Step by step
The question
Question: a sample of 64 has mean 3.65, and the SD of all the numbers is 0.6: at the 5% level, is the true mean more than 3.5?
Alpha, 0.05, is the level: a sample rarer than 5% counts as too rare.
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 3.5.
Hₐ is what we want to show.
Here: the mean is more than 3.5.
More than means only the right end counts.
One tail.
2 The test number
We act as if H zero is true: the mean is 3.5.
How far is our sample mean from it?
First the SD of the sample means: σ over the square root of n.
The square root of 64 is 8.
0.6/8 is 0.075.
z is the sample mean minus 3.5, over 0.075.
3.65 minus 3.5 is 0.15.
Divided by 0.075: 2.
3 The p-value
The p-value is the chance of getting a sample this far out, if H zero is true.
On the bell of H zero: 3.5 in the middle, our sample at z = 2.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.0, column .00.
For z = 2: the row 2.0 and the column .00 meet at 0.9772.
Careful!
0.9772 is the area to the left.
The p-value is the end past our z.
The right end is 1 minus 0.9772: 0.0228.
4 Decide
The p-value, 0.0228, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
5 Say it in words
In words: at the 5% level, there is enough evidence that the mean is more than 3.5.
6 Check
Let's check the other way, with the cut-off.
0.05 in the right end gives the cut-off 1.645.
z is 2: past the cut-off.
The same decision.
So it is right!
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.