Step by step
The question
Question: in a sample of 200, 70 say yes: at the 5% level, is the true proportion less 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 less than 0.4.
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.
200 times 0.4 is 80.
200 times 0.6 is 120.
Both are big enough.
3 The test number
p prime, the sample's part: 70 divided by 200 is 0.35.
The SD uses p zero from H zero: 0.4 times 0.6, over 200, then the square root.
About 0.0346.
Careful!
We act as if H zero is true.
So the SD uses 0.4, not 0.35.
z is p prime minus p zero, over the SD.
0.35 minus 0.4 is −0.05.
Divided by 0.0346: about −1.44.
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 = −1.44.
We shade the left end, from our z down.
The z-table gives the area to the left of z.
Row −1.4, column .04.
For z = −1.44: the row −1.4 and the column .04 meet at 0.0749.
The left end is the table number itself: 0.0749.
5 Decide
The p-value, 0.0749, 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 less than 0.4.
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 left end gives the cut-off −1.645.
z is −1.44: 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.