Step by step
The question
Question: in a sample of 1000, 470 say yes: at the 5% level, is the true proportion less than 0.5?
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
1000 times 0.5 is 500.
1000 times 0.5 is 500.
Both are big enough.
3 The test number
p prime, the sample's part: 470 divided by 1000 is 0.47.
The SD uses p zero from H zero: 0.5 times 0.5, over 1000, then the square root.
About 0.0158.
Careful!
We act as if H zero is true.
So the SD uses 0.5, not 0.47.
z is p prime minus p zero, over the SD.
0.47 minus 0.5 is −0.03.
Divided by 0.0158: about −1.9.
4 The p-value
The p-value is the chance of getting a sample at least this far out, if H zero is true.
On the bell of H zero: 0.5 in the middle, our sample at z = −1.9.
We shade the left end, from our z down.
The z-table gives the area to the left of z.
Row −1.9, column .00.
For z = −1.9: the row −1.9 and the column .00 meet at 0.0287.
The left end is the table number itself: 0.0287.
5 Decide
The p-value, 0.0287, 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, from the table, as in the cut-offs lesson.
z is −1.9: past the cut-off.
The same decision.
So it is right!
8 Pick the choice
Our answer is reject H₀.
That is choice A.
9 Why not the others
B is Do not reject H0.
Wrong: the p-value 0.0287 is less than 0.05.
C is Accept H0.
Wrong: a test never shows H0 is true.
D is Accept Ha because the sample is large.
Wrong: a large sample alone proves nothing: the p-value decides.
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.
Then: check.
Then: pick the choice.
Last: why not the others.