Worked examples
The question
Question: in group 1, 45 of 100 say yes, and in group 2, 30 of 100: at the 5% level, is the first proportion more than the second?
p1 > p2? 45 of 100, 30 of 100, α = 0.05
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the two proportions are the same, so the difference is 0.
H0 : p1 = p2
Hₐ is what we want to show.
Here: the first proportion is more than the second.
Ha : p1 > p2
More than: one tail, the right end.
one tail: the right end
2 The two parts
Group 1: 45/100 is 0.45.
p′1 = 45100 = 0.45
Group 2: 30/100 is 0.3.
p′2 = 30100 = 0.3
3 Pool the groups
If H zero is true, the two groups are the same.
So we put them together: the pooled proportion.
if H0 is true, the groups are the same: put them together
All the yeses over everyone: 75/200 is 0.375.
pc = 45 + 30100 + 100 = 75200 = 0.375
Careful!
We pool the counts, not the two proportions.
Careful: pool the counts: (x₁ + x₂) ÷ (n₁ + n₂)
And q c is 1 minus 0.375: 0.625.
qc = 1 − 0.375 = 0.625
4 Standard error
The standard error, the SD of the difference: the square root of p c times q c, times 1/n 1 plus 1/n 2.
SE = √pc × qc × (1n1 + 1n2)
0.375 times 0.625 is 0.2344.
1/100 plus 1/100 is 0.02.
Multiply, then the square root: about 0.0685.
SE = √0.375 × 0.625 × (1100 + 1100) ≈ 0.0685
5 The test number
z is the difference over the SE.
0.45 minus 0.3 is 0.15.
Divided by 0.0685: about 2.19.
z = p′1 − p′2SE = 0.45 − 0.30.0685 ≈ 2.19
6 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: 0 in the middle, our sample at z = 2.19.
We shade the right end, from our z up.
The z-table gives the area to the left of z.
Row 2.1, column .09.
For z = 2.19: the row 2.1 and the column .09 meet at 0.9857.
The right end is 1 minus 0.9857: 0.0143.
p-value = 1 − 0.9857 = 0.0143
7 Decide
The p-value, 0.0143, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
= p-value 0.0143 < α = 0.05 → reject H0
8 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 first proportion is more than the second.
the first proportion is more than the second.
9 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 2.19: past the cut-off.
The same decision.
So it is right!
z = 2.19 > 1.645: past the cut-off, reject
Let's look back.
First: write the hypotheses.
Then: the two parts.
Then: pool the groups.
Then: standard error.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.0143 < α = 0.05 → reject H0
▶ Watch it solved
The question
Question: in group 1, 120 of 200 say yes, and in group 2, 100 of 200: at the 5% level, is the first proportion different from the second?
p1 ≠ p2? 120 of 200, 100 of 200, α = 0.05
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the two proportions are the same, so the difference is 0.
H0 : p1 = p2
Hₐ is what we want to show.
Here: the first proportion is different from the second.
Ha : p1 ≠ p2
Different from: two tails.
two tails
2 The two parts
Group 1: 120/200 is 0.6.
p′1 = 120200 = 0.6
Group 2: 100/200 is 0.5.
p′2 = 100200 = 0.5
3 Pool the groups
If H zero is true, the two groups are the same.
So we put them together: the pooled proportion.
if H0 is true, the groups are the same: put them together
All the yeses over everyone: 220/400 is 0.55.
pc = 120 + 100200 + 200 = 220400 = 0.55
Careful!
We pool the counts, not the two proportions.
Careful: pool the counts: (x₁ + x₂) ÷ (n₁ + n₂)
And q c is 1 minus 0.55: 0.45.
qc = 1 − 0.55 = 0.45
4 Standard error
The standard error, the SD of the difference: the square root of p c times q c, times 1/n 1 plus 1/n 2.
SE = √pc × qc × (1n1 + 1n2)
0.55 times 0.45 is 0.2475.
1/200 plus 1/200 is 0.01.
Multiply, then the square root: about 0.0498.
SE = √0.55 × 0.45 × (1200 + 1200) ≈ 0.0498
5 The test number
z is the difference over the SE.
0.6 minus 0.5 is 0.1.
