Hypothesis Hypothesis TestingTesting

Variance known?Variance known?

Sampling DistributionSampling Distribution

Over-the-counter stock selling pricesOver-the-counter stock selling prices• calculate average price of all stocks listed [calculate average price of all stocks listed []]

take a sample of 25 stocks and record price take a sample of 25 stocks and record price • calculate average price of the 25 stocks [x-bar]calculate average price of the 25 stocks [x-bar]

take all possible samples of size 25 take all possible samples of size 25 • would all x-bars be equal?would all x-bars be equal?

average all the possible x-bars …equals average all the possible x-bars …equals

Levine, Prentice-Hall

Sampling DistributionSampling Distribution

Sample Mean = 5020

H0

Levine, Prentice-Hall

Sampling DistributionSampling Distribution

Sample Mean = 50

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

20

H0

Levine, Prentice-Hall

Sampling DistributionSampling Distribution

Sample Mean = 50

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

... if in fact this were the population mean

20

H0

Levine, Prentice-Hall

Sampling DistributionSampling Distribution

Sample Mean = 50

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

... if in fact this were the population mean

... therefore, we reject the hypothesis that = 50.

20

H0

Levine, Prentice-Hall

Null HypothesisNull Hypothesis

What is testedWhat is tested Always has equality sign: Always has equality sign: , , or or Designated HDesignated H00

• Example ………... HExample ………... H00: : 3 3

Levine, Prentice-Hall

Alternative Alternative HypothesisHypothesis

Opposite of null hypothesisOpposite of null hypothesis Always has inequality sign: Always has inequality sign: ,,, or, or Designated HDesignated H11

Example Example • HH11: : < 3 < 3

Levine, Prentice-Hall

DecisionDecision

Reject null hypothesisReject null hypothesis Retain, or, fail to reject, null Retain, or, fail to reject, null

hypothesishypothesis

Do not use the term Do not use the term “accept”“accept”

Levine, Prentice-Hall

p-valuep-value

Probability of obtaining a test statistic more Probability of obtaining a test statistic more extreme (extreme (or or than actual sample than actual sample value given Hvalue given H00 is true is true

Called observed level of significanceCalled observed level of significance• Smallest value of Smallest value of H H00 can be rejected can be rejected

Used to make rejection decisionUsed to make rejection decision

• If p-value If p-value , reject H, reject H00

Levine, Prentice-Hall

Level of Level of SignificanceSignificance

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 Designated Designated (alpha)(alpha)

• Typical values are .01, Typical values are .01, .05.05, .10, .10 Selected by researcher at startSelected by researcher at start

Levine, Prentice-Hall

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

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

Sampling DistributionSampling Distribution

1 -

Level of ConfidenceLevel of Confidence

Levine, Prentice-Hall

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

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

Sampling DistributionSampling Distribution

1 -

Level of ConfidenceLevel of Confidence

Observed sample statisticObserved sample statistic

Levine, Prentice-Hall

HoValueCritical

Value

Sample Statistic

RejectionRegion

NonrejectionRegion

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

Sampling DistributionSampling Distribution

1 -

Level of ConfidenceLevel of Confidence

ObservedObserved sample statistic sample statistic

Levine, Prentice-Hall

Rejection RegionsRejection Regions (two-(two-tailed test)tailed test)

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

Sampling Distribution

1 - 1 -

Level of Confidence

Levine, Prentice-Hall

Rejection RegionsRejection Regions (two-(two-tailed test)tailed test)

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

Sampling Distribution

1 - 1 -

Level of Confidence

ObservedObserved sample statistic sample statistic

Levine, Prentice-Hall

Rejection RegionsRejection Regions (two-(two-tailed test)tailed test)

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

Sampling Distribution

1 - 1 -

Level of Confidence

ObservedObserved sample statistic sample statistic

Levine, Prentice-Hall

Rejection RegionsRejection Regions (two-(two-tailed test)tailed test)

HoValue Critical

ValueCriticalValue

1/2 1/2

Sample Statistic

RejectionRegion

RejectionRegion

NonrejectionRegion

Sampling Distribution

1 - 1 -

Level of Confidence

ObservedObserved sample statistic sample statistic

Levine, Prentice-Hall

Risk of Errors in Making Risk of Errors in Making DecisionDecision

Type I errorType I error• Reject true null hypothesisReject true null hypothesis• Has serious consequencesHas serious consequences• Probability of Type I error is alpha [Probability of Type I error is alpha [

– Called level of significanceCalled level of significance Type II errorType II error

• Do not reject false null hypothesisDo not reject false null hypothesis• Probability of Type II error is beta [ Probability of Type II error is beta [

