sda 3e chapter 5

8/4/2019 SDA 3E Chapter 5

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2007 Pearson Education

Chapter 5: Hypothesis Testingand Statistical Inference


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Hypothesis Testing Hypothesis testing involves drawing

inferences about two contrasting propositions

(hypotheses) relating to the value of apopulation parameter, one of which isassumed to be true in the absence ofcontradictory data.

We seek evidence to determine if thehypothesis can be rejected; if not, we canonly assume it to be true but have notstatistically proven it true.


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Hypothesis Testing Procedure1. Formulate the hypothesis

2. Select a level of significance, which defines

the risk of drawing an incorrect conclusionthat a true hypothesis is false

3. Determine a decision rule

4. Collect data and calculate a test statistic5. Apply the decision rule and draw a

conclusion


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Hypothesis Formulation Null hypothesis, H0 a statement that is

accepted as correct

Alternative hypothesis, H1 a proposition thatmust be true if H0 is false

Formulating the correct set of hypothesesdepends on burden of proof what you

wish to prove statistically should be H1 Tests involving a single population parameter

are called one-sample tests; tests involvingtwo populations are called two-sample tests.


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Types of Hypothesis Tests One Sample Tests

H0: population parameter constant vs.

H1: population parameter < constant

H0: population parameter constant vs.H

1: population parameter > constant

H0: population parameter = constant vs.

H1: population parameter constant

Two Sample Tests H

0: population parameter (1) - population parameter (2) 0 vs.

H1: population parameter (1) - population parameter (2) < 0

H0: population parameter (1) - population parameter (2) 0 vs.

H1: population parameter (1) - population parameter (2) > 0

H0: population parameter (1) - population parameter (2) = 0 vs.H

1: population parameter (1) - population parameter (2) 0


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Four Outcomes1. The null hypothesis is actually true, and the

test correctly fails to reject it.

2. The null hypothesis is actually false, and thehypothesis test correctly reaches thisconclusion.

3. The null hypothesis is actually true, but the

hypothesis test incorrectly rejects it (Type Ierror).

4. The null hypothesis is actually false, but thehypothesis test incorrectly fails to reject it

(Type II error).


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Quantifying Outcomes Probability of Type I error (rejecting H0 when

it is true) = = level of significance Probability ofcorrectly failingto reject H0 = 1

= confidence coefficient Probability of Type II error (failing to reject H0

when it is false) = Probability ofcorrectly rejectingH0 when it is

false = 1 = power of the test


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Decision Rules Compute a test statistic from sample data and

compare it to the hypothesized sampling

distribution of the test statistic Divide the sampling distribution into a

rejection region and non-rejection region.

If the test statistic falls in the rejection region,reject H0 (concluding that H1 is true);

otherwise, fail to reject H0


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Rejection Regions


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Hypothesis Tests and

Spreadsheet SupportType of Test Excel/PHStatProcedure

One sample test for mean, unknown PHStat: One Sample Test Z-test for the

Mean, Sigma KnownOne sample test for mean, unknown PHStat: One Sample Test t-test for the

Mean, Sigma Unknown

One sample test for proportion PHStat: One Sample Test Z-test for theProportion

Two sample test for means, known Excel z-test: Two-Sample for Means

PHStat: Two Sample Tests Z-Test forDifferences in Two Means

Two sample test for means, unknown,unequal

Excel t-test: Two-Sample AssumingUnequal Variances


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Spreadsheet Support (contd)Type of Test Excel/PHStatProcedure

Two sample test for means, unknown,assumed equal

Excel t-test: Two-Sample Assuming EqualVariances

PHStat: Two Sample Tests t-Test forDifferences in Two Means

Paired two sample test for means Excel t-test: Paired Two-Sample for Means

Two sample test for proportions PHStat: Two Sample Tests Z-Test forDifferences in Two Proportions

Equality of variances Excel F-test Two-Sample for Variances

PHStat: Two Sample Tests F-Test forDifferences in Two Variances


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One Sample Tests for Means

Standard Deviation Unknown Example hypothesis

H0: 0 versus H1: < 0 Test statistic:

Reject H0if t < -t

n-1,

ns

xt

/

0=


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Example For the Customer Support Survey.xls data, test thehypotheses

H0: mean response time 30 minutes H1: mean response time< 30 minutes

Sample mean = 21.91; sample standard deviation =

19.49; n = 44 observations

Reject H0 because t = 2.75 < -t43,0.05 = -1.6811


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PHStatTool: t-Test for Mean PHStatmenu > One Sample

Tests > t-Test for the Mean,

Sigma Unknown

Enter null hypothesis and alpha

Enter sample statistics or datarange

Choose type of test


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Results


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One Sample Tests for

Proportions Example hypothesis

H0:

0versusH

1:


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Example For the Customer Support Survey.xls data, test the hypothesis that the

proportion of overall quality responses in the top two boxes is at least0.75

H0: .75 H

0: < .75

Sample proportion = 0.682; n = 44

For a level of significance of 0.05, the critical value ofzis -1.645;therefore, we cannot reject the null hypothesis


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PHStat Tool: One Sample z-

Test for Proportions PHStat> One Sample Tests> z-Tests

for the Proportion

Enter null hypothesis,significance level, numberof successes, and sample

sizeEnter type of test


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Results


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Type II Errors and the Power

of a Test The probability of a Type II error, , and the

power of the test (1 ) cannot be chosen by

the experimenter. The power of the test depends on the true

value of the population mean, the level ofconfidence used, and the sample size.

