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Two-Sample Tests Two-Sample Tests

Two-Sample Tests - PowerPoint Presentation

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Two-Sample Tests - PPT Presentation

of Hypothesis Chapter 11 Learning Objectives LO111 Test a hypothesis that two independent population means are equal assuming that the population standard deviations are known and equal ID: 487896

lo11 population standard means population lo11 means standard test comparing deviations samples step equal dependent hypothesis difference unknown decision sample state independent

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Slide1

Two-Sample Tests of Hypothesis

Chapter 11Slide2

Learning Objectives

LO11-1

Test a hypothesis that two independent population

means

are equal, assuming that the

population

standard deviations

are known and equal.

LO11-2

Test a hypothesis that two independent population

means

are equal, with unknown population

standard deviations

.

LO11-3

Test a hypothesis about the mean population

difference

between paired or dependent

observations

.

LO11-4

Explain the difference between dependent and

independent

samples

.Slide3

Comparing Two Populations – Examples

Is there a difference in the mean value of residential real estate sold by male agents and female agents in south Florida?

Is there a difference in the mean number of defects produced on the day and the afternoon shifts at Kimble Products?

Is there a difference in the mean number of days absent between young workers (under 21 years of age) and older workers (more than 60 years of age) in the fast-food industry?

Is there is a difference in the proportion of Ohio State University graduates and University of Cincinnati graduates who pass the state Certified Public Accountant Examination on their first attempt?

Is there an increase in the production rate if music is piped into the production area?Slide4

Comparing Two Population Means: Equal, Known Population Variances

No assumptions about the shape of the populations are required.

The samples are from independent populations.

The formula for computing the value of

z

is:

LO11-1

Test a hypothesis that two independent population means are equal, assuming that the

population standard

deviations are known and equal. Slide5

Comparing Two Population Means:

Equal, Known Population Variances

– Example

The

Fast Lane procedure

was recently installed at the

local food market.

The store manager would like to know if the

mean

checkout time

using the standard checkout method

is longer

than using the

Fast Lane procedure.

She gathered the following sample information. The time is measured from when the customer enters the line until their bags are in the cart.

Hence,

the time includes both waiting in line and checking out.

LO11-1Slide6

Comparing Two Population

Means:

Equal, Known Population Variances

– Example

Applying the six-step hypothesis testing procedure:

Step 1: State the null and alternate hypotheses.

(keyword: “longer than”)

H

0

:

µ

S

≤ µ

U

H

1

: µS >

µU

Step 2:

Select

the level of significance.

The .01 significance level is requested in the problem.Step 3: Determine the appropriate test statistic. Because both population standard deviations are known, we can use the z-distribution as the test statistic.

LO11-1Slide7

Comparing Two Population Means:

Equal, Known Population Variances – Example

Step 4:

Formulate a decision rule.

Reject

H

0

if

z

> z z> 2.326

LO11-1Slide8

Comparing Two Population Means:

Equal, Known Population Variances – Example

Step 5:

Take

a sample and make a decision.

The computed value of

3.123

is larger than the critical value of

2.326.

Our decision is to reject the null hypothesis.

LO11-1

Step 6: Interpret the result.

The

difference of .20 minutes between the

mean

checkout time

using

the standard method is too large to

have occurred

by chance.

We conclude the Fast Lane method is faster.Slide9

Comparing Population Means: Equal, Unknown Population Standard Deviations

(The Pooled t-test)

The

t

distribution is used as the test statistic if one or more of the samples have less than 30 observations. The required assumptions are:

Both populations must follow the normal distribution.

The populations must have equal standard deviations.

The samples are from independent populations.

LO11-2

Test a hypothesis that two independent population means are equal, with unknown population

standard deviations

.Slide10

Comparing Population Means: Equal, Unknown Population Standard Deviations (The Pooled

t-test)

Finding the value of the test statistic requires two steps:

Pool the sample standard deviations.

Use the pooled standard deviation to compute the

t

-statistic.

