Statistics final exam, 40 multiple choice questions.

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ha TwoTwo--Sample Hypothesis TestsSample Hypothesis Tests

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Chapter ContentsChapter Contents

10.1 Two10.1 Two--Sample TestsSample Tests 10.2 Comparing Two Means: Independent Samples10.2 Comparing Two Means: Independent Samples 10.3 Confidence Interval for the Difference of Two Means, 10.3 Confidence Interval for the Difference of Two Means, 11 -- 22 10.4 Comparing Two Means: Paired Samples10.4 Comparing Two Means: Paired Samples 10.5 Comparing Two Proportions10.5 Comparing Two Proportions 10.6 Confidence Interval for the Difference of Two Proportions, 10.6 Confidence Interval for the Difference of Two Proportions, 11 -- 22 10.7 Comparing Two Variances10.7 Comparing Two Variances

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TwoTwo--Sample Hypothesis TestsSample Hypothesis Tests

Chapter Learning Objectives (LO’s)Chapter Learning Objectives (LO’s)

Chapter Learning Objectives (LO s)Chapter Learning Objectives (LO s)

LO10LO10 1:1: R i d f t t f t ith kR i d f t t f t ith kLO10LO10--1: 1: Recognize and perform a test for two means with known Recognize and perform a test for two means with known 11 and and 2.2.

LO10LO10 22LO10LO10--2: 2: Recognize and perform a test for two means with unknown Recognize and perform a test for two means with unknown 11 and and 2.2.

LO10LO10--3:3: Recognize paired data and be able to perform a paired Recognize paired data and be able to perform a paired t test.t test. LO10LO10--4: 4: Explain the assumptions underlying the twoExplain the assumptions underlying the two--sample test of sample test of

means. means.

LO10LO10--5:5: Perform a test to compare two proportions using Perform a test to compare two proportions using z.z.

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TwoTwo--Sample Hypothesis TestsSample Hypothesis Tests

Chapter Learning Objectives (LO’s)Chapter Learning Objectives (LO’s)

Chapter Learning Objectives (LO s)Chapter Learning Objectives (LO s)

LO10LO10--6: 6: Check whether normality may be assumed for two Check whether normality may be assumed for two proportions.proportions.

LO10LO10--7: 7: Use Excel to find Use Excel to find pp--values for twovalues for two--sample tests using sample tests using z z oror t.t. LO10LO10--8: 8: Carry out a test of two variances using the Carry out a test of two variances using the F F distribution.distribution.y gy g LO10LO10--99: Construct a confidence interval for Construct a confidence interval for μμ11− − μμ22 or or ππ11− − ππ22

((optional).optional).((optional).optional).

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10.1 Two10.1 Two--Sample TestsSample Tests

•• A TwoA Two--sample test compares two sample estimates with eachsample test compares two sample estimates with each What is a TwoWhat is a Two--Sample TestSample Test

•• A TwoA Two--sample test compares two sample estimates with each sample test compares two sample estimates with each other.other.

•• A oneA one--sample test compares a sample estimate to a nonsample test compares a sample estimate to a non--sample sample p p pp p p pp benchmark.benchmark.

Basis of TwoBasis of Two--Sample TestsSample TestsBasis of TwoBasis of Two Sample TestsSample Tests

• Two-sample tests are especially useful because they possess a built-in point of comparison.

•• The logic of twoThe logic of two--sample tests is based on the fact that two sample tests is based on the fact that two l d f th l ti i ld diff tl d f th l ti i ld diff tsamples drawn from the same population may yield different samples drawn from the same population may yield different

estimates of a parameter due to chance.estimates of a parameter due to chance.

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10.1 Two10.1 Two--Sample TestsSample Tests

•• If the two sample statistics differ by more than the amountIf the two sample statistics differ by more than the amount What is a TwoWhat is a Two--Sample TestSample Test

•• If the two sample statistics differ by more than the amount If the two sample statistics differ by more than the amount attributable to chance, then we conclude that the samples came attributable to chance, then we conclude that the samples came from populations with different parameter values.from populations with different parameter values.

