business statistics - qbm117 testing hypotheses about a population mean
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Business Statistics - QBM117
Testing hypotheses about a population mean
Objectives
To test hypotheses about a population mean when sigma is known and when sigma is unknown.
Testing a hypothesis about the population mean, when is known
If the manager wants to know whether the average fill of the boxes differs from 100g, he would specify the null and alternative hypotheses to be
100:
100:0
AH
H
The operations manager is concerned with determining whether the filling process for filling 100g boxes of smarties is working properly. He believes the standard deviation of the filling process is 10g.
Do we know the population standard deviation, or do we only have the sample standard deviation s?
we know
Does the question ask us to test a hypothesis about a mean or a proportion?
a mean
n
xZ
/
In order to test this hypothesis the manager selected a random sample of 25 boxes and found their average weight to be 95g. Is their sufficient evidence at = 0.05 to conclude that the weight of the boxes differs from 100g?
100:
100:0
AH
HStep 1
Step 2
n
xZ
/
Step 3
05.0
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.000-1.96 1.96
Region of non-rejection
Region of rejection
/2 = 0.025
Region of rejection
/2 = 0.025
0.95
Z
100:
100:0
AH
HStep 1
Step 2
n
xz
/
Step 3
96.1025.02/05.0 025.0 zStep 4
96.1zor 96.1if Reject 0 samplesamplezH
952510 xnStep 5
n
xz
/
5.225/10
10095/
n
xz
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.000-1.96 1.96
Region of non-rejection
Region of rejection
/2 = 0.025
Region of rejection
/2 = 0.025
0.95
z-2.5
Since -2.5 < -1.96 we reject HA.
952510 xnStep 5
n
xz
/
5.225/10
10095/
n
xz
Step 6
There is sufficient evidence at = 0.05 to conclude that the average fill is different from 100g.
Do we know the population standard deviation, or do we only have the sample standard deviation s?
Does the question ask us to test a hypothesis about a mean or a proportion?
a mean
ns
xt
/
Testing a hypothesis about the population mean, when is unknown
Exercise 10.26 p346 (9.26 p312 abridged)
160:
160:0
AH
HStep 1
Step 2
ns
xt
/
Step 3
01.0
0
Critical valueRegion of rejection
α = 0.01
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
Region of non-rejection
0.99
t
-2.624
160:
160:0
AH
HStep 1
Step 2
ns
xt
/
Step 3
624.201.0 14,01.01, tt n
Step 4
624.2if Reject 0 sampletH
1501510 xnsStep 5
ns
xtsample
/
87.315/10
160150/
ns
xtsample
xns
0
Region of rejection
α = 0.01
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
Region of non-rejection
0.99
t
-2.624-3.87
Since -3.87 < -2.624 we reject H0.
1501510 xnsStep 5
87.315/10
160150/
ns
xtsample
Step 6
There is sufficient evidence at = 0.01 to conclude that the mean is less than 160.
Do we know the population standard deviation, or do we only have the sample standard deviation s?
Does the question ask us to test a hypothesis about a mean or a proportion?
a mean
ns
xt
/
Testing a hypothesis about the population mean, when is unknown
Exercise 10.30 p347 (9.30 p313 abridged)
32:
32:0
AH
HStep 1
Step 2
ns
xt
/
Step 3
05.0
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.000-2.776 2.776
Region of non-rejection
α/2 = 0.025 α/2 = 0.025
0.95
t
32:
32:0
AH
HStep 1
Step 2
ns
xt
/
Step 3
776.205.0 4,025.01,2/ tt nStep 4
776.2or 776.2if Reject 0 samplesample ttH
4.24591.6 xnsStep 5
ns
xtsample
/
46.25/91.6
324.24/
ns
xtsample
xns
-3 -2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.000-2.776 2.776
Region of non-rejection
α/2 = 0.025 α/2 = 0.025
0.95
t
-2.46
Since -2.46 > -2.776 we do not reject H0.
4.24591.6 xnsStep 5
46.25/91.6
324.24/
ns
xtsample
Step 6
There is insufficient evidence at = 0.05 to conclude that the mean is not equal to 32.
Strong and weak conclusions
Generally we will be presented with a null hypothesis, which we will try to reject.
Before carrying out the test, we know there is a possibility we may make a type I error.
This probability is preset to a small number, say 0.05.
Knowing that we have a small probability of committing a type I error, ie rejecting a null hypothesis when it is true, makes our rejection of the null hypothesis a strong conclusion.
Generally the same cannot be said about not rejecting the null hypothesis.
This is because the probability of , failing to reject a null hypothesis when it is should be rejected, is not preset to a known small number.
Therefore, failing to reject the null hypothesis is generally a fairly weak conclusion because we do not know the probability that we will fail to reject a null hypothesis when it should be rejected.
Reading for next lecture
Chapter 10 Sections 10.7 (Chapter 9 Section 9.7 abridged)
Exercises to be completed before next lecture
S&S 10.2 10.3 10.5 10.11 10.29 10.33
(9.2 9.3 9.5 9.11 9.29 9.33 abridged)
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