Step by step
The question
Question: in a sample of 400, 220 vote yes: at the 5% level, is the true proportion different from 0.5?
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 different from 0.5.
Different from means too big or too small: both ends of the bell count.
Two tails.
2 Big enough for the bell
The bell works if both counts are at least 5.
400 times 0.5 is 200.
400 times 0.5 is 200.
Both are big enough.
3 The test number
p prime, the sample's part: 220 divided by 400 is 0.55.
The SD uses p zero from H zero: 0.5 times 0.5, over 400, then the square root.
About 0.025.
Careful!
We act as if H zero is true.
So the SD uses 0.5, not 0.55.
z is p prime minus p zero, over the SD.
0.55 minus 0.5 is 0.05.
Divided by 0.025: about 2.
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: 0.5 in the middle, our sample at z = 2.
Two tails: we shade both ends, as far out as our sample or more.
The z-table gives the area to the left of z.
Row 2.0, column .00.
For z = 2: the row 2.0 and the column .00 meet at 0.9772.
One end is 1 minus 0.9772: 0.0228.
Two ends: 2 times that is 0.0456.
5 Decide
The p-value, 0.0456, 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 different from 0.5.
7 Check
Let's check the other way, with the cut-off.
0.05 split into two ends gives the cut-offs plus and minus 1.96.
z is 2: 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.