Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Hypothesis Testing for Population Proportions

Copyright © 2014 Pearson Education, Inc. All rights reserved

Chapter 8

Hypothesis Testing

for Population

Proportions

8 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved

Learning Objectives

Know how to test hypotheses concerning a population proportion and hypotheses concerning the comparison of two population proportions.

Understand the meaning of p-value and how it is used.

Understand the meaning of significance level and how it is used.

Know the conditions required for calculating a p-value and significance level.


8.1

The Main Ingredients of Hypothesis Testing


Hypothesis Testing

How do you determine if someone is cheating when they toss a coin?

If the toss is fair P(Heads) = p = 0.5

Otherwise P(Heads) = p ≠ 0.5

If only 15 Heads come up out of 40 tosses, is the person cheating?


Null and Alternative Hypotheses

H0: p = 0.5 “H naught” or “the Null Hypothesis” The status quo, no cheating, no surprises, no change, no

effect

Ha: p ≠ 0.5 “The Alternative Hypothesis” This is what we hope or guess is true.

Note that p is the population proportion. Hypothesis testing is always for the population parameter, never the statistic.


Choosing the Null and Alternative Hypothesis

First come up with an idea or hypothesis. The dealer is palming aces. Babies are born less often on Tuesdays.

Write down the null hypothesis. H0: p = 1/4

H0: p = 1/7

Write down the alternative hypothesis. Ha: p > 1/4

Ha: p < 1/7


Level of Significance

The Level of Significance, , is the probability of rejecting H0 when H0 is true. Concluding that the coin tossing is not fair when

it is fair. Saying that the defendant is guilty but he did not

commit the crime. Concluding that the person has psychic abilities

when she was just guessing.


Typical Levels of Significance

Since the level of significance is the probability of making an error, it should be small. = 0.05 or 5% is the most typical. = 0.01 or 1% is used when making this error

has very bad repercussions. = 0.10 or 10% is used when an error is less an

issue than making no conclusion when Ha is true.


The Test Statistic

In conducting a hypothesis test one takes a sample and finds .

The Central Limit Theorem for proportions:

The Test Statistic is the z-score:

p̂

0 00

(1 ),

p pN p

n

0

0 0

ˆ

(1 )

p pObserved Nullz

SE p p

n


The Meaning of the Test Statistic

The test statistic tells us how unlikely that sample proportion could have happened by random chance had the null hypothesis been true.

If the null hypothesis is true, then the test statistic should be close to 0. Therefore, the farther the test statistic is from 0, the more the null hypothesis is discredited.


The p-Value

The p-value is a probability. Assuming the null hypothesis is true, the p-value is the probability that if the experiment were repeated, you would get a test statistic as extreme as or more extreme than the one you actually got. A small p-value suggests that a surprising outcome has occurred and discredits the null hypothesis.


p-Value Example A hypothesis test was conducted to see if 10% of the time that

the weather report states a 10% chance of rain that there is rain. Of 300 randomly selected days with a 10% chance of rain, 6% had rain. The p-value was found to be 0.02. Interpret this p-value.

If another 300 days with 10% chance of rain were randomly selected, then there is only a 2% chance that the number of days of rain would be less than 18 (6%) or more than 42 (14%).


Hypothesis Testing Summary

1. Hypothesize Come up with an idea or hypothesis. Write down H0 and Ha.

2. Prepare Choose the level of significance . Select the test statistic Check Sampling Distributions’ Conditions


Hypothesis Testing Summary (Continued)

3. Compute to Compare Compute the test statistic. Compute the p-Value.

4. Interpret If p-Value < , reject H0.

If p-Value > , fail to reject H0. State the conclusion in the context of the study.


8.2

Characterizing p-values


Three Types of Hypothesis Tests

Left Tailed Hypothesis H0: p = p0

Ha: p < p0

Right Tailed Hypothesis H0: p = p0

Ha: p > p0

Two Tailed Hypothesis H0: p = p0

Ha: p ≠ p0


Left Tailed Hypothesis

H0: p = p0

Ha: p < p0

The p-value represents the probability that if p = p0 and another random sample is taken with the same sample size then the new sample proportion will be less than observed sample proportion.