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10.1 Two10.1 Two--Sample TestsSample Tests

Test ProcedureTest Procedure

•• State the hypothesesState the hypotheses •• Set up the decision ruleSet up the decision rule •• Insert the sample statisticsInsert the sample statistics •• Make a decision based on the critical values or using Make a decision based on the critical values or using pp--valuesvalues

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

LO10LO10--11 10

LO10LO10--1: 1: Recognize and perform a test for two means with known Recognize and perform a test for two means with known σσ11 andand σσ22

Format of HypothesesFormat of Hypotheses

σσ11 and and σσ22..

• The hypotheses for comparing two independent population

ypyp

yp p g p p p means µ1 and µ2 are:

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

LO10LO10--11

Case 1: Known VariancesCase 1: Known Variances

•• When the variances are known, use the normal distribution for the When the variances are known, use the normal distribution for the

Case 1: Known VariancesCase 1: Known Variances

,, test (assuming a normal population).test (assuming a normal population).

•• The test statistic is:The test statistic is:

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

LO10LO10--22 10

LO10LO10--2: 2: Recognize and perform a test for two means with unknown Recognize and perform a test for two means with unknown σσ11 andand σσ22

Case 2: Unknown Variances, Assumed EqualCase 2: Unknown Variances, Assumed Equal σσ11 and and σσ22..

•• Since the variances are unknown, they must be estimated Since the variances are unknown, they must be estimated and the Student’s and the Student’s tt distribution used to test the means.distribution used to test the means.

•• Assuming the population variances are equal, Assuming the population variances are equal, ss1122 and and ss2222 can be used to estimate a common pooled variance can be used to estimate a common pooled variance sspp22..

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

LO10LO10--22

Case 3: Unknown Variances, Assumed UnequalCase 3: Unknown Variances, Assumed Unequal

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

LO10LO10--22

Case 3: Unknown Variances, Assumed UnequalCase 3: Unknown Variances, Assumed Unequal

•• WelchWelch--Satterthwaite testSatterthwaite test

•• A Quick Rule for degrees of freedom is to use min(A Quick Rule for degrees of freedom is to use min(nn11 –– 1, 1, nn22 –– 1). 1).

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

If th l ti i 2 d 2 k th th

Summary for the Test StatisticSummary for the Test Statistic

• If the population variances 12 and 22 are known, then use the normal distribution.

• If population variances are unknown and estimated using s12 andIf population variances are unknown and estimated using s1 and s22, then use the Students t distribution.

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

Steps in Testing Two MeansSteps in Testing Two Means

• Step 1: State the hypotheses

• Step 2: Specify the decision rulep p y Choose  (the level of significance) and determine the critical value(s).

• Step 3: Calculate the Test Statistic

•• Step 4Step 4: : Make the decision Reject Make the decision Reject HH00 if the test statistic falls in the if the test statistic falls in the pp jj 00 rejection region(s) as defined by the critical value(s).rejection region(s) as defined by the critical value(s).

• Step 5: : Take action based on the decision. p

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

If th l i l thIf th l i l th C 2C 2 dd C 3C 3 t tt t

Which Assumption Is Best?Which Assumption Is Best?

•• If the sample sizes are equal, the If the sample sizes are equal, the Case 2Case 2 and and Case 3Case 3 test test statistics will be identical, although the degrees of freedom may statistics will be identical, although the degrees of freedom may differ.differ.

•• If the variances are similar, the two tests will usually agree.If the variances are similar, the two tests will usually agree. •• If no information about the population variances is available, then If no information about the population variances is available, then p p ,p p ,

the best choice is the best choice is Case 3Case 3.. •• The fewer assumptions, the better.The fewer assumptions, the better.

Must Sample Sizes Be Equal?Must Sample Sizes Be Equal? •• Unequal sample sizes are common and the formulas still applyUnequal sample sizes are common and the formulas still apply•• Unequal sample sizes are common and the formulas still apply.Unequal sample sizes are common and the formulas still apply.

