Testing Hypotheses About Proportions

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Testing Hypotheses

About Proportions

Chapter 11 Hypothesis testing method:

uses data from a sample to judge whether or not a statement about a population may be true.

Steps in Any Hypothesis Test

1.  Determine the null and alternative hypotheses.

2.  Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic.

3.  Assuming the null hypothesis is true, find the p-value.

4.  Decide whether or not the result is statistically significant based on the p-value.

5.  Report the conclusion in the context of the situation.

11.1 Formulating

Hypothesis Statements

•  Does a majority of the population favor a new legal standard for the blood alcohol level that constitutes drunk driving?

Hypothesis 1: The population proportion favoring the new standard is not a majority.

Hypothesis 2: The population proportion favoring the new standard is a

More on Formulating Hypotheses

•  Do female students study, on average,

more than male students do?

Hypothesis 1: On average, women do not study more than men do.

Hypothesis 2: On average, women do study more than men do.

Terminology for the Two Choices

Null hypothesis: Represented by H₀, is a statement that there is nothing happening. Generally thought of as the status quo, or no relationship, or no difference. Usually the researcher hopes to disprove or reject the null hypothesis.

Alternative hypothesis: Represented by H_a, is a statement that something is happening. Generally it is what the researcher hopes to prove. It may be a statement that the assumed status quo is false, or that there is a relationship, or that there is a difference.

Examples of H

₀

and H

_a

Null hypothesis examples:

•  There is no extrasensory perception.

•  There is no difference between the mean pulse rates of men and women.

•  There is no relationship between exercise intensity and the resulting aerobic benefit.

Alternative hypothesis examples:

•  There is extrasensory perception.

•  Men have lower mean pulse rates than women do.

•  Increasing exercise intensity increases the resulting aerobic benefit.

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Example 11.1 Are Side Effects

Experienced by Fewer than 20% of Patients?

Pharmaceutical company wants to claim that the proportion of patients who experience side effects is less than 20%.

Null: 20% (or more) of users will experience side effects.

Alternative: Fewer than 20% of users will experience side effects.

Notice that the claim that the company hopes to prove is used as the alternative hypothesis.

The alternative is one-sided.

Example 11.2 Does a Majority Favor the Proposed Blood Alcohol Limit?

Legislator’s plan is to vote for the proposal if there is conclusive evidence that a majority of her constituents favor the proposal.

H₀: p ≤ .5 (not a majority) H_a: p > .5 (a majority)

Note: p = the proportion of her constituents that favors the proposal.

The alternative is one-sided.

11.2 Logic of Hypothesis Testing What if the Null is True?

Similar to “presumed innocent until proven guilty” logic.

We assume the null hypothesis is a possible truth until the sample data conclusively demonstrate otherwise.

The Probability Question

on Which Hypothesis Testing is Based If the null hypothesis is true about the population, what is the probability of observing sample data like that observed?

Example 11.3 Psychic Powers

Cartoon: Two characters playing a coin-flipping game.

Character 1: correctly guesses outcome of 100 flips.

Character 2: “just a coincidence”

Null: Character 1 does not have Psychic Powers (is just guessing)

Alternative: Character 1 has Psychic Powers Q: If character only guessing, how likely is correctly

guessing 100 consecutive fair coin tosses?

A: (½)¹⁰⁰ => extraordinarily small.

We reject the null hypothesis because the sample results are extremely inconsistent with it.

We conclude character was using psychic powers.

11.3 Reaching a Conclusion About the Two Hypotheses

•  Data summary used to evaluate the two hypotheses is called the test statistic.

•  Likelihood of observing a test statistic as extreme as what we did, or something even more extreme, if the null hypothesis is true is called the p-value.

•  Decision: reject H

₀

if the p-value is smaller than a designated level of significance, denoted by α

(usually 0.05, sometimes 0.10 or 0.01). In this

case the result is statistically significant.

Stating the Two Possible Conclusions

•  When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative hypothesis.

“Small” is defined as a p-value ≤ α, where α = level of significance (usually 0.05).

