STATISTICS FOR MANAGEMENT-II
LEARNING OBJECTIVES:
1- To learn how to use samples to decide whether a population
possesses a particular characteristic.
2- To understand the two types of errors possible when testing
hypotheses.
3- To learn when to use one-tailed tests and when to use two-tailed tests.
4- To learn the steps for testing the hypotheses.
5- To understand how and when to use the normal and t distributions
for testing the hypotheses about population mean and proportions.
INSTRUCTOR’S NAME
1
CHAPTER - 8
Statistical inference has two major areas,
i)
Estimation of parameters.
ii)
Testing of hypothesis.
In this chapter we will discuss about testing of hypothesis.
Testing of hypothesis:
It is process which enables us to decide (on the basis of information obtained
from population) whether to accept or reject a statement or assumption about
unknown value of population parameter.
Statistical hypothesis:
It is statement or assumption about unknown value of population parameter.
This statement may or may not be true. For example statement that average
height of students of college is 62 is called statistical hypothesis.
There are two types of hypotheses.
NULL HYPOTHESIS (HO )
ALTERNATE HYPOTHESIS(H1 )
Null hypothesis:
The hypothesis which to be tested for possible rejection under assumption that
it is true. It is denoted by HO .We can take the above example as null
hypothesis and we write it as,
HO :
µ = 62
Alternate hypothesis:
2
It is any other hypothesis which we accept when the null hypothesis is
rejected. It is denoted by H1 .
For example for above null hypothesis we have three types of alternate
hypothesis as,
H1 :
µ
≠ 62
OR
H1 :
µ > 62
OR
H1 :
µ < 62
Significance level:
It is the probability of rejecting the null hypothesis (HO), when it is true. It is
denoted by α .Usually it is 1% or 5%.It is noted that higher the significance
level, higher is the probability of rejecting null hypothesis (HO), when it is true.
Test Statistic:
A statistic on which the decision can be based whether to accept or reject null
hypothesis (HO ), is called test statistic for example z-statistic, t-statistic etc.
Normal distribution
(z- table)
t-distribution
(t- table)
𝜎 Known n ≤ 30 or n > 30
𝜎 unknown n > 30
𝜎 unknown n ≤ 30
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Critical Region:
The region where we reject our null hypothesis (HO), when it is true is called
critical region.
Critical Value:
The value(s) which separates the acceptance and rejection region is called the
critical value(s).
Two-Tailed Test:
When rejection region is equally taken on both sides of the sampling
distribution, the test is called two-tailed test. So we can say, in this case we
have two rejection regions.
One-Tailed Test:
When rejection region is taken only on one side of sampling distribution (either
right or left), the test is called one-tailed test. So we can say, in this case we
have one rejection region.
.
NULL
HYPOTHESIS
HO : µ = µO
. HO : µ = µO
HO : µ = µO
ALTERNATE HYPOTHESIS
H1 : µ ≠ µO
(Two-Tailed Test)
H1 : µ > µO (One-Tailed Test)
(right tailed test)
H1 : µ < µO
(One-Tailed Test)
(left tailed test)
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Type-I error:
Rejecting null hypothesis (HO), when it is true is called type –I one error. It is
denoted by α.
For example, in the court about accused, we state our null hypothesis (HO) that
accused is innocent (this statement may or may not be true).Suppose that our
null hypothesis (Ho) is true. After hearing, the judge punished the accused. In
this case, the judge has committed type-I error by rejecting true null
hypothesis (HO).
Type-II error:
Accepting null hypothesis (Ho ), when it is false is called type-II error. It is
denoted by β.
For example, in the above mention example, we suppose that our null
hypothesis (HO ) is false. After hearing, the judge did not punish the accused. In
this case, the judge has committed type-II error by accepting false null
hypothesis (HO ).
Power of the test:
Rejecting null hypothesis, when it is false is called power of the test.
It is denoted by 1-β.
True situation
HO is true
HO is false
Decision
Accept HO
Accept HO
Type-II error (β )
Reject HO
Type-I error ( α)
Reject HO
Power of the test
(1- Β)
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EX: (8-20,pg-414)
For the following cases, specify which probability distribution to use in a
hypothesis test.
a) HO: µ = 15, H1 : 𝜇 ≠ 15 ,
x
b) HO: µ = 9.9, H1 : 𝜇 ≠ 9.9 ,
c) HO: µ = 42, H1 : 𝜇 > 42 ,
= 14.8, 𝜎̂
x
x
d) HO: µ = 148, H1 : 𝜇 > 148 ,
e) HO: µ = 8.6, H1 : 𝜇 < 8.6 ,
= 3.0, n
= 10.6, 𝜎 = 2.3, n
= 44, 𝜎 = 4.0, n
x
x
= 35
= 10
= 152, 𝜎̂ = 16.4, n
= 8.5, 𝜎̂
= 16
= 0.15, n
= 29
= 24
Solution:
a) Normal Distribution.
b) Normal Distribution.
c) Normal Distribution.
d) t -Distribution.
e) t - Distribution
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The procedure for testing hypothesis about population parameter, involves the
following steps.
