Step by step
The question
Question: in a sample of 500, 400 recommend it: at the 1% level, is the true proportion more than 0.75?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.75.
Hₐ is what we want to show.
Here: the proportion is more than 0.75.
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 0.75 is 375.
500 times 0.25 is 125.
Both are big enough.
3 The test number
p prime, the sample's part: 400 divided by 500 is 0.8.
The SD uses p zero from H zero: 0.75 times 0.25, over 500, then the square root.
About 0.0194.
Careful!
We act as if H zero is true.
So the SD uses 0.75, not 0.8.
z is p prime minus p zero, over the SD.
0.8 minus 0.75 is 0.05.
Divided by 0.0194: about 2.58.
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: 0.75 in the middle, our sample at z = 2.58.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.5, column .08.
For z = 2.58: the row 2.5 and the column .08 meet at 0.9951.
The right end is 1 minus 0.9951: 0.0049.
5 Decide
The p-value, 0.0049, 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 0.75.
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 2.58: 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.