LO11-2Slide11

Comparing Population Means: Equal, Unknown Population Standard Deviations (The

Pooled t-test) – Example

Owens Lawn Care, Inc., manufactures and assembles lawnmowers that are shipped to dealers throughout the United States and Canada. Two different procedures have been proposed for mounting the engine on the frame of the lawnmower. The question is:

Is there a difference in the mean time to mount the engines on the frames of the lawnmowers?

The first procedure was developed by longtime Owens employee Herb Welles (designated

“W”)

, and the other procedure was developed by Owens Vice President of Engineering William Atkins (designated

“A”)

. To evaluate the two methods, it was decided to conduct a time and motion study.

A sample of five employees was timed using the Welles method and six using the Atkins method. The results, in minutes, are shown on the right.

Is there a difference in the mean mounting times? Use the .10 significance level

.

LO11-2Slide12

Comparing Population Means: Equal, Unknown Population Standard Deviations

(The

Pooled

t

-test)

– Example

Step 1: State the null and alternate hypotheses.

(Keyword: “Is there a

difference

”)

H0: µ

W

= µ

A

H1: µW ≠ µA

Step 2: State the level of significance.

The

0.10 significance level is stated in the problem.

Step 3:

Select the appropriate test statistic. Because the population standard deviations are not known but are assumed to be equal, we use the pooled t-test.

LO11-2Slide13

Comparing Population Means: Equal, Unknown Population Standard Deviations

(The

Pooled

t

-test)

– Example

Step 4: State the decision rule.

Reject H0 if t > t

/2,nW+nA-2 or t < - t

/2, n

W

+n

A

-2

t > t

.05,9

or t < - t.05,9 t > 1.833 or t < - 1.833 LO11-2Slide14

Comparing Population Means: Equal, Unknown Population Standard Deviations (The Pooled

t-test) – Example

Step 5: Compute the value of

t

and make a decision.

14

(a) Calculate the sample standard

deviations.

(b) Calculate the

pooled

sample standard

deviation.

LO11-2Slide15

Comparing Population Means: Equal, Unknown Population Standard Deviations (The Pooled

t-test) – Example

Step 5 (continued): Take a sample and m

ake a decision.

(c) Determine the value of

t.

The decision is not to reject the null

hypothesis

because

0.662

falls in the region between -1.833 and 1.833.

Step 6: Interpret the Result.

The data show no evidence that

there

is a difference in

the mean times to mount the

engine

on the frame

between the Welles and Atkins methods.

LO11-2Slide16

Comparing Population Means: Equal, Unknown Population Standard Deviations (The Pooled

t-test) – Example

LO11-2Slide17

Comparing Population Means with Unknown AND

Unequal Population Standard Deviations

Use the formula for the

t

-statistic shown if it is not reasonable to assume the population standard deviations are equal.

The degrees of freedom are adjusted downward by a rather complex approximation formula. The effect is to reduce the number of degrees of freedom in the test, which will require a larger value of the test statistic to reject the null hypothesis.

LO11-2Slide18

Comparing Population Means with Unknown

AND

Unequal Population Standard

Deviations – Example

Personnel in a consumer testing laboratory are evaluating the absorbency of paper towels. They wish to compare a set of store brand towels to a similar group of name brand ones. For each brand they dip a ply of the paper into a tub of fluid, allow the paper to drain back into the vat for two minutes, and then evaluate the amount of liquid the paper has taken up from the vat. A random sample of 9 store brand paper towels absorbed the following amounts of liquid in milliliters.

8 8 3 1 9 7 5 5 12

An independent random sample of 12 name brand towels absorbed the following amounts of liquid in milliliters:

12 11 10 6 8 9 9 10 11 9 8 10

Use the .10 significance level and test if there is a

difference

in the mean amount of liquid absorbed by the two types of paper towels.

LO11-2Slide19

Comparing Population Means with Unknown AND Unequal Population Standard

Deviations – Example

The following dot plot provided by MINITAB shows the variances to be unequal.

