Statistical techniques in business ecohomics chap011

Comparing two populations Does the distribution of the differences in sample means have a mean of 0?. If both samples contain at least 30 observations we use the z distribution as the

Two Sample Tests of Hypothesis

Comparing two populations

Does the distribution of the differences in sample

means have a mean of 0?

If both samples

contain at least 30

observations we use

the z distribution as

the test statistic

No assumptions about the shape of the populations are required

The samples are

2 1

2 1

n

s n

s

X

X z







EXAMPLE 1

with a standard deviation

of $7,000 for a sample of

35 households At the 01 significance level can we conclude the mean income

in Bradford is more?

Bradford and Kane are separated only

by the Conewango River There is competition

between the two cities The local paper recently reported that

the mean household income

in Bradford is $38,000 with

a standard deviation of

$6,000 for a sample of 40

households The same

article reported the mean

income in Kane is $35,000

Example 1 continued

Step 2 State the level of significance The 01 significance level is

stated in the problem

Step 3 Find the appropriate test statistic Because both samples are more than 30, we

can use z as the test statistic.

Step 1 State the null and

alternate hypotheses

H0: µB < µK

H1: µB > µK

Step 4 State the decision rule

The null hypothesis is

rejected if z is greater

than 2.33 or p < 01

Example 1 continued

98 1 35

) 000 ,

7

($

40

) 000 ,

6 ($

000 ,

35

$ 000

, 38

Because the computed Z of 1.98

< critical Z of 2.33, the p-value

of 0239 >  of 01, the decision

is to not reject the null hypothesis

We cannot conclude that the mean household income in Bradford is larger

2 1

2

1

n n

Two Sample Tests of Proportions investigate

whether two samples came from populations with an

equal proportion of successes

The two samples

are pooled using

the following

formula

where X1 and X2 refer to

the number of successes

in the respective samples

of n1 and n2

The value of the test statistic is computed from the following formula

2 1

2 1

) 1

( )

p p

p

p z

c c

missed more than 5

days last year, while a

sample of 300

unmarried workers

showed 35 missed more than five days Use a 05 significance level

Example 2 continued

The null and the alternate hypotheses

H0 : U < M H1 : U > M

The null hypothesis is

rejected if the computed

value of z is greater than

1.65 or the p-value < 05.

The pooled proportion

250300

Example 2 continued

10.1250

)1036

1(1036

.300

)1036

1(1036

250

22300

unmarried workers miss more days in a year than the married workers.

Small Sample Tests of Means

The required assumptions

1 Both populations must follow the normal distribution.

2 The populations must have equal standard deviations.

3 The samples are from independent populations.

Small Sample Tests of Means

The t distribution is used as the test statistic if one or

more of the samples have less than 30 observations.

Small sample test of means

continued

2

)1(

)1(

2 1

2 2 2

2 1 1

s n

2

2 1

1

1

n n

s

X

X t

p

Step Two: Determine the value of t from the

following formula.

Finding the value of the test statistic requires two steps.

Step One: Pool the

sample standard

deviations.

the 05 significance level can the EPA conclude that the mpg is higher on the imported cars?

Example 3 continued

Step 1 State the null and

alternate hypotheses

H0: µD > µI

H1: µD < µI

Step 2 State the level of significance The 05 significance level is stated in the problem

Find the appropriate test

statistic Both samples

are less than 30, so we

use the t distribution.

Example 3 continued

918

92

1215

)9.3)(

112

()

4.2)(

115

(

2

))(

1(

))(

1(

2 2

2 1

2 2 2

2 1 1

s n

s

n

s p

Step 4 The decision rule is to reject

Example 3 continued

640

1 12

1 15

1 312

8

7 35 7

33

1 1

2 1

2

2 1

s

X

X t

p

We compute the value of t as follows.

> critical z of –1.71, the value of 0567 >  of 05, H0

p-is not rejected There p-is insufficient sample evidence

to claim a higher mpg on the imported cars

P(t < -1.64)

= 0567 for a

one-tailed t-test.

Example 3 continued

Hypothesis Testing Involving Paired Observations

samples that are paired or related in some fashion

are samples that are not

related in any way.

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

Hypothesis Testing Involving Paired

where is the mean of the differences

is the standard deviation of the differences

n is the number of pairs (differences)

EXAMPLE 4

agency is comparing the

daily rental cost for

renting a compact car

from Hertz and Avis A

random sample of eight

cities revealed the

following information

At the 05 significance

level can the testing

agency conclude that

a decision.

Example 4 continued

00

1 8

0

 

1623

31

8

8

878

1

2

2 2

s d

894

08

1623

3

00

d t

d

of 05, do not reject the

null hypothesis There

is no difference in the

mean amount charged

by Hertz and Avis.

Advantage of dependent samples:

Reduction in variation in the sampling distribution

in time

Two types of dependent samples

Matched or paired observations