Statistics for Managers Using Microsoft Excel/SPSS · 08/02/2016 · © 1999 Prentice-Hall, Inc....

© 1999 Prentice-Hall, Inc. Chap. 8 - 1

Statistics for Managers

Using Microsoft Excel/SPSS

Chapter 8

Fundamentals of Hypothesis Testing: One-Sample Tests


Chapter Topics

•Hypothesis Testing Methodology

•Z Test for the Mean (s Known)

• p-Value Approach to Hypothesis Testing

•Connection to Confidence Interval Estimation

•One Tail Test

• t Test of Hypothesis for the Mean

•Z Test of Hypothesis for the Proportion


A hypothesis is an

assumption about the

population parameter.

A parameter is a

Population mean or

proportion

The parameter must be

identified before

analysis.

I assume the mean GPA

of this class is 3.5!

© 1984-1994 T/Maker Co.

What is a Hypothesis?


• States the Assumption (numerical) to be tested

e.g. The average # TV sets in US homes is at

least 3 (H0: 3)

• Begin with the assumption that the null

hypothesis is TRUE.

(Similar to the notion of innocent until proven guilty)

The Null Hypothesis, H0

•Refers to the Status Quo

•Always contains the ‘ = ‘ sign

•The Null Hypothesis may or may not be rejected.


• Is the opposite of the null hypothesis e.g. The average # TV sets in US homes is

less than 3 (H1: < 3)

• Challenges the Status Quo

• Never contains the ‘=‘ sign

• The Alternative Hypothesis may or may

not be accepted

The Alternative Hypothesis, H1


Steps:

State the Null Hypothesis (H0: 3)

State its opposite, the Alternative

Hypothesis (H1: < 3)

Hypotheses are mutually exclusive &

exhaustive

Sometimes it is easier to form the

alternative hypothesis first.

Identify the Problem


Population

Assume the

population

mean age is 50.

(Null Hypothesis)

REJECT

The Sample

Mean Is 20

Sample Null Hypothesis

50?20 XIs

Hypothesis Testing Process

No, not likely!


Sample Mean = 50

Sampling Distribution

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 null

hypothesis that

= 50.

20 H0

Reason for Rejecting H0


• Defines Unlikely Values of Sample

Statistic if Null Hypothesis Is True

Called Rejection Region of Sampling

Distribution

• Designated a (alpha)

Typical values are 0.01, 0.05, 0.10

• Selected by the Researcher at the Start

• Provides the Critical Value(s) of the Test

Level of Significance, a


Level of Significance, a and

the Rejection Region

H0: 3

H1: < 3

0

0

0

H0: 3

H1: > 3

H0: 3

H1: 3

a

a

a/2

Critical

Value(s)

Rejection

Regions


• Type I Error

Reject True Null Hypothesis

Has Serious Consequences

Probability of Type I Error Is a

Called Level of Significance

• Type II Error

Do Not Reject False Null Hypothesis

Probability of Type II Error Is b (Beta)

Errors in Making Decisions


H0: Innocent

Jury Trial Hypothesis Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H 0 True H 0 False

Innocent Correct Error Do Not

Reject

H 0

1 - a Type II

Error ( b )

Guilty Error Correct Reject

H 0

Type I Error ( a )

Power

(1 - b )

Result Possibilities


a

b

Reduce probability of one error

and the other one goes up.

a & b Have an

Inverse Relationship


• True Value of Population Parameter

Increases When Difference Between Hypothesized

Parameter & True Value Decreases

• Significance Level a

Increases When a Decreases

• Population Standard Deviation s

Increases When s Increases

• Sample Size n

Increases When n Decreases

Factors Affecting

Type II Error, b

a

b

b s

b

n


• Convert Sample Statistic (e.g., ) to

Standardized Z Variable

• Compare to Critical Z Value(s)

If Z test Statistic falls in Critical Region,

Reject H0; Otherwise Do Not Reject H0

Z-Test Statistics (s Known)

Test Statistic

X

n

XXZ

X

X

s

s


• Probability of Obtaining a Test Statistic

More Extreme or ) than Actual

Sample Value Given H0 Is True

• Called Observed Level of Significance

Smallest Value of a H0 Can Be Rejected

• Used to Make Rejection Decision

If p value a Do Not Reject H0

If p value < a, Reject H0

p Value Test


1. State H0 H0 : 3

2. State H1 H1 : < 3

3. Choose a a = .05

4. Choose n n = 100

5. Choose Test: Z Test (or p Value)

Hypothesis Testing: Steps

Test the Assumption that the true mean #

of TV sets in US homes is at least 3.


6. Set Up Critical Value(s) Z = -1.645

7. Collect Data 100 households surveyed

8. Compute Test Statistic Computed Test Stat.= -2

9. Make Statistical Decision Reject Null Hypothesis

10. Express Decision The true mean # of TV set is less than 3 in the US households.

Hypothesis Testing: Steps

Test the Assumption that the average # of

TV sets in US homes is at least 3.

