Step by step
The question
Question: in a sample of 300, 52 say no: at the 10% level, is the true proportion different from 0.2?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.2.
Hₐ is what we want to show.
Here: the proportion is different from 0.2.
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.
300 times 0.2 is 60.
300 times 0.8 is 240.
Both are big enough.
3 The test number
p prime, the sample's part: 52 divided by 300 is 0.1733.
The SD uses p zero from H zero: 0.2 times 0.8, over 300, then the square root.
About 0.0231.
Careful!
We act as if H zero is true.
So the SD uses 0.2, not 0.1733.
z is p prime minus p zero, over the SD.
0.1733 minus 0.2 is −0.0267.
Divided by 0.0231: about −1.15.
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: 0.2 in the middle, our sample at z = −1.15.
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 −1.1, column .05.
For z = −1.15: the row −1.1 and the column .05 meet at 0.1251.
One end is 0.1251.
Two ends: 2 times that is 0.2502.
5 Decide
The p-value, 0.2502, is more than alpha, 0.1.
Our sample is not rare enough.
So we do not reject H zero.
6 Say it in words
In words: at the 10% level, there is not enough evidence that the proportion is different from 0.2.
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.
0.1 split into two ends gives the cut-offs plus and minus 1.645.
z is −1.15: 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.