Probability tests and distributions

8/13/2019 Probability tests and distributions

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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10

Cass Business School 1

Business Statistics

Week 18 - Lecture 1

Hypothesis Testing:

Summary of the basics

The prob-value approach

Significance, effect size and power

Tests: proportion, difference between twomeans

Reference: Quantitative Methods topic 9

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


One-Tail Test: recall the Franchise

Problem

)26

280,000,5(~

:NullthetoAccording

280,900,4:samplesrandom26

000,5:

000,5:

22

1

0

==

==

-1.96

So the null hypothesis cannot be rejected inthis case.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


The prob-value approach

Prob-value (P-value) is the significance levelof the calculated test statistic


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


The prob-value approach in the franchiseexample

Test statistic: ns

x

/2

= 26/2802

100

= -1.82

If we test whether the mean is less than 5,000

( 000,5:1


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


A two tailed test diagram

H0H1H1

Reject H0 Reject H0

2.5% 2.5%

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Solution1. H0: = 15

H1: 15

2. Calculate the test statistic:

3. Decision: we do not reject H0 since 0.625 < 1.96 anddoes not fall into the rejection region or

P-value= 2 (0.266) ~0.52 >0.05

4. The claim is acceptable, no reason to be rejected.

625.01008

155.1422

=

=

=ns

xz

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Significance and effect size

If the sample is large, the result of a test may be

statistically significant (have a low significance

level) even if the difference is small.

Example

We have data for 100 franchises, giving an

average weekly turnover of 4,975 with standard

deviation 143. Can we reject the hypothesis

that the average weekly turnover is 5,000?


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Significance and effect size

Z = 75.1100143

000,5975,4

/2 =

Since this value of z is less than Z*= -1.64 the null

hypothesis is rejected with 95% confidence. We

conclude that the true weekly turnover is less than

5,000. But the difference of 25 may beunimportant even if it is statistically significant.

We should look at the size of the difference (the

effect size) and ask whether it is important or not.

While statistically significant, a difference of 25 in

5,000 is not economically significant.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Power

Power of a test = 1 Pr (Type II error) = 1 -

is the probability of not rejecting H0 when it isfalse

The power of a test is the probability of rejecting

H0 when it is false.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


The power of a test

xD

H H01


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Maximising the power of a testis desirable, provided it does not increase prob(Type Ierror).Three possible ways:1. Avoid situations where the effect size is small (the

difference between the means under H0 and H1 issmall).2. Use a large sample size, so that the sampling

variance of x under both H0 and H1 is reduced andthe distributions under the two hypothesis are more

distinct.3. Use sampling methods that have small samplingvariances (same effect as increasing the sample

size).

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Testing a proportion

A car manufacturer claims that no more than 10%

of its cars should need repairs in the first three years

of their life. A random sample of 50 three-year-old

cars found that 8 had required attention. Does this

contradict the makers claim?

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


The sampling distribution of the sample proportion in

large samples is given by:

))1(

,(~n

Np

Under the null hypothesis (the makers claim):= 0.10

The sample data are:

p= 8/50 = 0.16

n= 50


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Solution

:0H 10.0=

:1H 10.0>

Significance level: 05.0= Critical value of a one-tail test at the 5% level is Z*= 1.64

41.110.016.0

50

9.01.0)1(=

=

=

n

pz

The test statistic is less than the critical value. It falls in the non-rejection region. We do not reject the null hypothesis. Weaccept the manufacturers claim.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


It is claimed that 5% of adults fall asleep

while watching TV.

In a recent survey of 60 adults, 16.4%

indicated that they had fallen asleep in front

of the television in the past month. Do we have support for the claim?

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


05.4

60

95.005.0

05.0164.0

60;164.0

05.0:

05.0:

1

0

=

=

==

=

z

np

H

H

We reject the claim.

Two-tail

P-value~0


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Testing the difference of two means

The random variable to examine is xx 21 , the difference

between the average output, whose distribution for large

samples is:

+nn

xx N2

2

1

2

2121

21,~

The population variances 12

and 22

which are not known

may be replaced by their sample estimates s12

and s22

:0H 021 = :

1H 021

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



H0: 1 = 2 or H0: 1 -2 = 0

H1: 1 2 or H0: 1 -2 0

The test statistic is

( ) ( )

2

2

2

1

2

1

2121

n

s

n

s

xxz

+

=

Assumption: independent

samples future lecture

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



A car company wishes to compare the performance of its two

factories producing an identical car model. Is there any

difference between the two factories at a confidence level of

99%? Output is monitored for 30 days, chosen at random,

with the following results:

Factory 1 Factory 2

Average daily output 420 408

Standard deviation of daily output 25 20

Do these provide evidence of difference between the two

means?


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



This is a two-tail test: there is no a priori reason tobelieve that the output of one factory is higher than thatof the other.