A power curve shows (1 ) as a function of

1.


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Example Power Curve


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Two Sample Tests for Means

Standard Deviation Known Example hypothesis

H0:1 20 versusH1:1 -2 < 0

Test Statistic:

Reject if z < -z

2

2

21

2

1

21

// nn

xxz

+

=


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Sigma Unknown and Equal Example hypothesis

H0:1 20 versusH1:1 -2 > 0

Test Statistic:

Reject if z > z

21

21

21

2

22

2

11

21

2

)1()1(

nn

nn

nn

snsn

xxz

+

+

=


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Sigma Unknown and Unequal Example hypothesis

H0:1 2=0 versusH1:1 -2 0

Test Statistic:

Reject if z > z/2 or z < - z/2

t = (x1

-

x2) / 2

2

2

1

2

1

n

s

n

s+

+

+

1

)/(

1

)/(

2

2

2

2

2

1

2

1

2

1

2

2

2

2

1

2

1

n

ns

n

ns

ns

ns

with df =


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Excel Data Analysis Tool: Two

Sample t-Tests Tools> Data Analysis> t-test: Two Sample

Assuming Unequal Variances, or t-test: TwoSample Assuming Equal Variances

Enter range of data, hypothesized meandifference, and level of significance

Tool allows you to test H0:

1-

2= d

Output is provided for upper-tail test only For lower-tail test, change the sign on t

Critical one-tail, and subtract P(T


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PHStatTool: Two Sample

t-Tests PHStat> Two Sample Tests> t-Test

for Differences in Two Means

Test assumes equal variances Must compute and enter the sample

mean, sample standard deviation, and

sample size


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Comparison of Excel and PHStat

Results Lower-Tail Test


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Two Sample Test for Means

With Paired Samples Example hypothesis

H0: average difference=0 versus

H1: average difference0

Test Statistic:

Reject if t > tn-1,/2 or t < - tn-1,/2

ns

Dt

D

D

/

=


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Two Sample Tests for

Proportions Example hypothesis

H0: 1 2=0 versusH1: 1 -2 0

Test Statistic:

Reject if z > z/2 or z < - z/2

+

=

21

21

11)1(

nnpp

ppz

where21 nn

samplesbothinsuccessesofnumberp+

=


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Confidence Intervals If a 100(1 )% confidence interval contains

the hypothesized value, then we would not

reject the null hypothesis based on this valuewith a level of significance . Example hypothesis

H0: 0 versus H1: < 0 If a 100(1-)% confidence interval does not

contain 0, then we can reject H0


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F-Test for Differences in Two

Variances Hypothesis

H0: 1

2 2

2=0 versusH

1: 1

2 -22 0

Test Statistic:

Assume s12 > s2

2

Reject if F > F/2,n1-1,n2-1 (see Appendix A.4)

Assumes both samples drawn from normaldistributions

2

2

21

s

sF =


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Excel Data Analysis Tool: F-

Test for Equality of Variances Tools> Data Analysis> F-test for

Equality of Variances

Specify data ranges Use /2 for the significance level! If the variance of Variable 1 is greater

than the variance of variable 2, theoutput will specify the upper tail;otherwise, you obtain the lower tailinformation.


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PHStat Tool: F-Test for

Differences in Variances PHStatmenu > Two Sample Tests> F-

test for Differences in Two Variances

Compute and enter sample standarddeviations

Enter the significance level , not /2as in Excel


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Excel and PHStatResults


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Analysis of Variance (ANOVA) Compare the means of m different

groups (factors) to determine if all are

equal H

0: 1 = 1 = ... m

H1: at least one mean is different from the

others


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ANOVA Theory nj = number of observations in sample j

SST = total variation in the data

SSB = variation between groups

SSW = variation within groups

SST = SSB + SSW

= =

=n

j

n

i

ij

j

XXSST

1 1

2)(

=

=n

j

jj XXnSSB1

2)(

= = =n

j

n

i

jij

j

XXSSW1 1

2)(


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ANOVA Test Statistic MSB = SSB/(m 1)

MSW = SSW/(n m)

Test statistic: F = MSB/MSW Has an F-distribution with m-1 and n-m

degrees of freedom

Reject H0 if F > F/2,m-1,n-m


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Excel Data Analysis Tool for

ANOVA Tools> Data Analysis>ANOVA: Single

Factor


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ANOVA Results


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ANOVA Assumptions The mgroups or factor levels being studied

represent populations whose outcome

measures are Randomly and independently obtained Are normally distributed Have equal variances

Violation of these assumptions can affect thetrue level of significance and power of thetest.