Levine, Prentice-Hall

Decision ResultsDecision ResultsHH00: Innocent: InnocentJury Trial H0 Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0 False

Innocent Correct Error Do NotReject H0

CorrectType II

Error ()

Guilty Error Correct Reject H0Type I

Error ()Correct

Levine, Prentice-Hall

Hypothesis TestingHypothesis Testing

State HState H00

State HState H11

Choose Choose

Choose Choose nn

Choose testChoose test

Levine, Prentice-Hall

Hypothesis TestingHypothesis Testing

Set up critical valuesSet up critical values

Collect dataCollect data

Compute test statisticCompute test statistic

Make statistical decisionMake statistical decision

Express decisionExpress decision

State HState H00

State HState H11

Choose Choose

Choose Choose nn

Choose testChoose test

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

Does an average box of Does an average box of cereal contain 368 grams of cereal contain 368 grams of cereal? A random sample of cereal? A random sample of 25 boxes has an average 25 boxes has an average weight = 372.5 grams. The weight = 372.5 grams. The company has specified company has specified to to be 15 grams. Test at the .05 be 15 grams. Test at the .05 level.level.

368 gm.368 gm.

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: :

HH11: :

nn

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

nn

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

.05.05

nn 25 25

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

.05.05

nn 25 25

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

.05.05

nn 25 25

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

372 5 368

1525

150.

.

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

.05.05

nn 25 25

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

372 5 368

1525

150.

.

Do not reject at = .05

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test

HH00: : = 368 = 368

HH11: : 368 368

.05.05

nn 25 25

Critical Value(s):Critical Value(s):

Test Statistic: Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

372 5 368

1525

150.

.

Do not reject at = .05

No evidence No evidence average is not 368average is not 368

Levine, Prentice-Hall

Z0 1.50-1.50

Two-tailed z-test Two-tailed z-test [p-value]][p-value]]

Z value of sample statistic

Levine, Prentice-Hall

Z0 1.50-1.50

Two-tailed z-test Two-tailed z-test [p-value][p-value]

p-value is P(z -1.50 or z 1.50)

Z value of sample statistic

Levine, Prentice-Hall

Z0 1.50-1.50

1/2 p-Value1/2 p-Value

Two-tailed z-test Two-tailed z-test [p-value][p-value]

p-value is P(z -1.50 or z 1.50)

Z value of sample statistic

Levine, Prentice-Hall

Z0 1.50-1.50

1/2 p-Value1/2 p-Value

Two-tailed z-test Two-tailed z-test [p-value][p-value]

p-value is P(z -1.50 or z 1.50)

Z value of sample statistic

From Z table: lookup 1.50

.4332

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test [p-value][p-value]

Z0 1.50-1.50

1/2 p-Value.0668

1/2 p-Value.0668

p-value is P(z -1.50 or z 1.50)

Z value of sample statistic

From Z table: lookup 1.50

.4332

.5000- .4332

.0668

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test [p-value][p-value]

Z0 1.50-1.50

1/2 p-Value.0668

1/2 p-Value.0668

p-value is P(z -1.50 or z 1.50) = .1336

Z value of sample statistic

From Z table: lookup 1.50

.4332

.5000- .4332

.0668

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test [p-value][p-value]

0 1.50-1.50 Z

RejectReject

1/2 p-value = .06681/2 p-value = .0668

1/2 = .0251/2 = .025

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test [p-value][p-value]

0 1.50-1.50 Z

RejectReject

(p-Value = .1336) ( = .05)

Do not reject. 1/2 p-Value = .06681/2 p-Value = .0668

1/2 = .0251/2 = .025

Test statistic is in ‘Do not reject’ region

Levine, Prentice-Hall

Two-tailed z-testTwo-tailed z-test (( known) known) challengechallenge

You are a Q/C inspector. You want to find You are a Q/C inspector. You want to find out if a new machine is making electrical out if a new machine is making electrical cords to customer specification: average cords to customer specification: average breaking strength of 70 lb. with breaking strength of 70 lb. with = 3.5 lb. = 3.5 lb. You take a sample of 36You take a sample of 36 cords & compute cords & compute a sample mean of 69.7a sample mean of 69.7 lb. At the .05lb. At the .05 level, is there evidence that the machine level, is there evidence that the machine is not meeting the average breaking is not meeting the average breaking strength?strength?

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: :

HH11: :

= =

nn = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= =

nn = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= .05= .05

nn = 36 = 36

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= .05= .05

nn = 36 = 36

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= .05= .05

nn = 36 = 36

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

69 7 70

3 536

51.

..

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= .05= .05

nn = 36 = 36

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

69 7 70

3 536

51.

..