Significance level: 01.0= Lower than normal: the management does not want tointerfere, unless it is really confident of some difference

between the factories.

The critical value of the test is z* = 2.57. This cuts off0.5% in each tail of the standard Normal distribution.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



The test statistic is:

05.20)408420(

21

)()

3020

3025

(22

2

2

1

2

2121=

+

=

+

=

ns

ns

xxz

Decision rule: Z No significant difference between the two factories.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


An employment opportunities committee suspects

that females are not as well paid as their male

counterparts in comparable jobs.

A random sample of 75 males and 64 females in

junior academic positions are selected and the

following annual salary data is obtained:

Male Female

Mean 32,530 31,250

Standard Deviation 780 810

What do you conclude?

Exercises (prepare for Fridays lecture) - 1


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Quantitative Methods:

Exercise 5.5 (page 696)

Problems 5.10 to 5.14 (page 712)

Exercises (prepare for tutorial) - 2

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Business StatisticsWeek18 - Lecture 2

Hypothesis Testing (cont.)

Testing the difference between 2 means

Testing the difference of two proportions

Reference: Quantitative Methods, topic 9

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Testing Difference of Two Means

H0: 1 = 2 or H0: 1 -2 = 0

H1: 1 2 or H0: 1 -2 0


( ) ( )

2

2

2

1

2

1

2121

n

s

n

s

xxz

+

=

Assumption: independent

samples future lecture


13/17

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Recall the example from the last lecture

A car company wishes to compare the performance of its twofactories producing an identical car model. Is there anydifference between the two factories? Output is monitoredfor 30 days, chosen at random, with the following results:

Factory 1 Factory 2

Average daily output 420 408

Stand. Dev. (daily output) 25 20

Is there evidence of difference between the two means?

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Two tail-test: Z*=1.96

The test statistic is:

05.20)408420(

21

)()

3020

3025

(22

2

2

1

2

2121 =

+

=

+

=

ns

ns

xxz

Hence, difference between the two factories

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


An employment opportunities committee suspects

that females are not as well paid as their male

counterparts in comparable jobs.

A random sample of 75 males and 64 females in

junior academic positions are selected and the

following annual salary data is obtained:

Male Female

Mean 32,530 31,250

Standard Deviation 780 810

What do you conclude?


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Sample Info:Male Female

Mean 32,530 31,250

S 780 810

N 75 64


( ) ( )!!45.9

64

810

75

780

250,31530,322222

=

+

=

+

=

F

F

M

M

FMFM

n

s

n

s

xxz

0:

0:

1

0

>

=

FM

FM

H

H

Hypotheses:

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


A union claims that the average income for itsmembers in the UK is below that of employees ofthe same company in Spain.

A survey of 60 employees in the UK showed anaverage income of 895/week with a standarddeviation of 120.

A survey of 100 workers in Spain, after makingadjustments for various differences between the

two countries and converting to sterling, gave anaverage income of 914 with a standard deviationof 90.

Test at the 1% level if Spanish workers earn morethan their British counterparts.

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


33.206.1

100

90

60

120

)914895(

33.201.0

0:

0:

22

*

1

0

>=

+

=

==


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Testing the difference of two

proportions - 1

In a comparison of two holiday companies customers,

of the 75 who went with Happy Days Tours, 45 said

they were satisfied, while 48 of the 90 who went with

Fly by Night Holidays said they were satisfied. Is there

a significant difference between the two companies?

The sample evidence is:

p1 = 45/75 n1 = 75

p2= 48/90 n2 = 90

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



proportions - 2

:0H 021 = :

1H 021 Significance level:

05.0= Critical value: Z*= 1.96

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10



proportions - 3

The distribution of pp 21 is:

+

nnNpp

2

22

1

11

2121

)1()1(,~

so the test statistic is:

z =

nn

pp

2

22

1

11

2121

)1()1(

)) ((

+


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QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


A market research company wants to know ifshoppers are sensitive to the prices of items sold in

a supermarket.

A random sample of 802 shoppers was obtained

and 387 were able to state the correct price of an

item immediately after placing it in the trolley.

Test at the 7% level the null hypothesis that at

least 50% of all shoppers can state the correct

price.

Exercises (prepare for next Mondays lecture) - 1

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


A manufacturer believes that a new promotional

campaign will increase favourable perception of

the product by 10%. Before the campaign, a

random sample of 500 consumers showed that

20% had favourable reactions to the product. After

the campaign, a second random sample of 400

consumers found favourable reactions among

28%.

Test at the 5% significance level whether there has

been a 10% improvement in favourable

perceptions.

Exercises (prepare for next Mondays lecture) - 2

QM Sem. 2 - Week 7 28/02/11 &

04/03/11 - CR10


Quantitative Methods:

Exercises 5.6 and 5.8 (page 701)

Problems 5.15 to 5.18 (page 713)

Exercises (prepare for tutorial) - 3

Probability tests and distributions

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