Hypothesis Testing for ProportionsLecture 29
Sections 9.1 - 9.2
Robb T. Koether
Hampden-Sydney College
Fri, Mar 5, 2010
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 1 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 2 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 3 / 46
Homework Review
Exercise 8.24, page 552.Suppose that for a population, the response variable X has a N(−1,2)distribution.(a) Draw the distribution of X . Clearly indicate in your drawing the
mean and the standard deviation.
-6 -4 -2 2 4
0.05
0.10
0.15
0.20
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Homework Review
Exercise 8.24, page 552.Suppose that for a population, the response variable X has a N(−1,2)distribution.(a) Draw the distribution of X . Clearly indicate in your drawing the
mean and the standard deviation.
-6 -4 -2 2 4
0.05
0.10
0.15
0.20
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 4 / 46
Homework Review
Exercise 8.24, page 552.(b) Suppose that a simple random sample of size n = 100 is selected
from this population. Let X represent the sample mean response.
(i) What is the distribution of X for a simple random sample of sizen = 100? Give all relevant features of the distribution.The CLT says that
µx = µ = −1
andσx =
σ√n
=2√100
= 0.2.
So the distribution of x is N(−1,0.2).
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 5 / 46
Homework Review
Exercise 8.24, page 552.(b) Suppose that a simple random sample of size n = 100 is selected
from this population. Let X represent the sample mean response.(i) What is the distribution of X for a simple random sample of size
n = 100? Give all relevant features of the distribution.
The CLT says thatµx = µ = −1
andσx =
σ√n
=2√100
= 0.2.
So the distribution of x is N(−1,0.2).
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 5 / 46
Homework Review
Exercise 8.24, page 552.(b) Suppose that a simple random sample of size n = 100 is selected
from this population. Let X represent the sample mean response.(i) What is the distribution of X for a simple random sample of size
n = 100? Give all relevant features of the distribution.The CLT says that
µx = µ = −1
andσx =
σ√n
=2√100
= 0.2.
So the distribution of x is N(−1,0.2).
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 5 / 46
Homework Review
Exercise 8.24, page 552.(ii) Draw the distribution of X . Clearly indicate in your drawing the
model, the mean, and the standard deviation.
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4
0.5
1.0
1.5
2.0
Calculate P(X > −0.9). Show all steps used to get your answer.normalcdf(-0.9,E99,-1,0.2) = 0.3085.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 6 / 46
Homework Review
Exercise 8.24, page 552.(iii) (ii) Draw the distribution of X . Clearly indicate in your drawing the
model, the mean, and the standard deviation.
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4
0.5
1.0
1.5
2.0
Calculate P(X > −0.9). Show all steps used to get your answer.normalcdf(-0.9,E99,-1,0.2) = 0.3085.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 6 / 46
Homework Review
Exercise 8.24, page 552.(iii) (ii) Draw the distribution of X . Clearly indicate in your drawing the
model, the mean, and the standard deviation.
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4
0.5
1.0
1.5
2.0
Calculate P(X > −0.9). Show all steps used to get your answer.
normalcdf(-0.9,E99,-1,0.2) = 0.3085.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 6 / 46
Homework Review
Exercise 8.24, page 552.(iii) (ii) Draw the distribution of X . Clearly indicate in your drawing the
model, the mean, and the standard deviation.
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4
0.5
1.0
1.5
2.0
Calculate P(X > −0.9). Show all steps used to get your answer.normalcdf(-0.9,E99,-1,0.2) = 0.3085.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 6 / 46
Homework Review
Exercise 8.24, page 552.
(iii) -6 -4 -2 0 2 4
0.5
1.0
1.5
2.0
The distributions of X and X
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 7 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 8 / 46
Introduction
Any question about a population must first be stated in terms of apopulation parameter.We will work with only two parameters:
I The population mean µ.I The population proportion p.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 9 / 46
Introduction
There are only two basic questions that we ask:What is the value of the parameter? (Estimation)Does the evidence support or refute a claim about the value of theparameter? (Hypothesis testing)
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 10 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 11 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.
1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.
2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.
3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.
4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.
5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.
6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.
7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
The Steps of Testing a Hypothesisp-Value Approach
The seven steps of hypothesis testing.1 State the null and alternative hypotheses.2 State the significance level.3 State the formula for the test statistic.4 Compute the value of the test statistic.5 Compute the p-value.6 Make a decision.7 State the conclusion.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 12 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 13 / 46
Example
Example (Hypothesis testing)Are male births more common than female births?Suppose a random sample of 1000 live births shows that 520 aremales and 480 are females.Test the hypothesis that male births are more common thanfemale births, at the 5% level of significance.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 14 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 15 / 46
The Hypotheses
Select the appropriate parameter (p) and describe what itrepresents.The null hypothesis should state a hypothetical value p0 for thepopulation proportion.
H0 : p = p0.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 16 / 46
The Hypotheses
The alternative hypothesis must contradict the null hypothesis inone of three ways:
I H1 : p < p0. (if direction of extreme is left.)I H1 : p > p0. (if direction of extreme is right.)I H1 : p 6= p0. (if direction of extreme is left and right.)
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 17 / 46
The Hypotheses
Example (Step 1)(1) Let p = proportion of births that are male.
H0 : p = 0.50.H1 : p > 0.50.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 18 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 19 / 46
The Level of Significance
Specify the level of significance α.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 20 / 46
The Level of Significance
Example (Step 2)(2) α = 0.05.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 21 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 22 / 46
The Test Statistic
According to the Central Limit Theorem, the statistic p̂ has anormal distribution with
µp̂ = p
and
σp̂ =
√p(1− p)
n.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 23 / 46
The Test Statistic
That is, p̂ is N(
p,√
p(1−p)n
).