Right Tailed Hypothesis

H0: p = p0

Ha: p > p0

The p-value represents the probability that if p = p0 and another random sample is taken with the same sample size then the new sample proportion will be greater than observed sample proportion.


Two Tailed Hypothesis

H0: p = p0

Ha: p ≠ p0

The p-value represents the probability that if p = p0 and another random sample is taken with the same sample size then the new sample proportion will be farther from p0 than observed sample proportion.


p-Value Example

45% of Americans have type “O” blood. A study was done to see if the percent differs for college students. 47% of the 1000 randomly selected students had type O blood. The p-Value was 0.2.

Two tailed test: “differs” If 45% of college students do have type “O” blood

and if another 1000 randomly selected students are tested, then there is a 20% chance that either fewer than 43% or greater than 47% of this new group of students will have type “O” blood.


Requirements for a Valid p-Value

1. Random sample: The sample is collected randomly.

2. Large enough sample size: The sample size, n, is large enough that the sample has at least 10 expected successes and 10 expected failures np0 ≥10 and n(1 - p0) ≥ 10

3. Without replacement: If the sample is collected without replacement, then the population size is at least 10 times bigger than the sample size.

4. Independence: Each observation or measurement must have no influence on any others.

5. Null hypothesis : The null hypothesis is true.


StatCrunch and the p-Value

Have more than 70% of all dogs had their rabies shots? 592 of the 800 dogs examined had their rabies shots.

H0: p = 0.7 Ha: p > 0.7 Stat→Proportions→One sample→with summary


StatCrunch and the p-Value

Have more than 70% of all dogs had their rabies shots? 592 of the 800 dogs examined had their rabies shots.

H0: p = 0.7 Ha: p > 0.7

If 70% of all dogs have had their rabies shots and if another 800 randomly selected dogs are examined, then there is a 0.68% chance that more than 592 of them will have had their rabies shots.


Summary of the p-Value

Tells us how surprising the sample data is if the null hypothesis is true.

A very small p-Value, less than 0.05 for example, indicates that the results that were obtained would be very surprising.

A larger p-Value, greater than 0.05 for example, indicates that the results that were obtained would not be very surprising.


8.3

Hypothesis Testing in Four Steps


Four Steps to Hypothesis Testing

1. Hypothesize: State your hypothesis about the population parameter.

2. Prepare: Choose a significance level and test statistic, check requirements, and state assumptions.

3. Compute to Compare: Compute the test statistic and find the p-value to measure the surprise.

4. Interpret: Reject or fail to reject the null hypothesis. State the conclusions in the context of the study.


Hypothesize

Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers. H0: p = 0.05

Ha: p < 0.05


Hypothesize

90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year. H0: p = 0.90

Ha: p ≠ 0.90


Prepare Each year 5% of the world tiger population is killed

by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.

Use the standard = 0.05. np0 = (300)(0.05) = 15 ≥ 10

n(1-p0) = (300)(0.95) = 285 ≥ 10 Can use the z-statistic. Assume that the 300 tigers were randomly selected

and that there are more than 3000 tigers in the world.


Prepare 90% of all restaurants fail after one year. Is that

number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.

Use the standard = 0.05. np0 = (108)(0.9) = 97.2 ≥ 10

n(1-p0) = (108)(0.1) = 10.8 ≥ 10 Can use the z-statistic. Assume that the 108 Chinese restaurants were

randomly selected and that there are more than 1080 Chinese restaurants in existence.


Compute to Compare

Each year 5% of the world tiger population is killed by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.

z ≈ 0.795 p-value ≈ 0.79 If 5% of the tigers are killed by poaching and if

another sample is observed, there is a 79% chance of getting a sample proportion less than 6%. 79% is very high.

