Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Outline
1
Hypothesis Testing for Proportions
2
Statistical Significance vs. Practical Significance
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Outline
1
Hypothesis Testing for Proportions
2
Statistical Significance vs. Practical Significance
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Hypotheses about Proportions
For qualitative data, just as for quantitative, we phrase our
questions in terms of null and alternative hypotheses.
Example
If a coin doesn’t come up heads half the time, you’ll call it unfair.
The population is all flips of the coin, ever.
The population proportion p is the probability that the coin
comes up heads.
Our hypotheses are:
H0 : p = 0.50
HA : p 6= 0.50.
We’ll take a sample by flipping the coin some number of
times and counting how often it did come up heads.
The percentage of times it came up heads in the sample is
b, the sample proportion.
p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Sometimes it’s easier to think about HA first.
Example
If you are confident candidate Smith has more than 50% of the
vote, you’ll predict him to win the election.
The population is all voters.
The population proportion p is the percentage of the
population that will vote for Smith.
Our hypotheses are:
HA : p > 0.50.
H0 : p ≤ 0.50.
We’ll take a random sample of some voters and ask them
for whom they’ll vote.
The percentage of the sample that says they’ll vote for
b, the sample proportion.
Smith is p
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Recall Hypothesis Testing for Means
When you’re doing a hypothesis test,
1
2
Formulate your hypotheses H0 and HA .
DRAW A PICTURE, and decide whether
You act if x is too small or too large,
you act only if x is too small,
or you act only if x is too large.
3
Use α to find the boundary(ies) between rejection and
nonrejection.
4
Decide which t-curve you need (or z-curve if n ≥ 30).
5
Convert x into a t-score and see into which region it falls.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Some correspondences
This. . .
µ
µ0
x
√σ
n
corresponds to . . .
p
p0
b
p
q
pq
n
(parameter)
(target parameter)
(statistic)
(standard error)
(Recall that q = 1 − p.)
Strategy
Our strategy will be to mimic what we did before, using these
correspondences.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Some correspondences
This. . .
µ
µ0
x
√σ
n
corresponds to . . .
p
p0
b
p
q
pq
n
(parameter)
(target parameter)
(statistic)
(standard error)
(Recall that q = 1 − p.)
Strategy
Our strategy will be to mimic what we did before, using these
correspondences.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Rejection and Non-Rejection Ranges
0
p0
b falls far away from p0 , we reject H0 and accept HA .
If p
b falls close to p0 , however, we don’t reject H0 .
If p
The question, then, is where exactly to draw the
boundaries between the rejection region and the
nonrejection region.
1
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Rejection and Non-Rejection Ranges
0
p0
rejection region
b falls far away from p0 , we reject H0 and accept HA .
If p
b falls close to p0 , however, we don’t reject H0 .
If p
The question, then, is where exactly to draw the
boundaries between the rejection region and the
nonrejection region.
1
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Rejection and Non-Rejection Ranges
nonrejection region
0
p0
rejection region
b falls far away from p0 , we reject H0 and accept HA .
If p
b falls close to p0 , however, we don’t reject H0 .
If p
The question, then, is where exactly to draw the
boundaries between the rejection region and the
nonrejection region.
1
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Rejection and Non-Rejection Ranges
nonrejection region
0
?
?
p0
rejection region
b falls far away from p0 , we reject H0 and accept HA .
If p
b falls close to p0 , however, we don’t reject H0 .
If p
The question, then, is where exactly to draw the
boundaries between the rejection region and the
nonrejection region.
1
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Probabilities
We think
H0
HA
Reality
H0
HA
1−α
β
α
1−β
Recall that α is the probability that we think HA , given that
H0 is really true.
In conditional probability terms,
α = Pr we think HA H0 .
So in order to work with alpha, we presume H0 is true.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Probabilities
We think
H0
HA
Reality
H0
HA
1−α
β
α
1−β
Recall that α is the probability that we think HA , given that
H0 is really true.
In conditional probability terms,
α = Pr we think HA H0 .
So in order to work with alpha, we presume H0 is true.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Probabilities
We think
H0
HA
Reality
H0
HA
1−α
β
1−β
α
Recall that α is the probability that we think HA , given that
H0 is really true.
In conditional probability terms,
α = Pr we think HA H0 .
So in order to work with alpha, we presume H0 is true.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Probabilities
We think
H0
HA
Reality
H0
HA
1−α
β
1−β
α
Recall that α is the probability that we think HA , given that
H0 is really true.