Therefore, if the null hypothesis is true, then
p̂ is N
(p0,
√p0(1− p0)
n
).
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 24 / 46
The Test Statistic
The z-score of p̂ is computed as
z =p̂ − p0√p0(1−p0)
n
.
Because p̂ is normal, the z-score is standard normal.That is, z is N(0,1).
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 25 / 46
The Test Statistic
Definition (Test statistic)The test statistic is the statistic that is used to perform the hypothesistest.
If the original statistic (e.g., p̂) is normal, then the test statistic isthe z-score of that statistic.Write the name of the statistic and state the formula to be used.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 26 / 46
The Test Statistic
Example (Step 3)(3) The test statistic is
Z =p̂ − p0√p0(1−p0)
n
.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 27 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 28 / 46
The Value of the Test Statistic
To compute the value of the test statistic, substitute the valuesobtained from the sample and from the null hypothesis.In this case, use the values of p̂, p0, and n.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 29 / 46
The Value of the Test Statistic
Example (Step 4)(4) Given
p0 = 0.50,
p̂ =520
1000= 0.52,
n = 1000,
Compute
Z =0.52− 0.50√(0.50)(1−0.50)
1000
=0.02
0.01581= 1.265.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 30 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 31 / 46
The p-Value
To find the p-value, use the normalcdf function and the value ofthe test statistic (z).Pay attention to the direction of extreme as indicated by thealternative hypothesis.
I To the left: p-value = normalcdf(-E99,z).I To the right: p-value = normalcdf(z,E99).I Two-sided: Find the area in the appropriate tail and then double it.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 32 / 46
The p-Value
Example (Step 5)(5) p-value = normalcdf(1.265,E99) = 0.1029.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 33 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 34 / 46
The Decision
The decision states whether to accept or reject the null hypothesis.If the p-value is less than α, then write “Reject H0.”If the p-value is greater than α, then write “Accept H0.”
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 35 / 46
The Decision
Example (Step 6)(6) Accept H0.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 36 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 37 / 46
The Conclusion
The conclusion restates the decision in the language of theoriginal problem, without using any statistical jargon.It is enough to restate in plain English the hypothesis that wasaccepted.
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The Conclusion
Example (Step 7)(7) The proportion of male births is equal to 50%.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 39 / 46
Summary
Example (The seven steps)(1) Let p = proportion of births that are male.
H0 : p = 0.50.H1 : p > 0.50.
(2) α = 0.05.
(3) The test statistic is Z =p̂ − p0√p0(1−p0)
n
.
(4) Z = 0.52−0.50√(0.50)(1−0.50)
1000
= 0.020.01581 = 1.265.
(5) p-value = normalcdf(1.265,E99) = 0.1029.(6) Accept H0.(7) The proportion of male births is equal to 50%.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 40 / 46
Outline
1 Homework Review
2 Introduction
3 The Hypothesis Testing Procedure
4 ExampleThe HypothesesThe Significance LevelThe Test StatisticThe Value of the Test StatisticThe p-ValueThe DecisionThe Conclusion
5 Assignment
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 41 / 46
Assignment
HomeworkRead Sections 9.1 - 9.2, pages 563 - 580.Let’s Do It! 9.1, 9.2, 9.3, 9.4.Exercises 1 - 12, 14, page 580.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 42 / 46
Assignment
Answers2. (a) True.
(b) False.(c) True.(d) False.
4. Type II.6. (a) Two sided.
(b) 0.6038.(c) 0.0306.(d) The percent of all shoppers in the U.S. that think that “Made in
America” means that 100% of labor plus materials are from the U.S.is not 70%.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 43 / 46
Assignment
Answers8. (a) The population is all pregnant women working with a computer 1 to
20 hours per week.(b) The seven steps:
(1) p is proportion of miscarriages among women in the abovepopulation.
H0 : p = 0.20.H1 : p > 0.20.
(2) α = 0.01.(3) Z = p̂−p0√
p0(1−p0)
n
.
(4) Z = 1.4772.(5) p-value = 0.0698.(6) Accept H0.(7) The proportion of miscarriages in the above population is 0.20.
(c) No. The p-value is greater than α, so the results are not statisticallysignificant.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 44 / 46
Assignment
Answers10. (a) p is proportion of all students who feel that encouraging team or
group work is important to good teaching.H0 : p = 0.50.H1 : p > 0.50.
(b) Let Z = p̂−p0√p0(1−p0)
n
. The value of Z is 4.0943.
(c) p-value = 2.1184× 10−5.(d) Reject H0. The proportion of all students who feel that encouraging
team or group work is important to good teaching is more than 0.50.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 45 / 46
Assignment
Answers12. The seven steps:
(1) p is the proportion of all workers willing to work fewer hours for lesspay to obtain more time for personal leisure activities.
H0 : p = 0.20.H1 : p < 0.20.
(2) α = 0.05.(3) Z = p̂−p0√
p0(1−p0)
n
.
(4) Z = −3.6742.(5) p-value = 1.1931× 10−4.(6) Reject H0.(7) Less than 20% of all workers are willing to work fewer hours for less
pay to obtain more time for personal leisure activities.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 46 / 46
Assignment
Answers14. (a) No. The p-value measures the likelihood that the observed value
would occur if H0 were true. It does not measure the likelihood thatH1 is true.
(b) 0.66.
Robb T. Koether (Hampden-Sydney College) Hypothesis Testing for Proportions Fri, Mar 5, 2010 47 / 46