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Transcript of Probability tests and distributions
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8/13/2019 Probability tests and distributions
1/17
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
Cass Business School 2
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
Cass Business School 9
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
Cass Business School 10
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
Cass Business School 16
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
Cass Business School 17
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
Cass Business School 18
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 &
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Cass Business School 19
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
Cass Business School 20
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
Cass Business School 21
The power of a test
xD
H H01
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QM Sem. 2 - Week 7 28/02/11 &
04/03/11 - CR10
Cass Business School 22
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
Cass Business School 23
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
Cass Business School 24
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 &
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Cass Business School 25
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
Cass Business School 26
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
Cass Business School 27
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
Cass Business School 28
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
Cass Business School 29
Testing the difference of two means
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
Cass Business School 30
Testing the difference of two means
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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04/03/11 - CR10
Cass Business School 31
Testing the difference of two means
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
Cass Business School 32
Testing the difference of two means
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
Cass Business School 33
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
Cass Business School 34
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
Cass Business School 35
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
Cass Business School 36
Testing Difference of Two Means
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
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QM Sem. 2 - Week 7 28/02/11 &
04/03/11 - CR10
Cass Business School 37
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
Cass Business School 38
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
Cass Business School 39
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
Cass Business School 40
Sample Info:Male Female
Mean 32,530 31,250
S 780 810
N 75 64
The test statistic is
( ) ( )!!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
Cass Business School 41
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
Cass Business School 42
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
Cass Business School 43
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
Cass Business School 44
Testing the difference of two
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
Cass Business School 45
Testing the difference of two
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 &
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Cass Business School 49
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
Cass Business School 50
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
Cass Business School 51
Quantitative Methods:
Exercises 5.6 and 5.8 (page 701)
Problems 5.15 to 5.18 (page 713)
Exercises (prepare for tutorial) - 3