 has mean 1.024, and the SD of all the numbers is 0.2: find the 95% confidence interval for the true mean.
x̄, the sample mean, is 1.024: the middle of our interval.
σ, the SD of all the numbers, is known: 0.2.
And n, the sample size, is 25.
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 25 is 5.
0.2/5 is 0.04.
Times 1.96: 0.0784.
Careful!
We divide by the square root of 25, not by 25.
3 The interval
Take the error bound away from the middle: 1.024 minus 0.0784 is 0.9456.
And add it: 1.024 plus 0.0784 is 1.1024.
The 95 percent confidence interval is (0.9456, 1.1024).
4 Say it in words
In words: we are 95 percent confident that the true mean is between 0.9456 and 1.1024.
5 Check
Let's check: the middle of the interval should be 1.024.
0.9456 plus 1.1024, divided by 2, is 1.024.
Half the width should be the error bound: 1.1024 minus 0.9456, divided by 2, is 0.0784.
So it is right!
Let's look back.
First: find z*.
Then: error bound.
Then: the interval.
Then: say it in words.
Last: check.