The following output provided by MINITAB shows the descriptive

statistics.

LO11-2Slide20

Comparing Population Means with Unknown

AND

Unequal Population Standard

Deviations – Example

Step 1: State the null and alternate hypotheses

.

H

0

:

1

= 2

H

1

:

1 ≠ 2

Step 2: State the level of significance. The .10 significance level is stated in the problem.

Step 3:

Find the

appropriate test statistic. A t-test adjusted for unequal variances.LO11-2Slide21

Comparing Population Means with Unknown

AND

Unequal Population Standard

Deviations – Example

Step 4: State the decision rule.

Reject H

0

if

t

>

t/2d.f. or t < -

t

/2,d.f.

t

>

t

.05,10 or t < - t.05, 10 t > 1.812 or t

< -1.812

Step

5:

Compute the value of t and make a decision.The computed value of t (-2.474) is less than the lower critical value, so our decision is to reject the null hypothesis.

Step 6: Interpret the result.

We

conclude that the mean absorption rate for the two

towels

is not the same.

LO11-2Slide22

Comparing Population Means with Unknown AND Unequal Population Standard

Deviations – Minitab

LO11-2Slide23

Comparing Population Means: Hypothesis Testing with Dependent Samples

Dependent samples

are samples that are paired or related in some fashion.

For example:

If you wished to buy a car you would look at the

same

car at two (or more)

different

dealerships and compare the prices.

If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.

LO11-3

Test a hypothesis about the mean population difference between paired or dependent observations.Slide24

Comparing Population Means: Hypothesis Testing with Dependent Samples

Use the following test when the samples are

dependent:

Where

is

the mean of the differences

s

d

is the standard deviation of the differences

n

is

the number of pairs (differences)

LO11-3Slide25

Comparing Population Means: Hypothesis Testing with Dependent Samples – Example

Nickel Savings and Loan wishes to compare the two companies,

Schadek

and Bowyer, it uses to appraise the value of residential homes. Nickel Savings selected a sample of 10 residential properties and scheduled both firms for an appraisal. The results, reported in $000, are shown in the table (right).

At the .05 significance level, can we conclude there is a difference in the mean appraised values of the homes?

LO11-3Slide26

Comparing Population Means: Hypothesis Testing with Dependent

Samples – Example

Step 1: State the null and alternate hypotheses.

H

0

:

d

= 0

H1:

d

0

Step 2: State the level of significance.

The .05 significance level is stated in the problem.Step 3: Select the appropriate test statistic.To test the difference between two population means with dependent samples, we use the t-statistic.

LO11-3Slide27

Comparing Population Means: Hypothesis Testing with Dependent

Samples – Example

Step 4: State the decision rule.

Reject

H

0

if

t

> t

/2, n-1 or t < - t/2,n-1

t

>

t

.025,9

or

t

< - t.025, 9 t > 2.262 or t < -2.262 LO11-3Slide28

Comparing Population Means: Hypothesis Testing with Dependent Samples – Example

Step 5: Take a sample and make a decision.

The computed value of

t,

3.305, is

greater than the higher critical

value, 2.262,

so our decision is to reject the null hypothesis.

LO11-3

Step 6: Interpret the result.

The data indicate that there is a

significant

statistical difference in the property appraisals

from the

two firms. We would hope that appraisals of a property

would

be similar. Slide29

Comparing Population Means: Hypothesis Testing with Dependent Samples – Excel Example

LO11-3Slide30

Dependent versus Independent Samples

How do we differentiate between dependent and independent samples?

Dependent samples are characterized

by a measurement

followed by

an intervention of some kind and then another measurement. This could

be called

a “before” and “after” study

.

Dependent samples are characterized by matching or pairing observations.

Why do we prefer dependent samples to independent samples? By using dependent samples

, we are able to reduce the variation in the sampling distribution.

LO11-4

Explain the difference between dependent and independent samples.