Do not reject at = .05

Levine, Prentice-Hall

Two-tailed z-test Two-tailed z-test (( known)known)

HH00: : = 70 = 70

HH11: : 70 70

= .05= .05

nn = 36 = 36

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.96-1.96

.025

Reject H 0 Reject H 0

.025

ZX

n

69 7 70

3 536

51.

..

Do not reject at = .05

No evidence average is not 70

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

AssumptionsAssumptions• Population is normally distributedPopulation is normally distributed• If not normal, can be approximated by If not normal, can be approximated by

normal distribution for large samplesnormal distribution for large samples

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

AssumptionsAssumptions• Population is normally distributedPopulation is normally distributed• If not normal, can be approximated by If not normal, can be approximated by

normal distribution for large samplesnormal distribution for large samples Null hypothesis has Null hypothesis has or or sign only sign only

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

AssumptionsAssumptions• Population is normally distributedPopulation is normally distributed• If not normal, can be approximated by If not normal, can be approximated by

normal distribution for large samplesnormal distribution for large samples Null hypothesis has Null hypothesis has or or sign only sign only Z-test statisticZ-test statistic

ZX X

n

x

x

Levine, Prentice-Hall

Z0

Reject H 0

One-tailed z-test One-tailed z-test (( known) known)

H0:0 H1: < 0

Must be significantly below

Levine, Prentice-Hall

Z0

Reject H 0

Z0

Reject H 0

One-tailed z-test One-tailed z-test (( known) known)

H0:0 H1: < 0

H0:0 H1: > 0

Must be significantly below

Small values satisfy H0 . Do not reject!

Levine, Prentice-Hall

Z0

= 1

One-tailed z-test One-tailed z-test (( known) known)

What is “z” given = .025?

= .025

Levine, Prentice-Hall

Z0

= 1

One-tailed z-test One-tailed z-test (( known) known)

.500 - .025

.475

What Is Z given = .025?

= .025

Levine, Prentice-Hall

Z0

= 1 Z .05 .07

1.6 .4505 .4515 .4525

1.7 .4599 .4608 .4616

1.8 .4678 .4686 .4693

.4744 .4756

One-tailed z-test One-tailed z-test (( known) known)

.500 - .025

.475

.06

1.9 .4750

Standardized Normal Probability Table (Portion)

What is “z” given = .025?

= .025

Levine, Prentice-Hall

Z0

= 1

1.96

Z .05 .07

1.6 .4505 .4515 .4525

1.7 .4599 .4608 .4616

1.8 .4678 .4686 .4693

.4744 .4756

One-tailed z-test One-tailed z-test (( known) known)

.500 - .025

.475

.06

1.9 .4750

Standardized Normal Probability Table (Portion)

What Is Z given = .025?

= .025

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known) known)

Does an average box of Does an average box of cereal contain more than cereal contain more than 368 grams of cereal? A 368 grams of cereal? A random sample of 25 random sample of 25 boxes showedboxes showedX = 372.5. X = 372.5. The company has The company has specified specified to be 15 grams. to be 15 grams. Test at the .05 level. Test at the .05 level. 368 gm.368 gm.

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: :

HH11: :

= =

n n = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= =

n n = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= .05= .05

n n = 25= 25

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= .05= .05

n n = 25= 25

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.645

.05

Reject

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= .05= .05

n n = 25= 25

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.645

.05

Reject

ZX

n

372 5 368

1525

150.

.

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= .05= .05

n n = 25= 25

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.645

.05

Reject

ZX

n

372 5 368

1525

150.

.

Do not reject at = .05

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test (( known)known)

HH00: : 368 368

HH11: : > 368 > 368

= .05= .05

n n = 25= 25

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Z0 1.645

.05

Reject

ZX

n

372 5 368

1525

150.

.

Do not reject at = .05

No evidence average is more than 368

Levine, Prentice-Hall

Z0 1.50

One-tailed z-test One-tailed z-test ((

known)known) p-value Solutionp-value Solution

Z value of sample statistic

Use alternative hypothesis to find direction

Levine, Prentice-Hall

Z0 1.50

p-Value.0668

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

Z value of sample statistic

Use alternative hypothesis to find direction

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

Z0 1.50

p-Value.0668

Z value of sample statistic

From Z table: lookup 1.50

..4332

Use alternative hypothesis to find direction

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

Z0 1.50

p-Value.0668

Z value of sample statistic

From Z table: lookup 1.50

..4332

Use alternative hypothesis to find direction

.5000- .4332

.0668

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

Z0 1.50

p-Value.0668

Z value of sample statistic

From Z table: lookup 1.50

..4332

Use alternative hypothesis to find direction

.5000- .4332

.0668

p-value = .0668

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

0 1.50 Z

Reject

0 1.50 Z

Rejectp-value = .0668

= .05

Levine, Prentice-Hall

One-tailed z-test One-tailed z-test ((

known)known) p-valuep-value

0 1.50 Z

Reject

0 1.50 Z

Reject

(p-value = .0668) ( = .05).