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

Large SamplesLarge Samples

•• For unknown variances, if both samples are large (For unknown variances, if both samples are large (nn11  30 and 30 and nn22  30) and the population is not badly skewed, use the following 30) and the population is not badly skewed, use the following formula with appendix C.formula with appendix C.pppp

Caution: Three IssuesCaution: Three IssuesCaution: Three IssuesCaution: Three Issues

1.1. Are the populations skewed? Are there outliers? Are the populations skewed? Are there outliers?

Check using histograms and/or dot plots of each sample. Check using histograms and/or dot plots of each sample. tt tests are OK if moderately skewed, especially if samples are tests are OK if moderately skewed, especially if samples are

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y , p y py , p y p large. Outliers are more serious.large. Outliers are more serious.

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10.2 Comparing Two Means: Independent 10.2 Comparing Two Means: Independent SamplesSamples

Caution: Three IssuesCaution: Three Issues 22 Are the sample sizes largeAre the sample sizes large (n(n  30)?30)?

2.2. Are the sample sizes large Are the sample sizes large (n(n  30)?30)?

If samples are small, the mean is not a reliable indicator of central If samples are small, the mean is not a reliable indicator of central tendency and the test may lack powertendency and the test may lack powertendency and the test may lack power.tendency and the test may lack power.

3.3. Is the difference Is the difference important important as well as significant?as well as significant?

A ll diff i ti ld b i ifi t ifA ll diff i ti ld b i ifi t ifA small difference in means or proportions could be significant if A small difference in means or proportions could be significant if the sample size is large.the sample size is large.

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10.3 Confidence Interval for the Difference of 10.3 Confidence Interval for the Difference of Two Means Two Means 11 -- 22

LO10LO10--99 10

LO10LO10--9: 9: Construct a confidence interval for Construct a confidence interval for 11 − − 22 or or 11 -- 22 ((optional)optional)

Confidence Intervals for the Difference of Two Means

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10.3 Confidence Interval for the Difference of 10.3 Confidence Interval for the Difference of Two Means Two Means 11 -- 22

LO10LO10--99 10

LO10LO10--9: 9: Construct a confidence interval for Construct a confidence interval for 11 − − 22 or or 11 -- 22 ((optional)optional)

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10 3 C fid I t l f th Diff f10 3 C fid I t l f th Diff f pter

10.3 Confidence Interval for the Difference of 10.3 Confidence Interval for the Difference of Two Means Two Means 11 -- 22

LO10LO10--99 10

LO10LO10--9: 9: Construct a confidence interval for Construct a confidence interval for 11 − − 22 or or 11 -- 22 ((optional)optional)

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10 4 Comparing Two Means:10 4 Comparing Two Means: pter LO10LO10--33 10.4 Comparing Two Means: 10.4 Comparing Two Means:

Paired SamplesPaired Samples

LO10LO10--3: 3: Recognize paired data and be able to perform a paired Recognize paired data and be able to perform a paired t test.t test.

Paired DataPaired Data

•• Data occurs in matched pairs when the same item is observed Data occurs in matched pairs when the same item is observed twice but under different circumstances.twice but under different circumstances.

•• For example blood pressure is taken before and after a treatmentFor example blood pressure is taken before and after a treatmentFor example, blood pressure is taken before and after a treatment For example, blood pressure is taken before and after a treatment is given.is given.

•• Paired data are typically displayed in columns.Paired data are typically displayed in columns.

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10 4 Comparing Two Means:10 4 Comparing Two Means: pter LO10LO10--33 10.4 Comparing Two Means: 10.4 Comparing Two Means:

Paired SamplesPaired Samples Paired t TestPaired t Test

•• Paired data typically come from a before/after experimentPaired data typically come from a before/after experiment

•• Paired data typically come from a before/after experiment.Paired data typically come from a before/after experiment.

•• In the paired In the paired tt test, the difference between test, the difference between xx11 and and xx22 is measured is measured asas dd == xx11 –– xx22as as dd xx11 xx22

•• The mean and standard deviation for the differences d are given The mean and standard deviation for the differences d are given below.below.

Th t t t ti ti i j t fTh t t t ti ti i j t f l tl t t tt t•• The test statistic is just for a oneThe test statistic is just for a one--sample tsample t--test.test.

