6 - 1 © 2000 Prentice-Hall, Inc. A First Course in Business Statistics Inferences Based on a Single Sample: Tests of Hypothesis Chapter 6

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© 2000 Prentice-Hall, Inc.© 2000 Prentice-Hall, Inc.

A First Course in A First Course in Business StatisticsBusiness Statistics

Inferences Based on a Single Sample: Inferences Based on a Single Sample: Tests of HypothesisTests of Hypothesis

Chapter 6Chapter 6

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Learning ObjectivesLearning Objectives

1.1. Distinguish Types of HypothesesDistinguish Types of Hypotheses

2.2. Describe Hypothesis Testing ProcessDescribe Hypothesis Testing Process

3.3. Solve Hypothesis Testing Problems Solve Hypothesis Testing Problems Based on a Single Sample Based on a Single Sample

4.4. Explain p-Value ConceptExplain p-Value Concept

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What’s a What’s a Hypothesis?Hypothesis?

1.1. A Belief about a A Belief about a Population ParameterPopulation Parameter

Parameter Is Parameter Is PopulationPopulation Mean, Mean, Proportion, VarianceProportion, Variance

Must Be StatedMust Be StatedBefore AnalysisBefore Analysis

I believe the mean GPA I believe the mean GPA of this class is 3.5!of this class is 3.5!

© 1984-1994 T/Maker Co.

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Null HypothesisNull Hypothesis

1.1. What Is TestedWhat Is Tested

2.2. Has Serious Outcome If Incorrect Decision Has Serious Outcome If Incorrect Decision MadeMade

3.3. Always Has Equality Sign: Always Has Equality Sign: , , , or , or 4.4. Designated HDesignated H00 (Pronounced H-oh) (Pronounced H-oh)

5.5. Specified as HSpecified as H00: : Some Numeric Value Some Numeric Value Specified with = Sign Even if Specified with = Sign Even if , or , or Example, HExample, H00: : 3 3

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Alternative Alternative HypothesisHypothesis

1.1. Opposite of Null HypothesisOpposite of Null Hypothesis

2.2. Always Has Inequality Sign:Always Has Inequality Sign: ,,, or , or

3.3. Designated HDesignated Haa

4.4. Specified HSpecified Haa: : < Some Value < Some Value Example, HExample, Haa: : < 3 < 3

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Basic IdeaBasic Idea

Sample Mean = 50 Sample Mean = 50

HH00HH00

Sampling DistributionSampling Distribution

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It is unlikely It is unlikely that we would that we would get a sample get a sample mean of this mean of this value ...value ...

20202020HH00HH00

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... if in fact this were... if in fact this were the population mean the population mean

20202020HH00HH00

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... if in fact this were... if in fact this were the population mean the population mean

... therefore, ... therefore, we reject the we reject the hypothesis hypothesis

that that = 50.= 50.

20202020HH00HH00

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Level of SignificanceLevel of Significance

1.1. ProbabilityProbability

2.2. Defines Unlikely Values of Sample Defines Unlikely Values of Sample Statistic if Null Hypothesis Is TrueStatistic if Null Hypothesis Is True Called Rejection Region of Sampling Called Rejection Region of Sampling

DistributionDistribution

3.3. Designated Designated αα(alpha)(alpha) Typical Values Are .01, .05, .10Typical Values Are .01, .05, .10

4.4. Selected by Researcher at StartSelected by Researcher at Start

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Rejection RegionRejection Region( one-tail test)( one-tail test)

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Rejection RegionRejection Region(Two-tailed test)(Two-tailed test)

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Errors in Errors in Making DecisionMaking Decision

1.1. Type I ErrorType I Error Reject True Null HypothesisReject True Null Hypothesis Has Serious ConsequencesHas Serious Consequences Probability of Type I Error Is Probability of Type I Error Is (Alpha)(Alpha)

Called Level of SignificanceCalled Level of Significance

2.2. Type II ErrorType II Error Fail to Reject False Null HypothesisFail to Reject False Null Hypothesis Probability of Type II Error Is Probability of Type II Error Is (Beta)(Beta)

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Jury Trial H0 Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0

False

Innocent Correct ErrorDo NotReject

H0

1 - Type IIError

()

Guilty Error Correct RejectH0

Type IError ()

Power(1 - )

Jury Trial H0 Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0

False

Innocent Correct ErrorDo NotReject

H0

1 - Type IIError

()

Guilty Error Correct RejectH0

Type IError ()

Power(1 - )

Decision ResultsDecision Results

HH00: Innocent: Innocent

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& & Have an Have an Inverse RelationshipInverse Relationship

You can’t reduce both errors simultaneously!

