Testing Hypotheses about a Population Proportion

Lecture 30

Sections 9.3

Wed, Oct 24, 2007

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ

Beforecollecting

data

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ

Aftercollecting

data

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ

Case Study 11

Male births vs. female births.

Testing Hypotheses on the TI-83 The TI-83 has special functions designed

for hypothesis testing. Press STAT. Select the TESTS menu. Select 1-PropZTest… Press ENTER.

A window with several items appears.

Testing Hypotheses on the TI-83 Enter the value of p0. Press ENTER and the

down arrow. Enter the numerator x of p^. Press ENTER and

the down arrow. Enter the sample size n. Press ENTER and the

down arrow. Select the type of alternative hypothesis. Press

the down arrow. Select Calculate. Press ENTER.

Testing Hypotheses on the TI-83 The display shows

The title “1-PropZTest”The alternative hypothesis.The value of the test statistic Z.The p-value.The value of p^.The sample size.

We are interested in the p-value.

Case Study 12

A recent study has shown that moderate exercise helps reduce the risk of catching a cold.

53 subjects were assigned to an exercise group that did moderate exercise.

62 subjects did only stretching exercises. In the first group, only 5 caught a cold. In the second group, 20 caught a cold.

Case Study 12

Use the TI-83 to test the hypothesis that a person who gets moderate exercise has less than a 1 in 3 chance of catching a cold.

The p-Value Approach

p-Value approach.Compute the p-value of the statistic.Report the p-value. If is specified, then report the decision.

Two Approaches for Hypothesis Testing Classical approach.

Specify .Determine the critical value and the rejection

region.See whether the statistic falls in the rejection

region.Report the decision.

Classical Approach

H0

Classical Approach

H0

Classical Approach

0z

c

H0

Critical value

Classical Approach

0z

c

Rejection RegionAcceptance Region

H0

Classical Approach

0z

c

Rejection RegionAcceptance Region

H0

Classical Approach

0z

c

Rejection RegionAcceptance Region

Reject

z

H0

Classical Approach

0z

c

Rejection RegionAcceptance Region

H0

Classical Approach

0z

c

Rejection RegionAcceptance Region

Accept

z

H0

The Classical Approach

The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Write the formula for the test statistic. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion.

(Do not compute the p-value.)

The Classical Approach

The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Write the formula for the test statistic. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion.

(Do not compute the p-value.)

Beforecollecting

data

The Classical Approach

The seven steps 1. State the null and alternative hypotheses. 2. State the significance level. 3. Write the formula for the test statistic. 4. State the decision rule. 5. Compute the value of the test statistic. 6. State the decision. 7. State the conclusion.

(Do not compute the p-value.)

Aftercollecting

data

Example of the Classical Approach

Test the hypothesis that there are more male births than female births.

Let p = the proportion of live births that are male.

Step 1: State the hypotheses.H0: p = 0.50

H1: p > 0.50

Example of the Classical Approach

Step 2: State the significance level.Let = 0.05.

Step 3: Define the test statistic.

npp

ppppZ

p )1(

ˆˆ

00

0

ˆ

0

Example of the Classical Approach Step 4: State the decision rule.

Find the critical value.On the standard scale, the value z0 = 1.645

cuts off an upper tail of area 0.05. This is a normal percentile problem. Use invNorm(0.95) on the TI-83 or use the table.

Therefore, we will reject H0 if z > 1.645.

The decision rule

Example of the Classical Approach Step 5: Compute the value of the test

statistic.

.265.101581.0

50.052.0

.01581.01000

)50.0)(50.0(1 00

z

n

pp

Example of the Classical Approach

Step 6: State the decision.Because z < 1.645, our decision is to accept H0.

Step 7: State the conclusion.The proportion of male births is not greater

than the proportion of female births.

Summary

H0: p = 0.50

H1: p > 0.50 = 0.05. Test statistic:

Reject H0 if z > 1.645.

Accept H0. The proportion of male births is the same as the

proportion of female births.

npp

ppppZ

p )1(

ˆˆ

00

0

ˆ

0

.265.101581.0

50.052.0

z

Case Study 12

Use the TI-83 and the classical approach to test the hypothesis that a person who gets moderate exercise has less than a 1 in 3 chance of catching a cold.