ˆ 0.06p


Compute to Compare

90% of all restaurants fail after one year. Is that number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.

z ≈ -3.27 p-value ≈ 0.001 If 90% of all Chinese restaurants fail after one year

and if another sample is observed, there is a 0.1% chance of getting a sample proportion less than 81%. 0.1% is a very small chance.

ˆ 0.81p


Rules to Interpret

1. If p-value < , reject H0 and accept Ha and state the conclusion.

2. If p-value > , fail to reject H0 and just state that we do not have statistically significant evidence to make a conclusion.


Interpret Each year 5% of the world tiger population is killed

by poachers. Now that there is a campaign to educate people on the problem, has this percent gone down? 300 tigers were observed and 18 were killed by poachers.

z ≈ 0.795, p-value ≈ 0.79, Since p-value ≈ 0.79 > 0.05 = , fail to reject H0.

There is statistically insignificant evidence to support the claim that since the campaign began, less than 5% of the world tiger population is killed by poachers.

ˆ 0.06p


Interpret 90% of all restaurants fail after one year. Is that

number different for Chinese restaurants. Of the 108 new Chinese restaurants 87 had failed after one year.

z ≈ -3.27, p-value ≈ 0.001, Since p-value ≈ 0.001 < 0.05 = , reject H0 and

accept Ha. There is statistically significant evidence to support the claim that the percent of Chinese restaurants that fail after one year is different from 90%.

ˆ 0.81p


8.4

Comparing Proportions from Two Populations


Comparing Two Populations

Comparing two groups, men vs. women, blacks vs. whites, with treatment vs. without treatment, etc.

Instead of the sample size, n, there will be two sample sizes, n1 and n2.

p1 and p2 instead of a single population proportion, p.

and instead of the sample proportion, 1p̂ 2p̂ p̂


Hypothesis Tests

The Null Hypothesis will be: H0: p1 = p2

Three possibilities for the Alternative Hypothesis: Ha: p1 < p2 (Left Tailed Test)

Ha: p1 > p2 (Right Tailed Test)

Ha: p1 ≠ p2 (Two Tailed Test)


The Test Statistic

The estimator is just the difference between the sample proportions:

The null value is typically 0.

StatCrunch is easier to use than the formula.

estimator null valuez

SE

1 2ˆ ˆp p

1 2

1 1 Total # of Successesˆ ˆ ˆ1 ,

Total # of TrialsSE p p p

n n


Checking Conditions

Large Sample: Note the use of the pooled proportion.

Random Sample or at least close to random. The two samples are independent of each other. Independent Within Samples: The observations

within each sample must be independent of one another.

The Null Hypothesis is true.

1 1 2 2ˆ ˆ ˆ ˆ10, 1 10, 10, 1 10n p n p n p n p


Using StatCrunch Stat→Proportions→Two sample→with summary


Caffeine Therapy For Premature Infants

Apnea of prematurity occurs when premature babies have shallow breathing or stopbreathing for more than 20 seconds.

Treatment group received caffeine therapy while the other group received a placebo.


Does caffeine therapy lower the rate of bad events? Of the 937 infants given the therapy, 377 suffered from death or disability. The placebo group had 932 infants, and of these, 431 suffered from death or disability.

1. Hypothesize H0: p1 = p2, Ha: p1 < p2

2. Prepare Use = 0.05. z-statistic 937 x 0.43, 937 x 0.57, 932 x 0.43, 932 x 0.57

are all greater than or equal to 10. Assume that the infants were randomly and independently selected.


Does caffeine therapy lower the rate of bad events? Of the 937 infants given the therapy, 377 suffered from death or disability. The placebo group had 932 infants, and of these, 431 suffered from death or disability.

3. Compute to Compare

StatCrunch → z = -2.62, p-value = 0.004

4. Interpret p-value = 0.004 < 0.05 = Reject H0, Accept Ha. There is statistically significant

evidence to conclude that a lower proportions of babies will die or suffer with this therapy than without this therapy.

