Worked examples
The question
Question: a sample of 36 has mean 52, and the SD of all the numbers is 6: at the 5% level, is the true mean more than 50?
μ > 50? x̄ = 52, σ = 6, n = 36, α = 0.05
Alpha, 0.05, is the level: a sample rarer than 5% counts as too rare.
α = 0.05: the level, how rare is too rare
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 50.
H0 : μ = 50
Hₐ is what we want to show.
Here: the mean is more than 50.
Ha : μ > 50
More than means only the right end counts.
One tail.
one tail: the right end
2 The test number
We act as if H zero is true: the mean is 50.
How far is our sample mean from it?
as if H0 is true: μ = 50
First the SD of the sample means: σ over the square root of n.
The square root of 36 is 6.
6/6 is 1.
σ√n = 6√36 = 66 = 1
z is the sample mean minus 50, over 1.
52 minus 50 is 2.
Divided by 1: 2.
z = x̄ − μ0SD = 52 − 501 = 21 = 2
3 The p-value
The p-value is the chance of getting a sample this far out, if H zero is true.
p-value: the chance of a sample this far out, if H0 is true
On the bell of H zero: 50 in the middle, our sample at z = 2.
We shade the right end, from our z up.
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.
Careful!
0.9772 is the area to the left.
The p-value is the end past our z.
Careful: not p-value = 0.9772 ✗
The right end is 1 minus 0.9772: 0.0228.
p-value = 1 − 0.9772 = 0.0228
4 Decide
The p-value, 0.0228, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
= p-value 0.0228 < α = 0.05 → reject H0
5 Say it in words
At the 5% level, there is enough evidence that
In words: at the 5% level, there is enough evidence that the mean is more than 50.
the mean is more than 50.
6 Check
Let's check the other way, with the cut-off.
0.05 in the right end gives the cut-off 1.645.
0.05 in one end: cut-off 1.645
z is 2: past the cut-off.
The same decision.
So it is right!
z = 2 > 1.645: past the cut-off, reject
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.0228 < α = 0.05 → reject H0
▶ Watch it solved
The question
Question: a sample of 36 has mean 21.5, and the SD of all the numbers is 6: at the 5% level, is the true mean different from 20?
μ ≠ 20? x̄ = 21.5, σ = 6, n = 36, α = 0.05
Alpha, 0.05, is the level: a sample rarer than 5% counts as too rare.
α = 0.05: the level, how rare is too rare
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 20.
H0 : μ = 20
Hₐ is what we want to show.
Here: the mean is different from 20.
Ha : μ ≠ 20
Different from means too big or too small: both ends of the bell count.
Two tails.
two tails: far on either side counts
2 The test number
We act as if H zero is true: the mean is 20.
How far is our sample mean from it?
as if H0 is true: μ = 20
First the SD of the sample means: σ over the square root of n.
The square root of 36 is 6.
6/6 is 1.
σ√n = 6√36 = 66 = 1
z is the sample mean minus 20, over 1.
21.5 minus 20 is 1.5.
Divided by 1: 1.5.
z = x̄ − μ0SD = 21.5 − 201 = 1.51 = 1.5
3 The p-value
The p-value is the chance of getting a sample this far out, if H zero is true.
p-value: the chance of a sample this far out, if H0 is true
On the bell of H zero: 20 in the middle, our sample at z = 1.5.
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 1.5, column .00.
For z = 1.5: the row 1.5 and the column .00 meet at 0.9332.
Careful!
0.0668 is only one end.
Two tails: we double it.
Careful: not p-value = 0.0668 ✗
One end is 1 minus 0.9332: 0.0668.
Two ends: 2 times that is 0.1336.
p-value = 2 × (1 − 0.9332) = 2 × 0.0668 = 0.1336
4 Decide
The p-value, 0.1336, is more than alpha, 0.05.
Our sample is not rare enough.
So we do not reject H zero.
= p-value 0.1336 > α = 0.05 → do not reject H0
5 Say it in words
At the 5% level, there is not enough evidence that
In words: at the 5% level, there is not enough evidence that the mean is different from 20.
the mean is different from 20.
This does not prove H zero is true.
We just cannot rule it out.
not rejecting H0 does not prove H0
6 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.
0.05 in two ends: cut-offs ±1.96
z is 1.5: not past the cut-off.
The same decision.
So it is right!
z = |1.5| < 1.96: not past it, do not reject
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.1336 > α = 0.05 → do not reject H0
▶ Watch it solved
The question
Question: a sample of 40 has mean 52, and the SD of all the numbers is 8: at the 5% level, is the true mean more than 50?
μ > 50? x̄ = 52, σ = 8, n = 40, α = 0.05
Alpha, 0.05, is the level: a sample rarer than 5% counts as too rare.
α = 0.05: the level, how rare is too rare
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the mean is 50.
H0 : μ = 50
Hₐ is what we want to show.
Here: the mean is more than 50.
Ha : μ > 50
More than means only the right end counts.
One tail.
one tail: the right end
2 The test number
We act as if H zero is true: the mean is 50.
How far is our sample mean from it?
as if H0 is true: μ = 50
First the SD of the sample means: σ over the square root of n.
8 over the square root of 40 is about 1.2649.
σ√n = 8√40 = ≈ 1.2649
z is the sample mean minus 50, over 1.2649.
52 minus 50 is 2.
Divided by 1.2649: about 1.58.
z = x̄ − μ0SD = 52 − 501.2649 = 21.2649 ≈ 1.58
3 The p-value
The p-value is the chance of getting a sample this far out, if H zero is true.
p-value: the chance of a sample this far out, if H0 is true
On the bell of H zero: 50 in the middle, our sample at z = 1.58.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 1.5, column .08.
For z = 1.58: the row 1.5 and the column .08 meet at 0.9429.
Careful!
0.9429 is the area to the left.
The p-value is the end past our z.
Careful: not p-value = 0.9429 ✗
The right end is 1 minus 0.9429: 0.0571.
p-value = 1 − 0.9429 = 0.0571
4 Decide
The p-value, 0.0571, is more than alpha, 0.05.
Our sample is not rare enough.
So we do not reject H zero.
= p-value 0.0571 > α = 0.05 → do not reject H0
5 Say it in words
At the 5% level, there is not enough evidence that
In words: at the 5% level, there is not enough evidence that the mean is more than 50.
the mean is more than 50.
This does not prove H zero is true.
We just cannot rule it out.
not rejecting H0 does not prove H0
6 Check
Let's check the other way, with the cut-off.
0.05 in the right end gives the cut-off 1.645.
0.05 in one end: cut-off 1.645
z is 1.58: not past the cut-off.
The same decision.
So it is right!
z = 1.58 < 1.645: not past it, do not reject
Let's look back.
First: write the hypotheses.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.0571 > α = 0.05 → do not reject H0
▶ Watch it solved
