Objective: To test claims about inferences for proportions, under

specific conditions

7.5: HYPOTHESES TESTS FOR PROPORTIONS

• The hypotheses for proportions are similar to those for means.

• In fact, they are the same. We use p instead of µ

H0: p= 0.20

HYPOTHESES

There are four basic parts to a hypothesis test:

1. Hypotheses

2. Model

3. Mechanics

4. Conclusion

Let’s look at these parts in detail…

REASONING OF HYPOTHESIS TESTING

1. Hypotheses

• The null hypothesis: To perform a hypothesis test, we must first translate our question of interest into a statement about model parameters.

• In general, we have

H0: parameter = hypothesized value.

• The alternative hypothesis: The alternative hypothesis, HA, contains the values of the parameter we consider plausible when we reject the null.

REASONING OF HYPOTHESIS TESTING (CONT.)

1. Hypotheses Example:A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. A city’s fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio and TV and in the newspaper. The city wonders if this concerted effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence that the local rate is higher than the national rate? Set up the hypotheses.

REASONING OF HYPOTHESIS TESTING (CONT.)

2. Model• To plan a statistical hypothesis test, specify the model you will

use to test the null hypothesis and the parameter of interest.

• All models require assumptions, so state the assumptions and check any corresponding conditions.

• Your conditions should conclude with a statement such as:

• Because the conditions are satisfied, I can model the sampling distribution of the proportion with a Normal model.

• Watch out, though. It might be the case that your model step ends with “Because the conditions are not satisfied, I can’t proceed with the test.” If that’s the case, stop and reconsider (proceed with caution)

REASONING OF HYPOTHESIS TESTING (CONT.)

2. Model

• Don’t forget to name your test!

• The test about proportions is called a one-proportion z-test.

REASONING OF HYPOTHESIS TESTING (CONT.)

2. Model (cont.)• The conditions for the one-proportion z-test are the

same as for the one proportion z-interval. We test the hypothesis H0: p = p0

• using the statistic

• where

• When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value.

REASONING OF HYPOTHESIS TESTING (CONT.)

z p̂ p0 SD p̂

SD p̂ p0q0n

2. Model (cont.)

Finding the P-ValueEither use the table provided, or you may use your calculator:

• normalcdf( Lower z, upper z)

• OR use the test feature in the calculator

REASONING OF HYPOTHESIS TESTING (CONT.)

3. Mechanics

• Under “mechanics” we place the actual calculation of our test statistic from the data.

• Different tests will have different formulas and different test statistics.

• Usually, the mechanics are handled by a statistics program or calculator, but it’s good to know the formulas.

REASONING OF HYPOTHESIS TESTING (CONT.)

3. Mechanics (continued)

• The ultimate goal of the calculation is to obtain a P-value.

• The P-value is the probability that the observed statistic value (or an even more extreme value) could occur if the null model were correct.

• If the P-value is small enough, we’ll reject the null hypothesis.

• Note: The P-value is a conditional probability—it’s the probability that the observed results could have happened if the null hypothesis is true.

REASONING OF HYPOTHESIS TESTING (CONT.)

4. Conclusion• The conclusion in a hypothesis test is always a

statement about the null hypothesis.

• The conclusion must state either that we reject or that we fail to reject the null hypothesis.

• And, as always, the conclusion should be stated in context.

REASONING OF HYPOTHESIS TESTING (CONT.)

1. Check Conditions and show that you have checked these!

• Random Sample: Can we assume this?

• 10% Condition: Do you believe that your sample size is less than 10% of the population size?

• Success/Failure:

and 2. State the test you are about to conduct

Ex) One-proportion z-test3. Set up your hypotheses

H0:

HA:

STEPS FOR HYPOTHESIS TESTING

4. Calculate your test statistic

5. Draw a picture of your desired area under the t-

model, and calculate your P-value.

6. Make your conclusion.When your P-value is small enough (or below α, if

given), reject the null hypothesis.

When your P-value is not small enough, fail to reject the null hypothesis.

STEPS FOR HYPOTHESIS TESTING (CONT.)

One Proportion Z-Test:

• Stat TESTS

• 5: 1-Prop ZTest

• Po = hypothesized proportion

• x = number of successes

• n = sample size

• Determine the tail

• Calculate

CALCULATOR TIPS

A large city’s DMV claimed that 80% of candidates pass driving tests, but a survey of 90 randomly selected local teens who had taken the test found only 61 who passed. Does this finding suggest that the passing rate for teenagers is lower than the DMV reported?

EXAMPLE 1

Home field advantage –teams tend to win more often when the play at home. Or do they? If there were no home field advantage, the home teams would win about half of all games played. In the 2007 Major League Baseball season, there were 2431 regular-season games. It turns out that the home team won 1319 of the 2431 games, or 54.26% of the time. Could this deviation from 50% be explained from natural sampling variability, or is it evidence to suggest that there really is a home field advantage, at least in professional baseball?

 

 

 

 

 

EXAMPLE 2 (PARTNERS)

• Day 1: 7.5-Set A Book Page # 1 – 3, 9, 11

• Day 2: 7.5-Set A Book Page # 4, 5, 12, 18

• Day 3: 7.5-Set A Book Page # 16, 22

7.5-Set B Book Page (Same as 6.5 B Book Page) # 6, 15, 17, 19

ASSIGNMENTS