testing hypotheses about a population proportion lecture 31 sections 9.1 – 9.3 wed, mar 22, 2006

Testing Hypotheses Testing Hypotheses about a Population about a Population

ProportionProportionLecture 31Lecture 31

Sections 9.1 – 9.3Sections 9.1 – 9.3

Wed, Mar 22, 2006Wed, Mar 22, 2006

Discovering Discovering Characteristics of a Characteristics of a

PopulationPopulation Any question about a population must Any question about a population must first be described in terms of a first be described in terms of a population parameter.population parameter.

Then the question about that Then the question about that parameter generally falls into one of parameter generally falls into one of two categories.two categories. EstimationEstimation

What is the value of the parameter?What is the value of the parameter? Hypothesis testingHypothesis testing

Does the evidence support or refute a claim Does the evidence support or refute a claim about the value of the parameter?about the value of the parameter?

ExampleExample A standard assumption is that a newborn A standard assumption is that a newborn

baby is as likely to be a boy as to be a girl. baby is as likely to be a boy as to be a girl. However, some people believe that boys However, some people believe that boys are more likely. are more likely.

Suppose a random sample of 1000 live Suppose a random sample of 1000 live births shows that 520 are boys and 480 are births shows that 520 are boys and 480 are girls.girls.

Use the data to estimate the proportion of Use the data to estimate the proportion of male births.male births.

Does this evidence support the claim that a Does this evidence support the claim that a greater proportion of births are boys?greater proportion of births are boys?

Two Approaches for Two Approaches for Hypothesis TestingHypothesis Testing

Classical approach.Classical approach. Specify Specify .. Determine the Determine the critical valuecritical value and the and the

rejection regionrejection region.. See whether the statistic falls in the See whether the statistic falls in the

rejection region.rejection region. Report the decision.Report the decision.

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

Classical ApproachClassical Approach

H0


0z

c

H0


0z

c

Rejection RegionAcceptance Region

H0


0z

c


Reject

z

H0


0z

c


H0


0z

c


Accept

z

H0

pp-Value Approach-Value Approach

H0


0z

H0


0z


H0


0z


H0

z


0z


Reject

p-value <

H0

z


0z


H0


0z


H0

z


0z


p-value >

H0

Accept

z

The Steps of Testing a The Steps of Testing a Hypothesis (Hypothesis (pp-Value -Value

Approach)Approach) The basic steps areThe basic steps are 1. State the null and alternative 1. State the null and alternative

hypotheses.hypotheses. 2. State the significance level.2. State the significance level. 3. State the formula for the test statistic.3. State the formula for the test statistic. 4. Compute the value of the test statistic.4. Compute the value of the test statistic. 5. Compute the 5. Compute the pp-value.-value. 6. Make a decision.6. Make a decision. 7. State the conclusion.7. State the conclusion.

See page 566. (The above steps are See page 566. (The above steps are modified from what is in the book.)modified from what is in the book.)

Step 1: State the Null Step 1: State the Null and Alternative and Alternative

HypothesesHypotheses Let Let pp = proportion of live births that = proportion of live births that

are boys.are boys. The null and alternative hypotheses The null and alternative hypotheses

areare HH00: : pp = 0.50. = 0.50.

HH11: : pp > 0.50. > 0.50.

State the Null and State the Null and Alternative HypothesesAlternative Hypotheses

The null hypothesis should state a The null hypothesis should state a hypothetical value hypothetical value pp00 for the population for the population proportion.proportion. HH00: : pp = = pp00..

The alternative hypothesis must contradict The alternative hypothesis must contradict the null hypothesis in one of three ways:the null hypothesis in one of three ways: HH11: : pp < < pp00. (Direction of extreme is left.). (Direction of extreme is left.)

HH11: : pp > > pp00. (Direction of extreme is right.). (Direction of extreme is right.)

HH11: : pp pp00. (Direction of extreme is left and . (Direction of extreme is left and right.)right.)

Explaining the DataExplaining the Data

The observation is 520 males out of The observation is 520 males out of 1000 births, or 52%. That is, 1000 births, or 52%. That is, pp^̂ = = 0.52.0.52.

Since we observed 52%, not 50%, Since we observed 52%, not 50%, how do we explain the discrepancy?how do we explain the discrepancy? Chance, orChance, or The true proportion is not 50%, but The true proportion is not 50%, but

something larger, maybe 52%.something larger, maybe 52%.

