Making Decisions about a Population Mean with Confidence Lecture 35 Sections 10.1 – 10.2 Fri, Mar 31, 2006

Making Decisions Making Decisions about a Population about a Population

Mean with Mean with ConfidenceConfidenceLecture 35Lecture 35

Sections 10.1 – 10.2Sections 10.1 – 10.2

Fri, Mar 31, 2006Fri, Mar 31, 2006

IntroductionIntroduction

In Chapter 10 we will ask the same In Chapter 10 we will ask the same basic questions as in Chapter 9, basic questions as in Chapter 9, except they will concern the mean.except they will concern the mean. Find an estimate for the mean.Find an estimate for the mean. Test a hypothesis about the mean.Test a hypothesis about the mean.

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

Approach)Approach) 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. Give the test statistic, including the 3. Give the test statistic, including the

formula.formula. 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. State the decision. 6. State the decision. 7. State the conclusion.7. State the conclusion.

The HypothesesThe Hypotheses

The null and altenative hypotheses The null and altenative hypotheses will be statements concerning will be statements concerning ..

Null hypothesis.Null hypothesis. HH00: : = = 00..

Alternative hypothesis (choose one).Alternative hypothesis (choose one). HH11: : < < 00.. HH11: : > > 00.. HH11: : 00..

Level of SignificanceLevel of Significance

The level of significance is the same The level of significance is the same as before.as before.

If the value is not given, assume that If the value is not given, assume that = 0.05. = 0.05.

The Test StatisticThe Test Statistic

The choice of test statistic will The choice of test statistic will depend on the sample size and what depend on the sample size and what is known about the population.is known about the population.

If we assume that If we assume that is is knownknown for the for the population and that eitherpopulation and that either The sample size The sample size nn is at least 30, or is at least 30, or The population is normal,The population is normal,

Then the Central Limit Theorem for Then the Central Limit Theorem for Means will apply.Means will apply.

The Sampling The Sampling Distribution ofDistribution ofxx

If the If the population is population is normalnormal, then the , then the distribution ofdistribution ofxx is also normal, with is also normal, with mean mean 00 and standard deviation and standard deviation //nn..

Note that this assumes that Note that this assumes that is known. is known. See p. 615, the Sampling Distribution See p. 615, the Sampling Distribution

of the Sample Mean.of the Sample Mean.

., is 0

n

Nx


Therefore, the Therefore, the test statistic istest statistic is

It is It is exactlyexactly standard normal. standard normal.

n

xZ

/0


On the other hand, if On the other hand, if the the population is not population is not normalnormal, , but the sample size is but the sample size is at leastat least 30, 30,

then the distribution ofthen the distribution ofxx is is approximatelyapproximately normal, with mean normal, with mean 00 and and standard deviation standard deviation //nn..

Note that we are still assuming that Note that we are still assuming that is is known.known.

.,ely approximat is 0

n

Nx


Therefore, the Therefore, the test statistic is test statistic is

It It is is approximatelyapproximately standard normal. standard normal. The approximation is good enough The approximation is good enough

that we can use the normal tables.that we can use the normal tables.

n

xZ

/0

Decision TreeDecision Tree

Is known?yes no


Is known?yes no

Is the population normal?

yes no


Is known?yes no


yes no

n

XZ

/


Is known?yes no


yes no

n

XZ

/

Is n 30?

yes no


Is known?yes no


yes no

n

XZ

/

Is n 30?

yes no

n

XZ

/


Is known?yes no


yes no

n

XZ

/

Is n 30?

yes no

n

XZ

/

Giveup


Is known?yes no


yes no

n

XZ

/

Is n 30?

yes no

n

XZ

/

Giveup

TBA

ExampleExample

See Example 10.1, p. 616 – Too See Example 10.1, p. 616 – Too Much Carbon Monoxide? (Much Carbon Monoxide? (zz-test with -test with known). known).

Hypothesis Testing on Hypothesis Testing on the TI-83the TI-83

Press Press STATSTAT.. Select Select TESTSTESTS. . Select Select Z-TestZ-Test. Press . Press ENTERENTER.. A window appears requesting A window appears requesting

information.information. Select Select DataData if you have the sample if you have the sample

data entered into a list.data entered into a list. Otherwise, select Otherwise, select StatsStats..


Assuming you selected Stats,Assuming you selected Stats, Enter Enter 00, the hypothetical mean., the hypothetical mean. Enter Enter . (Remember, . (Remember, is known.) is known.) EnterEnterxx.. Enter Enter nn, the sample size., the sample size. Select the type of alternative Select the type of alternative

hypothesis.hypothesis. Select Select CalculateCalculate and press and press ENTERENTER..


A window appears with the following A window appears with the following information.information. The title “Z-Test.”The title “Z-Test.” The alternative hypothesis.The alternative hypothesis. The value of the test statistic The value of the test statistic ZZ.. The The pp-value of the test.-value of the test. The sample mean.The sample mean. The sample size.The sample size.

ExampleExample

Re-do Example 10.1 on the TI-83 Re-do Example 10.1 on the TI-83 (using Stats).(using Stats).

The TI-83 reports thatThe TI-83 reports that zz = -2.575. = -2.575. pp-value = 0.005012.-value = 0.005012.


Suppose we had selected Suppose we had selected DataData instead of instead of StatsStats..

Then somewhat different information Then somewhat different information is requested.is requested. Enter the hypothetical mean.Enter the hypothetical mean. Enter Enter .. Identify the list that contains the data.Identify the list that contains the data. Skip Freq (it should be 1).Skip Freq (it should be 1). Select the alternative hypothesis.Select the alternative hypothesis. Select Select CalculateCalculate and press and press ENTERENTER..


Why enter Why enter if the TI-83 is capable of if the TI-83 is capable of computing the standard deviation computing the standard deviation from the data?from the data?

ExampleExample

Re-do Example 10.1 on the TI-83 Re-do Example 10.1 on the TI-83 (using Data).(using Data).

Enter the data in the chart on page Enter the data in the chart on page 616 into list L616 into list L11..

The TI-83 reports thatThe TI-83 reports that zz = -2.575. = -2.575. pp-value = 0.005012.-value = 0.005012. xx = 12.528. = 12.528. ss = 4.740 ( = 4.740 ( 4.8). 4.8).

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Making Decisions about a Population Mean with Confidence Lecture 35 Sections 10.1 – 10.2 Fri, Mar 31, 2006