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Nonparametric Tests Used when assumptions (usually

normality) are violated. Examples: Wilcoxon rank sum test for testing

difference between two medians Kurskal-Wallis rank test for determining

whether multiple populations have equalmedians. Both supported by PHStat


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Tukey-Kramer Multiple

Comparison ProcedureANOVA cannot identify which means

may differ from the rest

PHStatmenu > Multiple Sample Tests> Tukey-Kramer Multiple ComparisonProcedure

Enter Q Statistic from Table A.5


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Chi-Square Test for

Independence Test whether two categorical variables

are independent H0: the two categorical variables are

independent

H1: the two categorical variables are

dependent


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Example Is gender independent of holding a CPA

in an accounting firm?


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Chi-Square Test for

Independence Test statistic

Reject H0 if2 > 2, (r-1)(c-1) PHStattool available in Multiple Sample

Testsmenu

e

eo

f

ff 22 )( =

where f0= observed frequency

fe= expected frequency if H0 true

in the cells of the contingency table


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ExampleExpected No CPA CPA Total

Female 6.74 7.26 14

Male 6.26 6.74 13Total 13 14 27

Critical value with = 0.05 and (2 - 1)(2 - 1) - 1 df =3.841; therefore, we cannot reject the null hypothesisthat the two categorical variables are independent.


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PHStat Procedure Results


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Design of ExperimentsA test or series of tests that enables the

experimenter to compare two or more

methods to determine which is better,or determine levels of controllablefactors to optimize the yield of a

process or minimize the variability of aresponse variable.


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Factorial Experiments All combinations of levels of each factor are considered.

With m factors at k levels, there are km experiments. Example: Suppose that temperature and reaction time

are thought to be important factors in the percent yield ofa chemical process. Currently, the process operates at atemperature of 100 degrees and a 60 minute reactiontime. In an effort to reduce costs and improve yield, theplant manager wants to determine if changing the

temperature and reaction time will have any significanteffect on the percent yield, and if so, to identify the bestlevels of these factors to optimize the yield.


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Designed ExperimentAnalyze the effect of two levels of each

factor (for instance, temperature at 100

and 125 degrees, and time at 60 and90 minutes)

The different combinations of levels of

each factor are commonly calledtreatments.


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Treatment Combinations

Low

High

Low High


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Experimental Results


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Main Effects Measures the difference in the response that

results from different factor levels Calculations

Temperature effect = (Average yield at high level) (Average yieldat low level)

= (B + D)/2 (A + C)/2

= (90.5 + 81)/2 (84 + 88.5)/2

= 85.75 86.25 = 0.5 percent. Reaction effect = (Average yield at high level) (Average yield at

low level)

= (C + D)/2 (A + B)/2

= (88.5 + 81)/2 (84 + 90.5)/2

= 84.75 87.25 = 2.5 percent.


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Interactions When the effect of changing one factor

dependson the level of other factors.

When interactions are present, wecannotestimate response changes bysimply adding main effects; the effect

of one factor must be interpretedrelative to levels of the other factor.


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Interaction Calculations Take the average difference in response

when the factors are both at the high or lowlevels and subtracting the average difference

in response when the factors are at oppositelevels. Temperature Time Interaction

= (Average yield, both factors at same level)

(Average yield, both factors at opposite levels)= (A + D)/2 (B + C)/2

= (84 + 81)/2 (90.5 + 88.5)/2 = -7.0 percent


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Graphical Illustration ofInteractions


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Two-Way ANOVA

Method for analyzing variation in a 2-factorexperiment

SST = SSA + SSB + SSAB + SSWwhere

SST = total sum of squares

SSA = sum of squares due to factor A

SSB = sum of squares due to factor B

SSAB = sum of squares due to interaction

SSW = sum of squares due to random variation (error)


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Mean Squares

MSA = SSA/(r 1)

MSB = SSB/(c 1)

MSAB = SSAB/(r-1)(c-1)

MSW = SSW/rc(k-1),

where k = number of replications ofeach treatment combination.


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Hypothesis Tests

Compute F statistics by dividing each mean squareby MSW. F = MSA/MSW tests the null hypothesis that means for

each treatment level of factor A are the same against thealternative hypothesis that not all means are equal.

F = MSB/MSW tests the null hypothesis that means foreach treatment level of factor A are the same against the

alternative hypothesis that not all means are equal. F = MSAB/MSW tests the null hypothesis that the

interaction between factors A and B is zero against thealternative hypothesis that the interaction is not zero.


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ExcelAnova: Two-Factor withReplication


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Results

Examine p-values forsignificance

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