Step by step
The question
Question: in a sample of 400, 180 agree: at the 5% level, is the true proportion less than 0.5?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.5.
Hₐ is what we want to show.
Here: the proportion is less than 0.5.
Less than means only the left end counts.
One tail.
2 Big enough for the bell
The bell works if both counts are at least 5.
400 times 0.5 is 200.
400 times 0.5 is 200.
Both are big enough.
3 The test number
p prime, the sample's part: 180 divided by 400 is 0.45.
The SD uses p zero from H zero: 0.5 times 0.5, over 400, then the square root.
About 0.025.
Careful!
We act as if H zero is true.
So the SD uses 0.5, not 0.45.
z is p prime minus p zero, over the SD.
0.45 minus 0.5 is −0.05.
Divided by 0.025: about −2.
4 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: 0.5 in the middle, our sample at z = −2.
We shade the left end, from our z down.
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 0.0228.
The left end is the table number itself: 0.0228.
5 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.
6 Say it in words
In words: at the 5% level, there is enough evidence that the proportion is less than 0.5.
7 Check
Let's check the other way, with the cut-off.
0.05 in the left 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: big enough for the bell.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.