Step by step
The question
Question: in a sample of 200, 18 are broken: at the 5% level, is the true proportion more than 0.05?
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the proportion is 0.05.
Hₐ is what we want to show.
Here: the proportion is more than 0.05.
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.
200 times 0.05 is 10.
200 times 0.95 is 190.
Both are big enough.
3 The test number
p prime, the sample's part: 18 divided by 200 is 0.09.
The SD uses p zero from H zero: 0.05 times 0.95, over 200, then the square root.
About 0.0154.
Careful!
We act as if H zero is true.
So the SD uses 0.05, not 0.09.
z is p prime minus p zero, over the SD.
0.09 minus 0.05 is 0.04.
Divided by 0.0154: about 2.6.
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: 0.05 in the middle, our sample at z = 2.6.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.6, column .00.
For z = 2.6: the row 2.6 and the column .00 meet at 0.9953.
The right end is 1 minus 0.9953: 0.0047.
5 Decide
The p-value, 0.0047, 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 0.05.
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 2.6: 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.