Divided by 0.0498: about 2.01.
z = p′1 − p′2SE = 0.6 − 0.50.0498 ≈ 2.01
6 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: 0 in the middle, our sample at z = 2.01.
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 2.0, column .01.
For z = 2.01: the row 2.0 and the column .01 meet at 0.9778.
One end is 1 minus 0.9778: 0.0222.
Two ends: 2 times that is 0.0444.
p-value = 2 × (1 − 0.9778) = 2 × 0.0222 = 0.0444
7 Decide
The p-value, 0.0444, is less than alpha, 0.05.
Our sample would be rare if H zero were true.
So we reject H zero.
= p-value 0.0444 < α = 0.05 → reject H0
8 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 first proportion is different from the second.
the first proportion is different from the second.
9 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 2.01: past the cut-off.
The same decision.
So it is right!
z = |2.01| > 1.96: past the cut-off, reject
Let's look back.
First: write the hypotheses.
Then: the two parts.
Then: pool the groups.
Then: standard error.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.0444 < α = 0.05 → reject H0
▶ Watch it solved
The question
Question: in group 1, 256 of 400 say yes, and in group 2, 216 of 400: at the 1% level, is the first proportion different from the second?
p1 ≠ p2? 256 of 400, 216 of 400, α = 0.01
1 Write the hypotheses
H₀ is the "nothing new" idea: it has the equal sign.
Here: the two proportions are the same, so the difference is 0.
H0 : p1 = p2
Hₐ is what we want to show.
Here: the first proportion is different from the second.
Ha : p1 ≠ p2
Different from: two tails.
two tails
2 The two parts
Group 1: 256/400 is 0.64.
p′1 = 256400 = 0.64
Group 2: 216/400 is 0.54.
p′2 = 216400 = 0.54
3 Pool the groups
If H zero is true, the two groups are the same.
So we put them together: the pooled proportion.
if H0 is true, the groups are the same: put them together
All the yeses over everyone: 472/800 is 0.59.
pc = 256 + 216400 + 400 = 472800 = 0.59
Careful!
We pool the counts, not the two proportions.
Careful: pool the counts: (x₁ + x₂) ÷ (n₁ + n₂)
And q c is 1 minus 0.59: 0.41.
qc = 1 − 0.59 = 0.41
4 Standard error
The standard error, the SD of the difference: the square root of p c times q c, times 1/n 1 plus 1/n 2.
SE = √pc × qc × (1n1 + 1n2)
0.59 times 0.41 is 0.2419.
1/400 plus 1/400 is 0.005.
Multiply, then the square root: about 0.0348.
SE = √0.59 × 0.41 × (1400 + 1400) ≈ 0.0348
5 The test number
z is the difference over the SE.
0.64 minus 0.54 is 0.1.
Divided by 0.0348: about 2.88.
z = p′1 − p′2SE = 0.64 − 0.540.0348 ≈ 2.88
6 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: 0 in the middle, our sample at z = 2.88.
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 2.8, column .08.
For z = 2.88: the row 2.8 and the column .08 meet at 0.9980.
One end is 1 minus 0.9980: 0.0020.
Two ends: 2 times that is 0.0040.
p-value = 2 × (1 − 0.9980) = 2 × 0.0020 = 0.0040
7 Decide
The p-value, 0.0040, is less than alpha, 0.01.
Our sample would be rare if H zero were true.
So we reject H zero.
= p-value 0.0040 < α = 0.01 → reject H0
8 Say it in words
At the 1% level, there is enough evidence that
In words: at the 1% level, there is enough evidence that the first proportion is different from the second.
the first proportion is different from the second.
9 Check
Let's check the other way, with the cut-off.
0.01 split into two ends gives the cut-offs plus and minus 2.576.
0.01 in two ends: cut-offs ±2.576
z is 2.88: past the cut-off.
The same decision.
So it is right!
z = |2.88| > 2.576: past the cut-off, reject
Let's look back.
First: write the hypotheses.
Then: the two parts.
Then: pool the groups.
Then: standard error.
Then: the test number.
Then: the p-value.
Then: decide.
Then: say it in words.
Last: check.
Answer = p-value 0.0040 < α = 0.01 → reject H0
▶ Watch it solved
