Step by step
The question
Question: in a sample of 500, 270 say yes: at the 5% level, is the true proportion more than 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 more than 0.5.
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 0.5 is 250.
500 times 0.5 is 250.
Both are big enough.
3 The test number
p prime, the sample's part: 270 divided by 500 is 0.54.
The SD uses p zero from H zero: 0.5 times 0.5, over 500, then the square root.
About 0.0224.
Careful!
We act as if H zero is true.
So the SD uses 0.5, not 0.54.
z is p prime minus p zero, over the SD.
0.54 minus 0.5 is 0.04.
Divided by 0.0224: about 1.79.
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 = 1.79.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.7, column .09.
For z = 1.79: the row 1.7 and the column .09 meet at 0.9633.
The right end is 1 minus 0.9633: 0.0367.
5 Decide
The p-value, 0.0367, 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 more than 0.5.
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 1.79: 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.