Do not reject.p-Value = .0668

= .05

Test statistic is in ‘Fail to reject’ region

Levine, Prentice-Hall

p-valuep-value ChallengeChallenge

You’re an analyst for Ford. You want You’re an analyst for Ford. You want to find out if the average miles per to find out if the average miles per gallon of Escorts is at least 32 mpg. gallon of Escorts is at least 32 mpg. Similar models have a standard Similar models have a standard deviation of 3.8deviation of 3.8 mpg. You take a mpg. You take a sample of 60sample of 60 Escorts & compute a Escorts & compute a sample mean of 30.7sample mean of 30.7 mpg. What is mpg. What is the value of the observed level of the value of the observed level of significance (p-Value)?significance (p-Value)?

Levine, Prentice-Hall

p-valuep-value

Z0-2.65

p-Value.004

Z value of sample statistic

From Z table: lookup 2.645

.4960.4960

Use alternative hypothesis to find direction

.5000- .4960

.0040

p-value = .004 p-value < ( = .01) Reject H0.

Levine, Prentice-Hall

p-valuep-value

Probability of obtaining a test statistic more Probability of obtaining a test statistic more extreme (extreme (or or than actual sample than actual sample value given Hvalue given H00 is true is true

Called observed level of significanceCalled observed level of significance• Smallest value of Smallest value of H H00 can be rejected can be rejected

Used to make rejection decisionUsed to make rejection decision

• If p-value If p-value , reject H, reject H00

Levine, Prentice-Hall

One-tailed t-test One-tailed t-test (( unknown)unknown)

Does an average box of Does an average box of cereal contain less than cereal contain less than the 368 grams indicated on the 368 grams indicated on the package? A random the package? A random sample of 25 boxes sample of 25 boxes showedshowedX = 363.5 and X = 363.5 and s=15. Test at the .05 level.s=15. Test at the .05 level.

368 gr.368 gr.

Levine, Prentice-Hall

HH00: :

HH11: :

= =

n n = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

One-tailed t-test One-tailed t-test ((

unknown)unknown)

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= =

n n = =

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

One-tailed t-test One-tailed t-test ((

unknown)unknown)

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= .05= .05

n n = 25, d.f. = 24= 25, d.f. = 24

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

One-tailed t-test One-tailed t-test ((

unknown)unknown)

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= .05= .05

n n = 25, d.f. = 24= 25, d.f. = 24

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

One-tailed t-test One-tailed t-test ((

unknown)unknown)

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= .05= .05

n n = 25, d.f. = 24= 25, d.f. = 24

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

One-tailed t-test One-tailed t-test ((

unknown)unknown)

50.1

2515

3685 . 363

ns

Xt

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= .05= .05

n n = 25, d.f. = 24= 25, d.f. = 24

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do not reject at = .05

One-tailed t-test One-tailed t-test ((

unknown)unknown)

50.1

2515

3685 . 363

ns

Xt

Levine, Prentice-Hall

HH00: : 368 368

HH11: : < 368 < 368

= .05= .05

n n = 25, d.f. = 24= 25, d.f. = 24

Critical Value(s):Critical Value(s):

Test Statistic:

Decision:

Conclusion:

50.1

2515

3685 . 363

ns

Xt

Do not reject at = .05

No evidence average is less than 368

One-tailed t-test One-tailed t-test ((

unknown)unknown)

Levine, Prentice-Hall

One-tailed t-test One-tailed t-test ((

unknown)unknown) p-value Solutionp-value Solution

t value of sample statistic

Use alternative hypothesis to find direction

Levine, Prentice-Hall

One-tailed t-test One-tailed t-test ((

unknown)unknown) p-valuep-value

Use alternative hypothesis to find direction

t value of sample statistic

From t table: lookup -1.50 for 24 d.f.

P-value P-value = 0.075= 0.075

Levine, Prentice-Hall

p-value = .075

= .05

One-tailed t-test One-tailed t-test ((

unknown)unknown) p-valuep-value

Levine, Prentice-Hall

p-value = .075

= .05

One-tailed t-test One-tailed t-test ((

unknown)unknown) p-valuep-value

Test statistic is in ‘Fail to reject’ region

(p-value = .075) ( = .05).

Do not reject.

Reject

Questions?Questions?

ANOVAANOVA