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HH00 Testing Steps Testing Steps

1.1. State HState H00 and H and Haa

2.2. Determine Rejection Region (Critical Determine Rejection Region (Critical Value) Value)

3.3. Calculate Test StatisticCalculate Test Statistic

4.4. State Assumptions for TestState Assumptions for Test

5.5. Conclusion – Reject or Fail to RejectConclusion – Reject or Fail to Reject

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Overview of TestsOverview of Tests

Areas of Hypothesis Testing (Like Areas of Hypothesis Testing (Like Confidence Intervals)Confidence Intervals)

1.1. Test about Test about μμ σσ Known Known(z), large n(z), large n

2.2. Test about Test about μμ σσ Unknown Unknown(t), small n(t), small n

3.3. Test about pTest about p (large n)(large n)± 3s in interval (0-± 3s in interval (0-

1)1)

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Two-Tailed Z Test Two-Tailed Z Test for Mean (for Mean ( Known) Known)

1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by

Normal Distribution (Normal Distribution (nn 30) 30)

2.2. Alternative Hypothesis Has Alternative Hypothesis Has Sign Sign

3.3. Z-Test StatisticZ-Test Statistic

ZX X

n

x

x

Z

X X

n

x

x

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Two-Tailed Z TestTwo-Tailed Z Test Example Example

Does an average box of Does an average box of cereal contain cereal contain 368368 grams grams of cereal? A random of cereal? A random sample of sample of 2525 boxes boxes showedshowedX = 372.5X = 372.5. The . The company has specified company has specified to be to be 2525 grams. Test at grams. Test at the the .05.05 level. level. 368 gm.368 gm.

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Two-Tailed Z Test Two-Tailed Z Test Thinking ChallengeThinking Challenge

You’re a Q/C inspector. You want to You’re a Q/C inspector. You want to find out if a new machine is making find out if a new machine is making electrical cords to customer electrical cords to customer specification: specification: averageaverage breaking breaking strength of strength of 7070 lb. with lb. with = 3.5 = 3.5 lb. lb. You take a sample of You take a sample of 3636 cords & cords & compute a sample mean of compute a sample mean of 69.769.7 lb. lb. At the At the .05.05 level, is there evidence level, is there evidence that the machine is that the machine is notnot meeting the meeting the average breaking strength?average breaking strength?

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One-Tailed Z Test One-Tailed Z Test for Mean (for Mean ( Known) Known)

1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed If Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by

Normal Distribution (Normal Distribution (nn 30) 30)

2.2. Alternative Hypothesis Has Alternative Hypothesis Has or > Signor > Sign

3.3. Z-test StatisticZ-test Statistic

ZX X

n

x

x

Z

X X

n

x

x

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One-Tailed Z TestOne-Tailed Z Test Example Example

Does an average box of cereal Does an average box of cereal contain contain more thanmore than 368368 grams grams of cereal? A random sample of cereal? A random sample of of 2525 boxes showed boxes showedX = X = 372.5372.5. The company has . The company has specified specified to be to be 2525 grams. grams. Test at the Test at the .05.05 level. level.

368 gm.368 gm.

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Observed Significance Observed Significance Levels: p-ValuesLevels: p-Values

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p-Valuep-Value

1.1. Probability of Obtaining a Test Statistic Probability of Obtaining a Test Statistic More Extreme (More Extreme (or or than Actual than Actual Sample Value Given HSample Value Given H00 Is True Is True

2.2. Called Level of SignificanceCalled Level of Significance Smallest Value of Smallest Value of H H00 Can Be Rejected Can Be Rejected

3.3. Used to Make ________ DecisionUsed to Make ________ Decision If p-Value If p-Value , Do Not Reject H, Do Not Reject H00

If p-Value < If p-Value < , Reject H, Reject H00

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Two-Tailed Z Test Two-Tailed Z Test p-Value Example p-Value Example

Does an average box of Does an average box of cereal contain cereal contain 368368 grams grams of cereal? A random of cereal? A random sample of sample of 2525 boxes boxes showedshowedX = 372.5X = 372.5. The . The company has specified company has specified to be to be 2525 grams. Find the grams. Find the p-Value.p-Value. 368 gm.368 gm.

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One-Tailed Z Test One-Tailed Z Test p-Value Example p-Value Example

Does an average box of Does an average box of cereal contain cereal contain more thanmore than 368368 grams of cereal? A grams of cereal? A random sample of random sample of 2525 boxes showedboxes showedX = 372.5X = 372.5. . The company has The company has specified specified to be to be 2525 grams. Find the p-Value.grams. Find the p-Value. 368 gm.368 gm.