•  When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis.

“Not small” is defined as a p-value > α,where α = level of significance (usually 0.05).

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11.4 Testing Hypotheses About a Proportion

Possible null and alternative hypotheses:

1. H0: p = p0 versus Ha: p ≠ p0 (two-sided) 2. H₀: p ≥ p0 versus H_a: p < p₀ (one-sided) 3. H₀: p ≤ p0 versus H_a: p > p₀ (one-sided) p₀ = specific value called the null value.

Often H₀ for a one-sided test is written as H₀: p = p₀. Remember a p-value is computed assuming H₀ is true, and p0 is the value used for that computation.

•  Determine the sampling distribution of possible sample proportions when the true population proportion is p₀ (called the null value), the value specified in H₀.

•  Using properties of this sampling distribution, calculate a standardized score (z-score) for the observed sample proportion .

•  If the standardized score has a large magnitude, conclude that the sample proportion would be unlikely if the null value p₀ is true, and reject the null hypothesis.

The z-Test for a Proportion

1. The sample should be a random sample from the population.

Not always practical – most use test procedure as long as sample is representative of the population for the question of interest.

2.  The quantities np

₀

and n(1 – p

₀

) should both be at least 10.

A sample size requirement. Some authors say at least 5 instead of our conservative 10.

Conditions for Conducting the z-Test

Example 11.6 The Importance of Order

Survey of n = 190 college students.

About half (92) asked: “Randomly pick a letter - S or Q.”

Other half (98) asked: “Randomly pick a letter - Q or S.”

Is there a preference for picking the first?

Step 1: Determine the null and alternative hypotheses.

Let p = proportion of population that would pick first letter.

Null hypothesis: statement of “nothing happening.”

If no general preference for either first or second letter, p = .5 Alternative hypothesis: researcher’s belief or speculation.

A preference for first letter => p is greater than .5.

H₀: p = p₀ versus H_a: p > p₀ (one-sided)

Example 11.6 Importance of Order (cont)

Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic.

1. The sample should be a random sample from the population.

The sample is a convenience sample of students who were enrolled for a class. Does not seem this will bias results for this question, so will view the sample as a random sample.

2.  The quantities np₀ and n(1 – p₀) should both be at least 10.

With n = 190 and p₀= .5, both n p₀ and n(1 – p₀) equal 95, a quantity larger than 10, so the sample size condition is met.

Example 11.6 Importance of Order (cont)

Step 2: Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic.

Of 92 students asked “S or Q,” 61 picked S, the first choice.

Of 98 students asked “Q or S,” 53 picked Q, the first choice.

Overall: 114 students picked first choice => 114/190 = .60.

The sample proportion, .60, is used to compute the z-test statistic, the standardized score for measuring the difference between the = .60, and the null hypothesis value, p₀ = .50.

The z-statistic = 2.76 (formula comes later).

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Example 11.6 Importance of Order (cont)

Step 3: Assuming the null hypothesis is true,

find the p-value.

If the true p is .5, what is the probability that, for a sample of 190 people, the sample proportion could be as large as .60 (or larger)?

or equivalently

If the null hypothesis is true, what is the probability that the z-statistic could be as large as 2.76 (or larger)?

Using computer (or reading from print-out):

p-value = 0.003

Example 11.6 Importance of Order (cont)

Step 4: Decide whether or not the result is statistically significant based on the p-value.

Convention used by most researchers is to declare statistical significance when the

p-value is smaller than 0.05.

The p-value = 0.003 so the results are statistically significant

and we can reject the null hypothesis.

Example 11.6 Importance of Order (cont)

Step 5: Report the conclusion

in the context of the problem.

Statistical Conclusion =

Reject the null hypothesis that p = 0.50

Context Conclusion =

there is statistically significant evidence that the first letter presented is preferred.

The z-statistic for the significance test is

•  represents the sample estimate of the proportion

•  p₀ represents the specific value in null hypothesis

•  n is the sample size

Details for Calculating the z-Statistic

Example 11.1 Fewer than 20%? (cont)

Clinical Trial of n = 400 patients.

