Step by step
The question
Question: in a sample of 250, 60 own a cat: at the 5% level, is the true proportion more than 0.2?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.2.
Hₐ is what we want to show.
Here: the proportion is more than 0.2.
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.
250 times 0.2 is 50.
250 times 0.8 is 200.
Both are big enough.
3 The test number
p prime, the sample's part: 60 divided by 250 is 0.24.
The SD uses p zero from H zero: 0.2 times 0.8, over 250, then the square root.
About 0.0253.
Careful!
We act as if H zero is true.
So the SD uses 0.2, not 0.24.
z is p prime minus p zero, over the SD.
0.24 minus 0.2 is 0.04.
Divided by 0.0253: about 1.58.
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: 0.2 in the middle, our sample at z = 1.58.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.5, column .08.
For z = 1.58: the row 1.5 and the column .08 meet at 0.9429.
The right end is 1 minus 0.9429: 0.0571.
5 Decide
The p-value, 0.0571, is more than alpha, 0.05.
Our sample is not rare enough.
So we do not reject H zero.
6 Say it in words
In words: at the 5% level, there is not enough evidence that the proportion is more than 0.2.
This does not prove H zero is true.
We just cannot rule it out.
7 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 1.58: not 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.