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10 4 Comparing Two Means:10 4 Comparing Two Means: pter 10.4 Comparing Two Means: 10.4 Comparing Two Means:

Paired SamplesPaired Samples LO10LO10--33

St 1 St t th h th f l

Steps in Testing Paired DataSteps in Testing Paired Data

• Step 1: State the hypotheses, for example H0: µd = 0 H1: µd ≠ 01 µd

• Step 2: Specify the decision rule. Choose  (the level of significance) and ( g ) determine the critical values from Appendix D or with use of technology. St 3 C l l t th t t t ti tiSt 3 C l l t th t t t ti ti tt•• Step 3: Calculate the test statistic Step 3: Calculate the test statistic tt

•• Step 4: Make the decisionStep 4: Make the decision Reject Reject HH00 if the test statistic falls in the rejection region(s) as if the test statistic falls in the rejection region(s) as jj 00 j g ( )j g ( ) defined by the critical valuesdefined by the critical values

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10.4 Comparing Two Means: 10.4 Comparing Two Means: Paired SamplesPaired Samples

LO10LO10--33

A two tailed test for a zero difference is equivalent to asking

Analogy to Confidence IntervalAnalogy to Confidence Interval

• A two-tailed test for a zero difference is equivalent to asking whether the confidence interval for the true mean difference µd includes zero.

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--55 10

LO10LO10--5: 5: Perform a test to compare two proportions using Perform a test to compare two proportions using z.z.

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

• To compare two population proportions, 1, 2, use the following hypotheses

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--55 10

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

Sample ProportionsSample Proportions

• The sample proportion p1 is a point estimate of 1 and p2 is a point estimate of 2:

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--55 10

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

• If H0 is true, there is no difference between

Pooled ProportionPooled Proportion

0 , 1 and 2, so the samples are pooled (or averaged) in order to estimate the common population proportion.

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--55

T t St ti tiT t St ti ti

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

If th l l b d ll

Test StatisticTest Statistic

• If the samples are large, p1 – p2 may be assumed normally distributed.

• The test statistic is the difference of the sample proportionsThe test statistic is the difference of the sample proportions divided by the standard error of the difference.

• The standard error is calculated by using the pooled proportion.y g p p p • The test statistic for the hypothesis 1 - 2 = 0 is:

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--55 10

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

Steps in Testing Two ProportionsSteps in Testing Two Proportions

• Step 1: State the hypotheses • Step 2: Specify the decision rulep p y

Choose  (the level of significance) and determine the critical value(s).

• Step 3: Calculate the Test Statistic. Assuming that 1 = 2, use a pooled estimate of the common proportion.

•• Step 4: Make the decision RejectStep 4: Make the decision Reject HH if the test statistic falls in theif the test statistic falls in the•• Step 4: Make the decision Reject Step 4: Make the decision Reject HH00 if the test statistic falls in the if the test statistic falls in the rejection region(s) as defined by the critical value(s).rejection region(s) as defined by the critical value(s).

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10.5 Comparing Two Proportions10.5 Comparing Two ProportionsLO10LO10--66 10

LO10LO10--6: 6: Check whether normality may be assumed for two proportions.Check whether normality may be assumed for two proportions.

Testing for Zero Difference: Testing for Zero Difference: 11 -- 22 = 0= 0

• We have assumed a normal distribution for the statistic p1 – p2. Checking for NormalityChecking for Normality

p1 p2 • This assumption can be checked. • For a test of two proportions, the criterion for normality is n  10 and p p , y

n(1 − )  10 for each sample, using each sample proportion in place of . If ith l ti i t l th i diff t• If either sample proportion is not normal, their difference cannot safely be assumed normal.

• The sample size rule of thumb is equivalent to requiring that each e sa p e s e u e o u b s equ a e o equ g a eac sample contains at least 10 “successes” and at least 10 “failures.”

