hypothesis testing for proportions - hampden-sydney...

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


Outline

1 Homework Review

2 Introduction



5 Assignment


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


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).


Homework Review


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.



Homework Review


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.



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.


Homework Review

Exercise 8.24, page 552.(iii) (ii) Draw the distribution of X . Clearly indicate in your drawing the


-1.4 -1.2 -1.0 -0.8 -0.6 -0.4

0.5

1.0

1.5

2.0



Homework Review



-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.


Homework Review



-1.4 -1.2 -1.0 -0.8 -0.6 -0.4

0.5

1.0

1.5

2.0



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


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


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)


Outline

1 Homework Review

2 Introduction



5 Assignment


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.



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.



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.



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.



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.



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.



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.



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.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


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.)


The Hypotheses

Example (Step 1)(1) Let p = proportion of births that are male.

H0 : p = 0.50.H1 : p > 0.50.


Outline

1 Homework Review

2 Introduction



5 Assignment


The Level of Significance

Specify the level of significance α.


The Level of Significance

Example (Step 2)(2) α = 0.05.


Outline

1 Homework Review

2 Introduction



5 Assignment


The Test Statistic

According to the Central Limit Theorem, the statistic p̂ has anormal distribution with

µp̂ = p

and

σp̂ =

√p(1− p)

n.


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

).


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).


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.


The Test Statistic

Example (Step 3)(3) The test statistic is

Z =p̂ − p0√p0(1−p0)

n

.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


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.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


The p-Value

Example (Step 5)(5) p-value = normalcdf(1.265,E99) = 0.1029.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.”


The Decision

Example (Step 6)(6) Accept H0.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


The Conclusion

Example (Step 7)(7) The proportion of male births is equal to 50%.


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%.


Outline

1 Homework Review

2 Introduction



5 Assignment


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.


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%.


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.


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.


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.


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.


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