Step by step
The question
Question: in a sample of 200, 90 agree: at the 5% level, is the true proportion different from 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 different from 0.4.
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.
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: 90 divided by 200 is 0.45.
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 0.45.
z is p prime minus p zero, over the SD.
0.45 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.
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 1.4, column .04.
For z = 1.44: the row 1.4 and the column .04 meet at 0.9251.
One end is 1 minus 0.9251: 0.0749.
Two ends: 2 times that is 0.1498.
5 Decide
The p-value, 0.1498, 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 different from 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 split into two ends gives the cut-offs plus and minus 1.96.
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.