si- hypothesis testing fis

7/31/2019 SI- Hypothesis Testing FIS

1/39

Fundamentals of Hypothesis

Testing: One-Sample Tests


2/39

Chapter Topics

Hypothesis Testing Methodology

Z Test for the Mean ( 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


3/39

A hypothesis is an

assumption about the

population parameter.

A parameteris acharacteristic of the

population, like its mean

or variance.

The parameter must beidentified before

analysis.

I assume the mean GPA

of this class is 3.5!

1984-1994 T/Maker Co.

What is a Hypothesis?


4/39

States the Assumption (numerical) to be tested

e.g. The grade point average of juniors is at least 3.0 (H0: 3.0) 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 theStatus Quo

Always contains the = sign

TheNull Hypothesismay or may not berejected.


5/39

Is theopposite of the null hypothesis e.g. The grade pointaverage of juniors is less than 3.0 (H1: < 3.0)

Challenges the Status Quo

Nevercontains the= sign

The Alternative Hypothesis may or may not beaccepted

Is generally the hypothesis that is believed to betruebythe researcher

The Alternative Hypothesis,H1


6/39

Steps: State the Null Hypothesis (H0: 3.0)

State its opposite, the Alternative Hypothesis (H1: 3H0: = 3H1: 3

/2

Critical

Value(s)

RejectionRegions


12/39

Type I Error Reject True Null Hypothesis (False Positive)

Has Serious Consequences

Probability of Type I Error IsCalledLevel of SignificanceSet by researcher

Type II Error

Do Not Reject False Null Hypothesis (False Negative)

Probability of Type II Error Is (Beta)

Errors in Making Decisions


13/39

H0: Innocent

Jury Trial Hypothesis Test

Actual Situation Actual Situation

Verdict Innocent Guilty Decision H0 True H0 False

Innocent Correct ErrorDo Not

Reject

H0

1 - Type IIError ( )

Guilty Error CorrectReject

H0

Type IError( )

Power

(1 - )

Result Possibilities


14/39

Reduce probability ofone error

and theother onegoes up.

& Have an Inverse Relationship


15/39

True Value of Population Parameter Increases When Difference Between Hypothesized Parameter & True

Value Decreases

Significance Level Increases When Decreases

Population Standard Deviation Increases When Increases

Factors Affecting Type II Error,


16/39

True Value of Population Parameter Increases When Difference Between Hypothesized Parameter &

True Value Decreases

Significance Level Increases When Decreases

Population Standard Deviation Increases When Increases

Sample Size n IncreasesWhen n Decreases

Factors Affecting Type II Error,

n


17/39

Choice depends on the cost of the error

Choose little type I error when the cost of rejecting the

maintained hypothesis is high A criminal trial: convicting an innocent person

The Exxon Valdise: Causing an oil tanker to sink

Choose large type I error when you have an interest in

changing the status quo A decision in a startup company about a new piece of software

A decision about unequal pay for a covered group.

How to choose between Type I and Type II errors


18/39

Deregulation leads to lower air travel prices. The university discriminates against women faculty

members

Stock prices increase when a firm announces a layoff.

Example Hypotheses: How do you test them?


19/39

Hiring Policy Hypotheses

A recruiter must decide who to hire. This is like forming a

hypothesis about whether or not the individual predicts to be

a good employee. Assume that an employee is either good

or poor.

WHAT ARE THE NULL AND ALTERNATIVE HYPOTHESES?


20/39


NULL: THE INDIVIDUAL DOES NOT MEET THE STANDARDS

(MEANZ)


21/39


WHAT ARE THE POSSIBLE ERRORS THAT THE RECRUITER

CAN MAKE?

HOW DO THESE RELATE TO TYPE I AND TYPE II ERRORS?


22/39


FAILURE TO HIRE A GOOD EMPLOYEE (failure to reject a

false null=type II error)

FAILURE TO REJECT A POOR EMPLOYEE(rejecting a null

when it is really true is type I error)


23/39


A positive decision is a decision to reject the null. A false

positive is therefore a type I error (hiring a poor person).