In conditional probability terms,
α = Pr we think HA H0 .
So in order to work with alpha, we presume H0 is true.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Relating α to the rejection region
Suppose H0 is true, so that p really equals p0 .
b has a z-distribution with mean p0 and standard
Then q
p
error
p0 q0
n .
µ0
Suppose we set our rejection region and nonrejection
region as shown.
Then the shaded areas are the probabilities of rejection
and nonrejection.
We want to set the regions so that α is a specific (small)
number.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Relating α to the rejection region
Suppose H0 is true, so that p really equals p0 .
b has a z-distribution with mean p0 and standard
Then q
p
error
p0 q0
n .
µ0
Suppose we set our rejection region and nonrejection
region as shown.
Then the shaded areas are the probabilities of rejection
and nonrejection.
We want to set the regions so that α is a specific (small)
number.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Relating α to the rejection region
Suppose H0 is true, so that p really equals p0 .
b has a z-distribution with mean p0 and standard
Then q
p
error
p0 q0
n .
µ0
Suppose we set our rejection region and nonrejection
region as shown.
Then the shaded areas are the probabilities of rejection
and nonrejection.
We want to set the regions so that α is a specific (small)
number.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Relating α to the rejection region
Suppose H0 is true, so that p really equals p0 .
b has a z-distribution with mean p0 and standard
Then q
p
error
p0 q0
n .
1−α
α
µ0
Suppose we set our rejection region and nonrejection
region as shown.
Then the shaded areas are the probabilities of rejection
and nonrejection.
We want to set the regions so that α is a specific (small)
number.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Relating α to the rejection region
Suppose H0 is true, so that p really equals p0 .
b has a z-distribution with mean p0 and standard
Then q
p
error
p0 q0
n .
1−α
α
µ0
Suppose we set our rejection region and nonrejection
region as shown.
Then the shaded areas are the probabilities of rejection
and nonrejection.
We want to set the regions so that α is a specific (small)
number.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
Example
You want to test whether a die is fair, so you roll it 100 times. 25
of those times it turns up . Can you conclude that the die is
unfair? Use α = 0.05.
Solution
Our hypotheses are
H0 : p = 16 .
HA : p 6= 16 .
So our “target” proportion is p0 = 61 .
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
Example
You want to test whether a die is fair, so you roll it 100 times. 25
of those times it turns up . Can you conclude that the die is
unfair? Use α = 0.05.
Solution
Our hypotheses are
H0 : p = 16 .
HA : p 6= 16 .
So our “target” proportion is p0 = 61 .
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
Example
You want to test whether a die is fair, so you roll it 100 times. 25
of those times it turns up . Can you conclude that the die is
unfair? Use α = 0.05.
Solution
Our hypotheses are
H0 : p = 16 .
HA : p 6= 16 .
So our “target” proportion is p0 = 61 .
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
Example
You want to test whether a die is fair, so you roll it 100 times. 25
of those times it turns up . Can you conclude that the die is
unfair? Use α = 0.05.
Solution
Our hypotheses are
H0 : p = 16 .
HA : p 6= 16 .
So our “target” proportion is p0 = 61 .
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
0.95
0.05
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
0.025
0.95
0.05
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
0.025
0.95
0.05
−1.96
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
1
We want α = 0.05.
2
Since n = 100, we draw the z-curve.
0.025
0.95
−1.96
0.05
1.96
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Then the area of the left tail is 0.025.
5
The z-table tells us the left tail ends at -1.96.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
0.95
6
7
8
−1.96
1.96
25
b = 100
We got 25 ’s out of 100 rolls, so p
= 0.25.
1
5
Now p0 = 6 = 0.167, so q0 = 6 = 0.833; also, n = 100.
q
q
(0.167)(0.833)
p0 q0
So the standard error is
= 0.037.
n =
100
b, namely
We calculate the z-score of p
b
p−p0
q
=
p0 q0
n
9
0.05
0.25−0.167
0.037
= 2.24
Since this z-score lies in the rejection region,
we reject H0 and accept HA .
We believe the die is unfair.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
0.95
6
7
8
−1.96
1.96
25
b = 100
We got 25 ’s out of 100 rolls, so p
= 0.25.
1
5
Now p0 = 6 = 0.167, so q0 = 6 = 0.833; also, n = 100.
q
q
(0.167)(0.833)
p0 q0
So the standard error is
= 0.037.
n =
100
b, namely
We calculate the z-score of p
b
p−p0
q
=
p0 q0
n
9
0.05
0.25−0.167
0.037
= 2.24
Since this z-score lies in the rejection region,
we reject H0 and accept HA .