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t Test for Mean t Test for Mean (( Unknown) Unknown)

1.1. AssumptionsAssumptions Population Is Normally DistributedPopulation Is Normally Distributed If Not Normal, Only Slightly Skewed & If Not Normal, Only Slightly Skewed &

Large Sample (Large Sample (nn 30) Taken 30) Taken

2.2. Parametric Test ProcedureParametric Test Procedure

3.3. t Test Statistict Test Statistic

tXSn

tXSn

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Two-Tailed t TestTwo-Tailed t TestThinking ChallengeThinking Challenge

You work for the FTC. A You work for the FTC. A manufacturer of detergent manufacturer of detergent claims that the mean weight claims that the mean weight of detergent is of detergent is 3.253.25 lb. You lb. You take a random sample of take a random sample of 6464 containers. You calculate the containers. You calculate the sample average to be sample average to be 3.2383.238 lb. with a standard deviation lb. with a standard deviation of of .117.117 lb. At the lb. At the .01.01 level, is level, is the manufacturer correct?the manufacturer correct?

3.25 lb.3.25 lb.

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One-Tailed t TestOne-Tailed t Test Thinking Challenge Thinking Challenge

You’re a marketing analyst for You’re a marketing analyst for Wal-Mart. Wal-Mart had teddy Wal-Mart. Wal-Mart had teddy bears on sale last week. The bears on sale last week. The weekly sales ($ 00) of bears weekly sales ($ 00) of bears sold in sold in 1010 stores was: stores was: 8 11 0 8 11 0 4 7 8 10 5 8 34 7 8 10 5 8 3. . At the At the .05.05 level, is there level, is there evidence that the average bear evidence that the average bear sales per store is sales per store is moremore thanthan 5 5 ($ 00)?($ 00)?

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Z Test of ProportionZ Test of Proportion

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Qualitative DataQualitative Data

1.1. Qualitative Random Variables Yield Qualitative Random Variables Yield Responses That ClassifyResponses That Classify e.g., Gender (Male, Female)e.g., Gender (Male, Female)

2.2. Measurement Reflects # in CategoryMeasurement Reflects # in Category

3.3. Nominal or Ordinal ScaleNominal or Ordinal Scale

4.4. ExamplesExamples Do You Own Savings Bonds? Do You Own Savings Bonds? Do You Live On-Campus or Off-Campus?Do You Live On-Campus or Off-Campus?

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ProportionsProportions

1.1. Involve Qualitative VariablesInvolve Qualitative Variables

2.2. Fraction or % of Population in a CategoryFraction or % of Population in a Category

3.3. If Two Qualitative Outcomes, Binomial If Two Qualitative Outcomes, Binomial DistributionDistribution Possess or Don’t Possess CharacteristicPossess or Don’t Possess Characteristic

4.4. Sample Proportion (Sample Proportion (pp))

pxn

number of successes

sample sizep

xn

number of successes

sample size

^̂

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One-Sample Z Test One-Sample Z Test for Proportionfor Proportion

1.1. AssumptionsAssumptions Two Categorical OutcomesTwo Categorical Outcomes Population Follows Binomial DistributionPopulation Follows Binomial Distribution Normal Approximation Can Be UsedNormal Approximation Can Be Used

Does Not Contain 0 or nDoes Not Contain 0 or n

2.2. Z-test statistic for proportionZ-test statistic for proportion

Zp pp p

n

( )0

0 01Z

p pp p

n

( )0

0 01Hypothesized Hypothesized population proportionpopulation proportion

ˆ1ˆ3ˆ ppnpn ˆ1ˆ3ˆ ppnpn

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One-Proportion Z Test One-Proportion Z Test

Example Example The present packaging The present packaging system produces system produces 10%10% defective cereal boxes. defective cereal boxes. Using a new system, a Using a new system, a random sample of random sample of 200200 boxes hadboxes had1111 defects. defects. Does the new system Does the new system produce produce fewerfewer defects? defects? Test at the Test at the .05.05 level. level.

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One-Proportion Z One-Proportion Z Test Thinking Test Thinking

ChallengeChallengeYou’re an accounting manager. A You’re an accounting manager. A year-end audit showed year-end audit showed 4%4% of of transactions had errors. You transactions had errors. You implement new procedures. A implement new procedures. A random sample of random sample of 500500 transactions transactions had had 2525 errors. Has the errors. Has the proportionproportion of incorrect transactions of incorrect transactions changedchanged at the at the .05.05 levellevel? ?

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6 - 1 © 2000 Prentice-Hall, Inc. A First Course in Business Statistics Inferences Based on a Single Sample: Tests of Hypothesis Chapter 6