A negative decision is a failure to reject the null. A false

negative is therefore a type II error (not hiring a good

person)


24/39

The addition of more criteria should increase your ability to

distinguish poor candidates => type I error falls

However, more criteria mean that more good employees are cut

accidently => type II error increases.

When do you use more criteria?



25/39

Convert Sample Statistic (e.g., ) to test statistic, for

example a Z, t or F-statistic

Compare to Critical value obtained from a table.

If the test statistic falls in the Critical Region, RejectH0;Otherwise Do Not RejectH0

Critical Value of the test statistic

X


26/39

Probability of Obtaining a Test StatisticMore Extreme( or ) than Actual Sample ValueGivenH0 Is True CalledObserved Level of Significance

Smallest Value of a H0 Can Be Rejected

Used toMake Rejection Decision Ifp value , Do Not Reject H0 Ifp value < , Reject H0

p Value Test


27/39

1. StateH0 H0 : 3.02. StateH1 H1 : < 3 . 03. Choose = .054. Choose n n = 100

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

Hypothesis Testing: Steps

Test the Assumption that the true mean

grade point average of juniors is at least 3.


28/39

6. Set Up Critical Value(s) t = -1.7

7. Collect Data 100 students sampled

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

(computed P value=.04, two-tailed test)

9. Make Statistical Decision Reject Null Hypothesis

10. Express Decision The true mean grade point is less than 3.0

Hypothesis Testing: Steps

Test the Assumption that grade point average of juniors is at least3.

(continued)


29/39

Z0

RejectH0

Z0

RejectH0

H0: 0 H1: < 0

H0: 0H1: > 0

Must BeSignificantly

Below = 0 Small values dont contradictH0Dont RejectH0!

Rejection Region


30/39

Assumptions Population isnormally distributed

If not normal, onlyslightly skewed& a large sample taken

(Central limit theorem applies)

Parametric test procedure

t test statistic, with n-1 degrees of freedom

t-Test: Unknown

n

SXt

=


31/39

# in sample - number of parameters that must be

estimated before test statistic can be computed.

For a single sample t-test, we must first estimate the

mean before we can estimate the standard deviation.

Once the mean is estimated, n-1 of the values are left

since we know that the nth value is equal to

Degrees of Freedom

f

=

1

1

n

iixxn


32/39

Example: One Tail t-Test

Does an average box of cerealcontain

more than368grams of cereal? A

random sample of36boxes showed

X = 372.5, and = 15. Test at the= 0.01 level.

368 gm.

H0: 368H1: > 368

is not given,


33/39

PH Stat Entries


34/39

80.1

36

15

3685.372=

=

=

nS

Xt

Example Solution: One Tail

P ti


35/39

Involvescategorical variables

Fraction or%of population in a category

Iftwo categorical outcomes, binomial

distribution Either possesses or doesnt possess the characteristic

Sample proportion(ps)

Proportions

sizesample

successesofnumber

n

Xps =

Z test for proportions


36/39

The null hypothesis for the proportion also implies we know the variance,

since the variance is just P times (1-P).

This is a good approximation when the sample size is large.

If the sample size is small, we could use the binomial distribution to compute

the exact p value that a sample of size n would yield a sample proportionps

given the population proportion P. Using the normal approximation is much

easier.

Z test for proportions

E l Z T t f P ti


37/39

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 = .05significancelevel.

Z T t f P ti S l ti


38/39

Z Test for Proportion: Solution

Critical Values: 1.96Decision:

Conclusion:We do not have sufficient

evidence to reject the companys

claim of 4% response rate.Z0

Reject Reject

.025.025

a

= 1.14

Ch t S


39/39

Chapter Summary

Addressed Hypothesis Testing Methodology

Performed t- Test for the Mean ( unknown)Discussed p-Value Approach to Hypothesis TestingPerformed One Tail and Two Tail Tests

Performed Z Test of Hypothesis for the Proportion

si- hypothesis testing fis

Documents