377 431ˆ 0.43

937 932p


Is there a difference between the proportions of men and women who play guitar? 82 of the 400 men and 54 of the 300 women surveyed played guitar.

Hypothesize H0: p1 = p2, Ha: p1 ≠ p2

Prepare Use = 0.05. Large Sample?

400 x 0.19, 400 x 0.81, 300 x 0.19, 300 x 0.81are all greater than or equal to 10.

Assume that the men and women were randomly and independently selected.


Is there a difference between the proportions of men and women who play guitar? 82 of the 400 men and 54 of the 300 women surveyed played guitar.

Compute to Compare StatCrunch → z = 0.83, p-value = 0.41

Interpret p-value = 0.41 > 0.05 = Fail to Reject H0. There is statistically insufficient

evidence to make a conclusion about there being a difference between the proportions of men and women who own a guitar.

82 54ˆ 0.19

400 300p


8.5

Understanding Hypothesis Testing


What to do if Conditions Fail

The sample size is too small Redo the study with a larger sample size. Use an advanced test, e.g. Fisher’s Exact Test.

The samples are not independent Take an advance statistics class or Consult a statistician.

Samples not random State the conclusion for the sample only.


Power

The Power of a hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false.

Example: If the alternative hypothesis is that the person cheated, then the power is the probability that if the person is cheating then we will conclude correctly that cheating occurred.

Always strive for a large power.


Hypotheses, Power, and Level of Significance

Before the data is collected you can choose to make it more likely to reject H0 or more likely to fail to reject H0, but not both.

Both choices lead to an increased chance of one of the “Bad” outcomes.

Only increasing the sample size decreases the chance of both “Bad” outcomes.


Power vs. Level of Significance

Increasing the level of significance decreases the power.

If you want a better chance of rejecting H0 when H0 is false, you can decrease the level of significance.

The only way to have a large power and a large level of significance is to have a large sample size.


Statistical Significance vs. Practical Significance

The term Statistical Significance is used when the null hypothesis is rejected. There is a small probability that if H0 is correct then results as extreme as were obtained would happen randomly.

The term Practical Significance means that the results obtained are clearly far from the hypothesized value.


Statistical Significance and Practical Significance

A poll to see if the incumbent will get more than 50% of the votes results resulted in a sample proportion 58%. = 0.05 and the p-value = 0.06. This is practically significant but statistically

insignificant. A study to see if there is a difference in pass rates

for male vs. female statistics students resulted in a sample proportion of men: 72.1% and women: 72.3%. = 0.05 and the p-value = 0.04. This is practically insignificant but statistically

significant.


Hypotheses Should Never Be Changed After the Data is Analyzed

The null and alternative hypotheses must be written down before the data is analyzed.

Never adjust H0 and Ha to fit your results. If you change your mind based on the data,

you must collect a new data set in order to support the adjusted hypotheses.


Using Proper Language

If the p-value is less than the level of significance, , then state that we “reject H0” and “accept Ha”. Do not state that the alternative hypothesis is proven or true.

If the p-value is greater than then state that we “fail to reject H0” and that no conclusion can be made. Do not state that H0 is accepted or true. It is possible that the power is too small and that H0 is still false.


Conclude Only the Inequality

A two tailed test was used to see whether there is a difference between the proportion of genetically modified seeds that germinate and natural seeds that germinate. The sample proportions were 0.68 and 0.43 and H0 was rejected.

State that there is evidence for a difference in the population proportions. Do not state that for all seeds, the ones that are genetically modified are 25% more likely to germinate.


Chapter 8

Case Study


TV Violence Leads to Actual Violence?

A study was done to see if children who view violent TV shows are more likely to become violent towards their spouses when they grow up.

1. Hypothesize H0: p1 = p2

Ha: p1 > p2

2. Prepare = 0.05


TV Violence Leads to Actual Violence?


Checking Conditions

We will assume that the participants were selected randomly and are independent from each other.

The population sizes are certainly large compared to the sample sizes.

