Step by step
The question
Question: in a sample of 500, 230 say yes: at the 1% level, is the true proportion more than 0.4?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.4.
Hₐ is what we want to show.
Here: the proportion is more than 0.4.
More than means only the right end counts.
One tail.
2 Big enough for the bell
The bell works if both counts are at least 5.
500 times 0.4 is 200.
500 times 0.6 is 300.
Both are big enough.
3 The test number
p prime, the sample's part: 230 divided by 500 is 0.46.
The SD uses p zero from H zero: 0.4 times 0.6, over 500, then the square root.
About 0.0219.
Careful!
We act as if H zero is true.
So the SD uses 0.4, not 0.46.
z is p prime minus p zero, over the SD.
0.46 minus 0.4 is 0.06.
Divided by 0.0219: about 2.74.
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: 0.4 in the middle, our sample at z = 2.74.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.7, column .04.
For z = 2.74: the row 2.7 and the column .04 meet at 0.9969.
The right end is 1 minus 0.9969: 0.0031.
5 Decide
The p-value, 0.0031, is less than alpha, 0.01.
Our sample would be rare if H zero were true.
So we reject H zero.
6 Say it in words
In words: at the 1% level, there is enough evidence that the proportion is more than 0.4.
7 Check
Let's check the other way, with the cut-off.
0.01 in the right end gives the cut-off 2.326.
z is 2.74: 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.