We believe the die is unfair.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
0.95
6
7
8
−1.96
1.96
25
b = 100
We got 25 ’s out of 100 rolls, so p
= 0.25.
1
5
Now p0 = 6 = 0.167, so q0 = 6 = 0.833; also, n = 100.
q
q
(0.167)(0.833)
p0 q0
So the standard error is
= 0.037.
n =
100
b, namely
We calculate the z-score of p
b
p−p0
q
=
p0 q0
n
9
0.05
0.25−0.167
0.037
= 2.24
Since this z-score lies in the rejection region,
we reject H0 and accept HA .
We believe the die is unfair.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
0.95
6
7
8
−1.96
1.96
25
b = 100
We got 25 ’s out of 100 rolls, so p
= 0.25.
1
5
Now p0 = 6 = 0.167, so q0 = 6 = 0.833; also, n = 100.
q
q
(0.167)(0.833)
p0 q0
So the standard error is
= 0.037.
n =
100
b, namely
We calculate the z-score of p
b
p−p0
q
=
p0 q0
n
9
0.05
0.25−0.167
0.037
= 2.24
Since this z-score lies in the rejection region,
we reject H0 and accept HA .
We believe the die is unfair.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Fairness of a Die
0.95
6
7
8
−1.96
1.96
25
b = 100
We got 25 ’s out of 100 rolls, so p
= 0.25.
1
5
Now p0 = 6 = 0.167, so q0 = 6 = 0.833; also, n = 100.
q
q
(0.167)(0.833)
p0 q0
So the standard error is
= 0.037.
n =
100
b, namely
We calculate the z-score of p
b
p−p0
q
=
p0 q0
n
9
0.05
0.25−0.167
0.037
= 2.24
Since this z-score lies in the rejection region,
we reject H0 and accept HA .
We believe the die is unfair.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
Example
Your newspaper wants to predict the results of the election
between candidates Smith and Jones. You poll 1000 voters at
random; 513 say they will vote for Smith, and 487 plan to vote
for Jones. Can you conclude that Smith will win the election?
Use α = 0.10.
Solution
Our hypotheses are
HA : p > 0.5.
H0 : p ≤ 0.5.
So our “target” proportion is p0 = 0.5.
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
Example
Your newspaper wants to predict the results of the election
between candidates Smith and Jones. You poll 1000 voters at
random; 513 say they will vote for Smith, and 487 plan to vote
for Jones. Can you conclude that Smith will win the election?
Use α = 0.10.
Solution
Our hypotheses are
HA : p > 0.5.
H0 : p ≤ 0.5.
So our “target” proportion is p0 = 0.5.
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
Example
Your newspaper wants to predict the results of the election
between candidates Smith and Jones. You poll 1000 voters at
random; 513 say they will vote for Smith, and 487 plan to vote
for Jones. Can you conclude that Smith will win the election?
Use α = 0.10.
Solution
Our hypotheses are
HA : p > 0.5.
H0 : p ≤ 0.5.
So our “target” proportion is p0 = 0.5.
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
Example
Your newspaper wants to predict the results of the election
between candidates Smith and Jones. You poll 1000 voters at
random; 513 say they will vote for Smith, and 487 plan to vote
for Jones. Can you conclude that Smith will win the election?
Use α = 0.10.
Solution
Our hypotheses are
HA : p > 0.5.
H0 : p ≤ 0.5.
So our “target” proportion is p0 = 0.5.
First, let’s figure out our rejection regions.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
1
We want α = 0.10.
2
We draw the z-curve.
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Looking up 0.90 in the z-table,
we see the boundary is at 1.28.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
1
We want α = 0.10.
2
We draw the z-curve.
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Looking up 0.90 in the z-table,
we see the boundary is at 1.28.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
1
We want α = 0.10.
2
We draw the z-curve.
0.90
0.10
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Looking up 0.90 in the z-table,
we see the boundary is at 1.28.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
1
We want α = 0.10.
2
We draw the z-curve.
0.90
0.10
1.28
3
We draw the rejection and nonrejection regions, and label
them with their probabilities.
4
Looking up 0.90 in the z-table,
we see the boundary is at 1.28.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
0.90
0.10
6
7
1.28
b = 0.513.