Step 2: State the Step 2: State the Significance LevelSignificance Level

The significance level The significance level should be should be given in the problem.given in the problem.

If it isn’t, then use If it isn’t, then use = 0.05. = 0.05. In this example, we will use In this example, we will use = =

0.050.05..

The Sampling The Sampling Distribution of Distribution of pp^̂

To decide whether the sample To decide whether the sample evidence is significant, we will evidence is significant, we will compare the compare the pp-value to -value to ..

If we were using the classical If we were using the classical approach, we would use approach, we would use to find the to find the critical value(s).critical value(s).

is the probability that the value of is the probability that the value of the test statistic is at least as the test statistic is at least as extreme as the critical value(s), extreme as the critical value(s), if if the null hypothesis is truethe null hypothesis is true..


Therefore, when we compute the Therefore, when we compute the pp--value, we do it under the assumption value, we do it under the assumption that that HH00 is true, i.e., that is true, i.e., that pp = = pp00..


We know that the sampling We know that the sampling distribution of distribution of pp^̂ is normal with is normal with mean mean pp and standard deviation and standard deviation

Thus, we assume that Thus, we assume that pp^̂ has mean has mean pp00 and standard deviation:and standard deviation:

n

ppp

1ˆ

n

ppp

00ˆ

1

Step 3: The Test StatisticStep 3: The Test Statistic

Test statistic – The Test statistic – The zz-score of -score of pp^̂, , under the assumption that under the assumption that HH00 is true. is true.

Thus,Thus,

npp

pppZ

p

p

00

0

ˆ

ˆ

1

ˆˆ

Write this

The Test StatisticThe Test Statistic

In our example, we computeIn our example, we compute

Therefore, the test statistic isTherefore, the test statistic is

Now, to find the value of the test statistic, Now, to find the value of the test statistic, all we need to do is to collect the sample all we need to do is to collect the sample data and substitute the value of data and substitute the value of pp^̂..

.01581.0

1000

50.1)50(.ˆ

p

01581.0

50.0ˆ p

Z

Step 4: Compute the Test Step 4: Compute the Test StatisticStatistic

In the sample, In the sample, pp^̂ = 0.52. = 0.52. Thus,Thus,

265.101581.0

50.052.0

Z

Step 5: Compute the Step 5: Compute the pp--valuevalue

To compute the To compute the pp-value, we must first -value, we must first check whether it is a one-tailed or a two-check whether it is a one-tailed or a two-tailed test.tailed test.

We will compute the probability that We will compute the probability that ZZ would be at least as extreme as the value would be at least as extreme as the value of our test statistic.of our test statistic.

If the test is two-tailed, then we must If the test is two-tailed, then we must take into account take into account both tailsboth tails of the of the distribution to get the distribution to get the pp-value. (Double -value. (Double the value in one tail.)the value in one tail.)

Compute the Compute the pp-value-value

In this example, the test is one-In this example, the test is one-tailed, with the direction of extreme tailed, with the direction of extreme to the right.to the right.

So we computeSo we compute

pp-value =-value = PP((ZZ > 1.265) = 0.1029 > 1.265) = 0.1029..

Compute the Compute the pp-value-value

An alternative is to evaluateAn alternative is to evaluate

normalcdf(0.52, E99, 0.50, 0.01581)normalcdf(0.52, E99, 0.50, 0.01581)

on the TI-83.on the TI-83. It should give the same answer It should give the same answer

(except for round-off).(except for round-off).

Step 6: Make a DecisionStep 6: Make a Decision

Since the Since the pp-value is greater than -value is greater than , , our our decisiondecision is: is: Do not reject the null Do not reject the null hypothesis.hypothesis.

The decision is stated in statistical The decision is stated in statistical jargon.jargon.

Step 7: State the Step 7: State the ConclusionConclusion

State the State the conclusionconclusion in a sentence: in a sentence: The data do not support the claim, that The data do not support the claim, that

more than 50% of live births are male.more than 50% of live births are male. The conclusion must relate the The conclusion must relate the

decision to the context of the decision to the context of the problem. It should not use problem. It should not use statistical jargon.statistical jargon.