68 patients experienced side effects.

Can the company claim that fewer than 20%

will experience side effects?

.

Hypothesis testing steps:

Step 1: Determine the null and alternative hypotheses Step 2: Verify necessary data conditions, and if met,

summarize the data into an appropriate test statistic.

Step 3: Assuming the null hypothesis is true, find the p-value.

Step 4: Decide whether or not the result is statistically significant based on the p-value.

Step 5: Report the conclusion in the context of the problem.

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Using Minitab:

•  To test hypotheses about a proportion, use Stat>Basic Statistics>1 Proportion.

•  If the raw data are in a column of the worksheet, specify the column. If not, enter the

summarized data.

•  Click on Options, select confidence level, alternative, and check “Use test and …” box.

•  Check “Perform hypothesis test…” and enter p

₀

.

•  Click OK and read off results.

Example 11.7 Left and Right Foot Lengths

Sample: 112 of 215 college students with unequal right and left foot measurements.

Let p = population proportion with a longer right foot.

Are Left and Right Foot Lengths Equal or Different?

11.5 Role of Sample Size in Statistical Significance

Cautions about Sample Size and Statistical Significance

•  If a small to moderate effect in the population, a small sample has little chance of being statistically significant.

•  With a large sample, even a small and unimportant effect in the population may be statistically significance.

Example 11.8 Same Sample Proportion Can Produce Different Conclusions

Let p = proportion in population that would prefer Drink A.

H₀: p = .5 (no preference)

H_a: p ≠ .5 (preference for one or other) Taste Test: Sample of people taste both drinks and record how many like taste of Drink A better than B.

Results based on two sample sizes: n = 60 and n = 960 and the sample proportion for both is 0.55.

Example 11.8 Different Conclusions (cont)

Results when n = 60

33 of the 60 preferred Drink A;

Results when n = 960

528 or the 960 preferred Drink A;

The z-value changes because the

sample size affects the standard error.

Why more significant for larger n?

•  When n =60, the null standard error = .065.

•  When n = 960, the null standard error = .016.

Increasing n decreases null standard error =>

an absolute difference between the sample

proportion and null value is more significant

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11.6 Real Importance versus Statistical Significance

The p-value does not provide information about the magnitude of the effect.

The magnitude of a statistically significant effect can be so small that the practical effect is not

important.

If sample size large enough,

almost any null hypothesis can be rejected.

Example 11.9 Birth Month and Height

Austrian study of heights of 507,125 military recruits.

Men born in spring were, on average, about 0.6 cm taller than men born in fall (Weber et al., Nature, 1998, 391:754–755).

A small difference: 0.6 cm = about 1/4 inch.

Sample size so large that even a very small difference was statistically significant.

Headline: Spring Birthday Confers Height Advantage

Case Study 11.1 Internet and Loneliness

A closer look: actual effects were quite small.

“one hour a week on the Internet was associated, on average, with an increase of 0.03, or 1 percent on the depression scale” (Harman, 30 August 1998, p. A3).

“greater use of the Internet was associated with declines in participants’ communication with family members in the household, declines in size of their social circle, and increases in their depression and loneliness” (Kraut et al., 1998, p. 1017)

11.7 What Can Go Wrong?

A type 1 error can only occur when the null hypothesis is actually true. The error occurs by concluding that the alternative hypothesis is true.

A type 2 error can only occur when the alternative hypothesis is actually true.

The error occurs by concluding that the null hypothesis cannot be rejected.

Null hypothesis: You do not have the disease.

Alternative hypothesis: You do have the disease.

Type 1 Error: You are told you have the disease, but you actually don’t. The test result was a false positive.

Consequence: You will be unnecessarily concerned about your health and you may receive unnecessary treatment.

Type 2 Error : You are told that you do not have the disease, but you actually do. The test result was a false negative.

Consequence: You do not receive treatment for a disease that you have. If this is a contagious disease, you may infect others.

Testing Hypotheses About Proportions