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10.5 Comparing Two Proportions10.5 Comparing Two Proportions 10

Testing for NonTesting for Non--Zero DifferenceZero Difference

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10.6 Confidence Interval for the Difference 10.6 Confidence Interval for the Difference of Two Proportions of Two Proportions 11 -- 22 10

•• If the confidence interval does not include 0, then we will reject If the confidence interval does not include 0, then we will reject the null hypothesis of no difference in the proportions.the null hypothesis of no difference in the proportions.e u ypo es s o o d e e ce e p opo o se u ypo es s o o d e e ce e p opo o s

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

F t f H thF t f H th

10LO10LO10--8: 8: Carry out a test of two variances using the Carry out a test of two variances using the F F distributiondistribution

•• To test whether two population means are equal, we may also To test whether two population means are equal, we may also need to test whether two population variances are equalneed to test whether two population variances are equal

Format of HypothesesFormat of Hypotheses

need to test whether two population variances are equal.need to test whether two population variances are equal.

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

•• The test statistic is the ratio of the sample variances:The test statistic is the ratio of the sample variances:

The F TestThe F Test

•• The test statistic is the ratio of the sample variances:The test statistic is the ratio of the sample variances:

• If the variances are equal, this ratio should be near unity: F = 1

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

The F TestThe F Test

• If the test statistic is far below 1 or above 1, we would reject the hypothesis of equal population variances.

• The numerator s12 has degrees of freedom df1 = n1 – 1 and the denominator s22 has degrees of freedom df2 = n2 – 1.

• The F distribution is skewed with the mean > 1 and its mode < 1• The F distribution is skewed with the mean > 1 and its mode < 1.

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

The F Test: Critical ValuesThe F Test: Critical Values

• Critical values for the F test are denoted FL (left tail) and FR (right tail). L ( ) R ( g )

• A right-tail critical value FR may be found from Appendix F using df1 and df2 degrees of freedom. FR = Fdf1, df2

• A left-tail critical value FR may be found by reversing the d d i d f f d fi di hnumerator and denominator degrees of freedom, finding the

critical value from Appendix F and taking its reciprocal: F = 1/FFL = 1/Fdf2, df1

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

The F Test: Critical ValuesThe F Test: Critical Values

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

Steps in Testing Two VariancesSteps in Testing Two Variances

• Step 1: State the hypotheses, for example H0: 12 = 22 H 2 ≠ 2H1: 12 ≠ 22

• Step 2: Specify the decision rule D f f dDegrees of freedom are: Numerator: df1 = n1 – 1 Denominator: df2 = n2 – 1 2 2 Choose a and find the left-tail and right-tail critical values from Appendix F.

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

Steps in Testing Two VariancesSteps in Testing Two Variances

•• Step 3: Calculate the test statistic Step 3: Calculate the test statistic FFcalccalc = = ss1122//ss2222.. •• Step 4: Make the decisionStep 4: Make the decision

Reject Reject HH00 if the test statistic falls in the rejection regions as if the test statistic falls in the rejection regions as defined by the critical values defined by the critical values FFLL and and FFUU..

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

Comparison of Variances: One Tailed TestComparison of Variances: One Tailed Test

• Step 1: State the hypotheses, for example H0: 12 = 220 1 2 H1: 12 < 22

• Step 2: State the decision rulep Degrees of freedom are: Numerator: df1 = n1 – 1 D i t df 1Denominator: df2 = n2 – 1 Choose a and find the left-tail critical value from Appendix F.

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

Comparison of Variances: One Tailed TestComparison of Variances: One Tailed Test

•• Step 3: Calculate the Test Statistic Step 3: Calculate the Test Statistic FFcalccalc = = ss1122//ss2222.. •• Step 4: Make the decisionStep 4: Make the decisionStep 4: Make the decisionStep 4: Make the decision

Reject Reject HH00 if the test statistic falls in the leftif the test statistic falls in the left--tail rejection region as tail rejection region as defined by the critical value.defined by the critical value.

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

EXCEL’s F TestEXCEL’s F Test

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10.7 Comparing Two Variances10.7 Comparing Two VariancesLO10LO10--88

Assumptions of the F TestAssumptions of the F Test

•• TheThe FF test assumes that the populations being sampled aretest assumes that the populations being sampled are

•• The The FF test assumes that the populations being sampled are test assumes that the populations being sampled are normal.normal.

•• It is sensitive to nonIt is sensitive to non--normality of the sampled populations.normality of the sampled populations.y p p py p p p •• MINITAB reports both the MINITAB reports both the FF test and an alternative test and an alternative Levene’s testLevene’s test

and and pp--values.values.

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