SummarySummary

1.1. HH00: : pp = 0.50 = 0.50HH11: : pp > 0.50 > 0.50

2.2. = 0.05. = 0.05.3.3. Test statistic: Test statistic:

4.4. zz = (0.52 – 0.50)/0.0158 = 1.26. = (0.52 – 0.50)/0.0158 = 1.26.5.5. pp-value = -value = PP((ZZ > 1.26) = 0.1038. > 1.26) = 0.1038.6.6. Do not reject Do not reject HH00..7.7. The data do not indicate that the The data do not indicate that the

proportion of boys is greater than the proportion of boys is greater than the proportion of girls among newborns.proportion of girls among newborns.

npp

pppZ

p

p

00

0

ˆ

ˆ

1

ˆˆ

SummarySummary

1.1. HH00: : pp = 0.50 = 0.50HH11: : pp > 0.50 > 0.50




npp

pppZ

p

p

00

0

ˆ

ˆ

1

ˆˆ

Before

collectingdata

SummarySummary

1.1. HH00: : pp = 0.50 = 0.50HH11: : pp > 0.50 > 0.50




npp

pppZ

p

p

00

0

ˆ

ˆ

1

ˆˆ

Before

collectingdata

Aftercollecting

data

Testing Hypotheses on Testing Hypotheses on the TI-83the TI-83

The TI-83 has special functions The TI-83 has special functions designed for hypothesis testing.designed for hypothesis testing.

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

A window with several items appears.A window with several items appears.


Enter the value of Enter the value of pp00. Press ENTER and . Press ENTER and the down arrow.the down arrow.

Enter the numerator Enter the numerator xx of of pp^̂. Press . Press ENTER and the down arrow. ENTER and the down arrow.

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

Select the type of alternative hypothesis. Select the type of alternative hypothesis. Press the down arrow.Press the down arrow.

Select Calculate. Press ENTER.Select Calculate. Press ENTER. (You may select Draw to see a picture.)(You may select Draw to see a picture.)


The display showsThe display shows The title “1-PropZTest”The title “1-PropZTest” The alternative hypothesis.The alternative hypothesis. The value of the test statistic The value of the test statistic ZZ.. The The pp-value.-value. The value of The value of pp^̂.. The sample size.The sample size.

We are interested in the We are interested in the pp-value.-value.

The Classical ApproachThe Classical Approach

The seven stepsThe seven steps 1. State the null and alternative 1. State the null and alternative

hypotheses.hypotheses. 2. State the significance level.2. State the significance level. 3.3. Write the formula for the test statistic.Write the formula for the test statistic. 4. State the decision rule.4. State the decision rule. 5. Compute the value of the test statistic.5. Compute the value of the test statistic. 6. State the decision.6. State the decision. 7. State the conclusion.7. State the conclusion.

(Do not compute the (Do not compute the pp-value.)-value.)





Beforecollecting

data





Beforecollecting

data

Aftercollecting

data

Example of the Classical Example of the Classical ApproachApproach

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

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

Step 1: State the hypotheses.Step 1: State the hypotheses. HH00: : pp = 0.50 = 0.50

HH11: : pp > 0.50 > 0.50


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

Step 3: Define the test statistic.Step 3: Define the test statistic.

n

pp

ppppZ

p )1(

ˆˆ

00

0

ˆ

0


Step 4: State the decision rule.Step 4: State the decision rule. Find the critical value.Find the critical value. On the standard scale, the value On the standard scale, the value zz00 = =

1.645 cuts off an upper tail of area 0.05.1.645 cuts off an upper tail of area 0.05. This is a normal percentile problem.This is a normal percentile problem. Use invNorm(0.95) on the TI-83 or use the Use invNorm(0.95) on the TI-83 or use the

table.table.

Therefore, we will reject Therefore, we will reject HH00 if if zz > 1.645. > 1.645.The decision rule


Step 5: Compute the value of the test Step 5: Compute the value of the test statistic.statistic.

.265.101581.0

50.052.0

.01581.01000

)50.0)(50.0(1 00

z

n

pp


Step 6: State the decision.Step 6: State the decision. Since Since zz < 1.645, our decision is to < 1.645, our decision is to

accept accept HH00..

Step 7: State the conclusion.Step 7: State the conclusion. Our conclusion is that the proportion of Our conclusion is that the proportion of

male births is the same as the male births is the same as the proportion of female births.proportion of female births.

testing hypotheses about a population proportion lecture 31 sections 9.1 – 9.3 wed, mar 22, 2006

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