Our sample had 513 successes out of 1000, so p
Now p0 = 0.5, so q0 = 0.5;
also,
n
=
1000.
q
q
So the standard error is
8
pq
n
=
(0.5)(0.5)
1000
= 0.0158.
b, namely
We calculate the z-score of p
0.513 − 0.500
b
p−p0
q
=
= 0.82.
p0 q0
0.0158
n
9
Since this z-score lies in the nonrejection region,
we do not have enough evidence to reject H0 .
The race is too close to call.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
0.90
0.10
6
7
1.28
b = 0.513.
Our sample had 513 successes out of 1000, so p
Now p0 = 0.5, so q0 = 0.5;
also,
n
=
1000.
q
q
So the standard error is
8
pq
n
=
(0.5)(0.5)
1000
= 0.0158.
b, namely
We calculate the z-score of p
0.513 − 0.500
b
p−p0
q
=
= 0.82.
p0 q0
0.0158
n
9
Since this z-score lies in the nonrejection region,
we do not have enough evidence to reject H0 .
The race is too close to call.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
0.90
0.10
6
7
1.28
b = 0.513.
Our sample had 513 successes out of 1000, so p
Now p0 = 0.5, so q0 = 0.5;
also,
n
=
1000.
q
q
So the standard error is
8
pq
n
=
(0.5)(0.5)
1000
= 0.0158.
b, namely
We calculate the z-score of p
0.513 − 0.500
b
p−p0
q
=
= 0.82.
p0 q0
0.0158
n
9
Since this z-score lies in the nonrejection region,
we do not have enough evidence to reject H0 .
The race is too close to call.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
0.90
0.10
6
7
1.28
b = 0.513.
Our sample had 513 successes out of 1000, so p
Now p0 = 0.5, so q0 = 0.5;
also,
n
=
1000.
q
q
So the standard error is
8
pq
n
=
(0.5)(0.5)
1000
= 0.0158.
b, namely
We calculate the z-score of p
0.513 − 0.500
b
p−p0
q
=
= 0.82.
p0 q0
0.0158
n
9
Since this z-score lies in the nonrejection region,
we do not have enough evidence to reject H0 .
The race is too close to call.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Example: Predicting an Election
0.90
0.10
6
7
1.28
b = 0.513.
Our sample had 513 successes out of 1000, so p
Now p0 = 0.5, so q0 = 0.5;
also,
n
=
1000.
q
q
So the standard error is
8
pq
n
=
(0.5)(0.5)
1000
= 0.0158.
b, namely
We calculate the z-score of p
0.513 − 0.500
b
p−p0
q
=
= 0.82.
p0 q0
0.0158
n
9
Since this z-score lies in the nonrejection region,
we do not have enough evidence to reject H0 .
The race is too close to call.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Summary
Do hypothesis tests for proportions as you did for means.
Only do these for large sample sizes, using the z-table.
(Different methods are needed for small sample sizes.)
q
Use p0nq0 for the standard error.
Important
For this test to work, both np and nq should be at least 10.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Summary
Do hypothesis tests for proportions as you did for means.
Only do these for large sample sizes, using the z-table.
(Different methods are needed for small sample sizes.)
q
Use p0nq0 for the standard error.
Important
For this test to work, both np and nq should be at least 10.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
Outline
1
Hypothesis Testing for Proportions
2
Statistical Significance vs. Practical Significance
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Questions
Is this statistically significant?
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Questions
Is this statistically significant? Yes, with more than 99% confidence.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Questions
Is this statistically significant? Yes, with more than 99% confidence.
Is this of practical importance?
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
An Interesting Example
Your company manufactures children’s 12-inch school rulers. You want to
make sure the rulers do average exactly 12 inches long.
H0 : µ = 1200 .
HA : µ =
6 1200 .
A sample of 35 rulers gives x = 12.0100 and s = 0.005.
You choose α = 0.01, which leads to cutoffs of ±2.58.
x − µ0
12.01 − 12.0
√ =
√
Your z-score is
= 11.83.
s/ n
0.005/ 35
This lies far, far into the rejection region, so you are more than 99%
confident that the true mean µ is not 1200 .
Questions
Is this statistically significant? Yes, with more than 99% confidence.
Is this of practical importance? No, not at all.
Hypothesis Testing for Proportions
Statistical Significance vs. Practical Significance
A distinction
When we reject the null hypothesis using statistics, we are
confident the null hypothesis is false.
That is not remotely the same question as whether the null
hypothesis is close to being true.
We can be very, very sure from statistics that µ 6= µ0 , even
if µ is very close to µ0 .
In short, statistical significance does not necessarily imply
practical significance.