(continued)


• Assumptions

Population Is Normally Distributed

If Not Normal, use large samples

Null Hypothesis Has or Sign Only

• Z Test Statistic:

One-Tail Z Test for Mean

(s Known)

n

xxz

x

x

s

s


Z 0

a

Reject H 0

Z 0

Reject H 0

a

H0:

H1: < 0 H0: 0

H1: > 0

Must Be Significantly

Below = 0

Small values don’t contradict H0

Don’t Reject H0!

Rejection Region


Does an average box of

cereal contain more than

368 grams of cereal? A

random sample of 25 boxes

showed X = 372.5. The

company has specified s to

be 15 grams. Test at the

a0.05 level.

368 gm.

Example: One Tail Test

H0: 368

H1: > 368

_


Z .04 .06

1.6 .5495 .5505 .5515

1.7 .5591 .5599 .5608

1.8 .5671 .5678 .5686

.5738 .5750

Z 0

s Z = 1

1.645

.50

-.05

.45

.05

1.9 .5744

Standardized Normal

Probability Table (Portion) What Is Z Given a = 0.05?

a = .05

Finding Critical Values:

One Tail

Critical Value

= 1.645


a = 0.025

n = 25

Critical Value: 1.645

Test Statistic:

Decision:

Conclusion:

Do Not Reject at a = .05

No Evidence True Mean

Is More than 368 Z 0 1.645

.05

Reject

Example Solution: One Tail

H0: 368

H1: > 368

50.1

n

XZ

s


Z 0 1.50

p Value .0668

Z Value of Sample

Statistic From Z Table:

Lookup 1.50

.9332

Use the

alternative

hypothesis

to find the

direction of

the test.

1.0000

- .9332

.0668

p Value is P(Z 1.50) = 0.0668

p Value Solution


0 1.50 Z

Reject

(p Value = 0.0668) (a = 0.05).

Do Not Reject.

p Value = 0.0668

a = 0.05

Test Statistic Is In the Do Not Reject Region

p Value Solution


Does an average box of

cereal contains 368 grams of

cereal? A random sample of

25 boxes showed X = 372.5.

The company has specified

s to be 15 grams. Test at the

a0.05 level.

368 gm.

Example: Two Tail Test

H0: 368

H1: 368


a = 0.05

n = 25

Critical Value: ±1.96

Test Statistic:

Decision:

Conclusion:


No Evidence that True

Mean Is Not 368 Z 0 1.96

.025

Reject

Example Solution: Two Tail

-1.96

.025

H0: 386

H1: 386

50.1

2515

3685.372

n

XZ

s


Connection to

Confidence Intervals

For X = 372.5oz, s = 15 and n = 25,

The 95% Confidence Interval is:

372.5 - (1.96) 15/ 25 to 372.5 + (1.96) 15/ 25

or

366.62 378.38

If this interval contains the Hypothesized mean

(368), we do not reject the null hypothesis.

It does. Do not reject.

_


Assumptions

Population is normally distributed

If not normal, only slightly skewed & a large

sample taken

Parametric test procedure

t test statistic

t-Test: s Unknown

nS

Xt


Example: One Tail t-Test

Does an average box of cereal

contain more than 368 grams

of cereal? A random sample of

36 boxes showed X = 372.5,

and s 15. Test at the a0.01

level. 368 gm.

H0: 368

H1: > 368 s is not given,


a = 0.01

n = 36, df = 35

Critical Value: 2.4377

Test Statistic:

Decision:

Conclusion:


No Evidence that True

Mean Is More than 368 Z 0 2.4377

.01

Reject

Example Solution: One Tail

H0: 368

H1: > 368

80.1

3615

3685.372

nS

Xt


• Involves categorical variables

• Fraction or % of population in a category

• If two categorical outcomes, binomial

distribution

Either possesses or doesn’t possess the characteristic

• Sample proportion (ps)

Proportions

sizesample

successesofnumber

n

Xps


Example:Z Test for Proportion

•Problem: A marketing company claims

that it receives 4% responses from its

Mailing.

•Approach: To test this claim, a random

sample of 500 were surveyed with 25

responses.

•Solution: Test at the a = .05 significance

level.


a = .05

n = 500

Do not reject at a = .05

Z Test for Proportion:

Solution

H0: p .04

H1: p .04

Critical Values: 1.96

Test Statistic:

Decision:

Conclusion: We do not have sufficient

evidence to reject the company’s

claim of 4% response rate.

Z p - p

p (1 - p) n

s =

.04 -.05

.04 (1 - .04) 500

= 1.14

Z 0

Reject Reject

.025 .025


Chapter Summary

•Addressed Hypothesis Testing Methodology

•Performed Z Test for the Mean (s Known)

• Discussed p-Value Approach to Hypothesis Testing

•Made Connection to Confidence Interval

Estimation

•Performed One Tail and Two Tail Tests

• Performed t Test of Hypothesis for the Mean

•Performed Z Test of Hypothesis for the Proportion

Statistics for Managers Using Microsoft Excel/SPSS · 08/02/2016 · © 1999 Prentice-Hall, Inc....

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Transcript of Statistics for Managers Using Microsoft Excel/SPSS · 08/02/2016 · © 1999 Prentice-Hall, Inc....