66ˆ 0.25 :

32966 0.25,66 .75,263 0.25,263 .75 10

p


3. Compute to Compare


4. Interpret Since p-value = 0.0033 < 0.05 = , we reject the

null hypothesis and accept the alternative hypotheses.

There is statistically significant evidence to conclude that children who grow up watching violent TV are more likely to become violent towards their spouses when they grow up.

We cannot conclude based solely on the data that watching violent TV as children causes adults to become violent towards their spouses.


Chapter 8

Guided Exercise 1


Gun Control

Historically, the percentage of U.S. residents who support stricter gun control laws has been 52%. A recent Gallup Poll of 1011 people showed 495 in favor of stricter gun control laws . Assume the poll was given to a random sample of people.

QUESTION Test the claim that the proportion of those favoring stricter gun control has changed from 0.52. Perform a hypothesis test, using a significance level of 0.05.


Hypothesize

Test the claim that the proportion of those favoring stricter gun control has changed from 0.52. H0: The population proportion that supports gun

control is 0.52, p = 0.52 Ha: p ≠ 0.52


Prepare

Choose the one-proportion z-test. Random sample: yes Sample size:

np0 = 1011(0.52) ≈ 526 ≥ 10

n(1 - p0) = 1011(0.48) ≈ 526 ≥ 10

Population size is more than 10 times 1011.


Compute and Compare


Interpret

Reject H0 (if the p-value is 0.05 or less) or do not reject H0 and choose one of the following conclusions:

1. The percentage is not significantly different from 52%. (A significant difference is one for which the p-value is less than or equal to 0.05.)

2. The percentage is significantly different from 52%.

Fail to reject H0. There is insufficient evidence to make a conclusion about the percentages being different.


Chapter 8

Guided Exercise 2


Perform a hypothesis test to test whether those who take Vioxx have a greater rate of heart attack than those who take a placebo. Use a level of significance of 0.05. Can we conclude that Vioxx causes an increased risk of heart attack?


The Study

In the fall of 2004, drug manufacturer Merck Pharmaceutical withdrew Vioxx, a drug that had been used for arthritis pain, from the market after a study revealed that its use was associated with an increase in the risk of heart attack. The experiment was placebo-controlled, randomized, and double-blind. Out of 1287 people taking Vioxx there were 45 heart attacks, and out of 1299 people taking the placebo there were 25 heart attacks (Source: Los Angeles Times , October 23, 2004).


Sample Proportions

Find the proportion of people in the sample taking Vioxx who had a heart attack and the proportion of people in the sample taking a placebo who had a heart attack. Compare these proportions.

There is a clear practical significance with 3.5% much higher than 1.9%.

1

45ˆ 0.035

1287p 2

25ˆ 0.019

1299p


Hypotheses

Let pV be the population proportion of those taking Vioxx who had a heart attack, and let pp be the population proportion of those taking the placebo who had a heart attack. H0: pV = pp

Ha: pV > pp


Prepare

Choose the two-proportion z -test. Although we don’t have a random sample, we have random assignment to groups. The pooled proportion of heart attacks is 45 25 70

ˆ 0.0271287 1299 2586

p

1 ˆ 1287 0.027 34.7 10n p

1 ˆ1 1287 0.973 10n p

2 ˆ 1299 0.027 10n p

2 ˆ1 1299 0.973 10n p


Compute and Compare

z ≈ 2.46 p-value = 0.0069


Interpret

Reject or do not reject H0 and choose i or ii: Those taking Vioxx did not have a significantly different

rate of heart attack than those taking the placebo. Those who took Vioxx had a significantly higher heart

attack rate than those who took the placebo.

Since p-value = 0.0069, we reject H0 and accept Ha. There is statistically significant evidence to support the claim that Vioxx users have a higher heart attack rate compared to non-Vioxx users.

Note that this does not mean that Vioxx causes heart attacks.

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Copyright © 2014 Pearson Education, Inc. All rights reserved Chapter 8 Hypothesis Testing for Population Proportions