STATISTICS FOR BUSINESS Chapter 8. Hypothesis testing for a single population - PowerPoint Presentation

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STATISTICS FOR BUSINESS

Chapter 8. Hypothesis testing for a single population

A nuclear power plant adjacent to a residential area

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)A nuclear power plant

adjacent to a residential area

Two

local

residents die

of leukemia

It is wrong to accept,

or reject, a hypothesis about a population parameter simply by intuition. One needs to decide objectively on the basis of measured sample information.

Does this event make us conclude that the government is giving wrong information?

"Radiation levels around a nuclear power plant are well below levels considered harmful".

An hypothesis is giving an opinion or making a decision without objective information

Consider: Government

announces

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)A nuclear power plant

adjacent to a residential area

Select a random sample

Measure the appropriate

statistic - the mean or proportion

Decide on the desired

level of significance: (Say 5%) Determine if the statistic falls within an appropriate region of acceptance

Accept the hypothesis if the statistic falls into the acceptance region. Otherwise, reject it

Even if a sample statistic does fall in the area of acceptance, it does not prove that the null hypothesis, Ho, is true. There is

simply no statistical evidence to reject it.,

Procedure for hypothesis testing

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)A nuclear power plant

adjacent to a residential area

Hypothesis:

In a certain country, average

age of population is

35 within a given significance level

Written as: H

o:µx = 35Null hypothesis is that population mean is equal to 35

Alternative hypothesis:Population mean is not equal to 35.That is the mean or average age is significantly different from 35

Written as: H1:µx  35

Whenever the null hypothesis is rejected, accepted conclusion is the alternative hypothesis

Binomial either “accept” or “reject”

Nomenclature in hypothesis testing

Exam grades - Case 1John has an A in the course on Business Statistic: Susan has an AIs the difference significant? NOExam grades - Case 2Sarah has an A in the course on Business Statistics: Derek has a C-

Is the difference significant? YES

Concept of significance

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Ages - Case 1

Joan, Susan, and Mike are in the same class at university.

Is there a significance difference in their age? PROBABLY NOTAges- Case 2Angela is the granddaughter of KennethIs there a significant difference in their ages? YES

Automobile prices - Case 1Erin has just bought a new red Austin Mini automobile. Peter has just bought the same model, but green.

Is there a significance difference in their purchase price? PROBABLY NOT

Automobile prices - Case 2

Pauline has just bought a new Austin Mini automobile. Jeffrey has just bought a Porsche.

Is there a significant difference in their purchase price? YES

Question

asked: "Is there evidence of a difference?"Null hypothesis: Average age

of a certain group is 35 years: Ho:µx =

35

If sample means falls

within the non shaded area,

accept the

null hypothesis

Reject the null hypothesis ifsample mean falls in either of the shaded regions

At 10% significance, there is 5% in each tailTwo-tailed hypothesis test

STATISTICS FOR BUSINESS(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Alternative: Is there evidence that average

age

the group is different than 35 years:

H

1

:

µ

x

≠ 35

Question

asked: "Is there evidence of a being greater than?"Null hypothesis is that average

age of a certain group is not greater than 35 years: Ho:µx ≤ 35

If sample means falls

within the non shaded area,

accept the

null hypothesis

Reject

the null hypothesis ifsample mean falls in the shaded region

At 10% significance, there is 10% in right hand tail

One-tailed right hand hypothesis testSTATISTICS FOR BUSINESS(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Alternative: Is there evidence that average

age

the group is greater than 35 years:

H

1

:

µ

x

> 35

Question

asked: "Is there evidence of being less than?"Null hypothesis: Average

age of a certain group is not less than 35 years: Ho:µx ≥ 35

If sample means falls

within the non shaded area,

accept the

null hypothesis

Reject

the null hypothesis ifsample mean falls in the shaded region

At 10% significance, there is 10% in left tail

One-tailed left hand hypothesis testSTATISTICS FOR BUSINESS(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Alternative hypothesis: Is there evidence that average

age

the group is less than 35 years:

H

1

:

µ

x

< 35

99%

90%

50%

Significance level of 1

%

(0.5% in

each tail

)

Significance level of

50%

(25% in

each tail

)

Significance level of

10%

(5% in

each tail

)

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Selecting a significance level

Significance

level Is total

area in the

tails

Higher

the significance level

for testing

the hypothesis,

greater

is probability

of rejecting a null

hypothesis when

it is true

.

However

,

in this case we

would rarely

accept

a null

hypothesis

when it is not true.

Population

standard deviation sx is known. Large samples

Test statistic is:

Numerator measures how far the observed mean is from hypothesized mean.

Denominator is standard error

z represents how many standard

errors observed mean is from hypothesized mean

hypothesized meanobserved mean

z can be + or -

Hypothesis testing for the mean with sample sizes greater than 30. Using Normal distribution

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Population

standard deviation sx is unknown.

Only standard deviation available is sample standard deviation, s. Small samples

Test statistic is:

If population is assumed to be normally distributed, sampling distribution of mean

will follow a t distribution with (n - 1) degrees of freedom

. n is the sample size less than 30

hypothesized mean

observed mean

t can be + or -

Hypothesis testing for the mean with sample sizes less than 30. Using Student-t distribution

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

The p-value (probability value) is

the observed level of significanceIt is

the smallest level at which H0 can be rejected for a given set of data.

The

p-value

answers the question,

“

If

H0 is true, what is the probability of obtaining x-bar or ps, this far or more from H

0 ?”  If the p-value from sample

is greater than, or equal to a the null hypothesis should be accepted

If the p-value is less than

a

the null hypothesis should be

rejected

p-value approach to hypothesis testing

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

From the established significant level determine the limits. Either Normal z, or Student t

Established whether the sample value lies within these limits ;

If it does accept the null hypothesis. If not reject the null hypothesis.

Comparing sample value with critical limits

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

TYPE I Error

Rejecting a null hypothesis when it is in fact true: Probability of a Type 1 error is alphaAlpha is the level of significance

TYPE II ErrorAccepting a null hypothesis, when it is in fact false

Probability of a Type II error is Beta

Types of errors

A

Type I error involves time

& cost of reworking a batch of chemicals that should have been accepted

A Type II error, means taking a chance than an entire group of users of the chemical will be poisoned Management would prefer a Type I error. Potential risk is lower

Making a Type I error involves shutting down and modifying an entire assembly line

at a work center. Making a Type II error, involves less expensive warranty repairs at the dealers

Management would prefer a Type II

error. Less costly! (Ethics?)

Under

Anglo Saxon criminal law

an

individual is

considered innocent

of a certain crime

.

Guilt must be

proven.

Preferable to

commit a Type II error (Accepting a null hypothesis when it

is false) and

let a guilty

person

go

free, rather than perhaps sentence an

innocent

person for a crime they did not commit.

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Binomial is correct distribution

Success Failure If n.p and n.q are both

 5: Normal distribution can be used to approximate the sampling distribution;

As for the mean there can be a two tail test, or a one tail test

Hypothesis testing of proportions

STATISTICS FOR BUSINESS

(Hypothesis testing for a single population)

A nuclear power plant

adjacent to a residential area

Hypothesized

proportion

Test value of proportion, p-bar