1- State your problem and formulate an appropriate null hypothesis(HO) with
an alternate hypothesis( H1 ).
2- Decide significance level (α) of the test, which is the probability of the
rejecting the null hypothesis, if it is true.
3- Choose an appropriate test-statistic.
4- Compute the value of the test-statistic from the sample data, in order to
decide whether to accept or reject the null hypothesis
5- Determine the critical region in such a way that probability of rejecting null
hypothesis, if it is true, is equal to significance level ( α ).
6- Reject null hypothesis (HO ),if computed value of test-statistic fall in
rejection region and accept alternate hypothesis( H1 ).Otherwise accept null
hypothesis.
EX :
An electric firm makes light bulbs that have a length of life that is
approximately normally distributed with a mean of 812 hours and standard
deviation of 40 hours. Test the hypothesis that µ = 812 hours against
alternative µ ≠ 812 hours, if random sample of 36 bulbs has an average
life of 800 hours. Use 5% level of significance.
Solution:
𝜇 = 812
𝜎 = 40
n = 36
x = 800
σx̄ =
=
𝜎
√𝑛
40
√36
=
40
6
= 6.67
7
HO: µ = 812
Two-tailed test
H1 : 𝜇 ≠ 812
α = 5%
𝜎 Known
n > 30
x −𝜇
Z =
=
normal distribution (z- table)
σx̄
800− 812
=
6.67
−12
6.67
= -1.80
Z = ±1.96
As -1.80 > -1.96 ,so accept HO i.e.
µ = 812
EX :
Past records show that the average marks of students in mathematics is 57
with standard deviation 10.A new method of teaching introduce and a
random sample of 70 students is selected. The sample average is 60.Can we
say that the average marks has increased, at 0.05 level of significance.
Solution:
𝜇 = 57
𝜎 = 10
n = 70
x = 60
𝜎
σx̄ =
√𝑛
=
10
√70
=
10
8.37
= 1.20
HO: µ = 57
H1: µ > 57
α
One tailed test (right tailed)
= 0.05
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𝜎 Known n >30
Z =
z =
normal distribution (z- table)
x −𝜇
σx̄
60−57
1.20
=
3
1.20
= 2.50
z = +1.65
As 2.50 > +1.65, so reject HO and accept H1 i.e. average marks increased.
H.W: Do EX.
EX.
Sc 8-5
Sc 8-6
page 422
page 422
EX:
A company making tires claims that the average life of its product is 35000
miles. A random sample of 16 tires was selected, and it was found that the
mean life was 34000 miles with a standard deviation of 2000 miles. Test the
hypothesis that 𝜇 = 35000 against alternate µ < 35000
at 5% level of significance.
Solution:
µ = 35000
x = 34000
n = 16
𝜎̂ = s = 2000
̂
σx̄
=
=
̂
𝜎
√𝑛
2000
√16
=
2000
4
= 500
9
HO : µ = 35000
One tailed test
(left tailed)
H1 : µ < 35000
α = 0.05
𝜎 unknown n<30
t =
=
t distribution (t table)
x −𝜇
̂
σx̄
34000− 35000
500
=
−1000
500
= -2
df = n-1 = 16- 1 = 15
t 0.05(15) = -1.753
As -2 < -1.753 so reject HO and accept H1 i.e. µ < 35000
EX:
(EX Sc 8-12 pg 436)
Solution:
µ = 13
x = 11.6
n = 7
𝜎̂ = s = 1.3
̂
σx̄
=
̂
𝜎
√𝑛
10
=
1.3
=
√7
1.3
2.65
= 0.49
HO : µ = 13
Two-tailed test
H1 : µ ≠ 13
α
= 0.02
𝜎 Unknown n < 30 t distribution(t table)
t =
=
x −𝜇
̂
σx̄
11.6−13
0.49
=
−1.4
0.49
= - 2.857
df = n – 1 = 7 – 1 = 6
t0.02(6) = ± 3.143
AS - 2.857 > -3.143 SO accept Ho i.e.
µ = 13
H.W: Do EX. Sc 8-11 page 436
EX.
8-45
page 436
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OBJECTIVE SECTION
Q-1 Write short answers for the following.
1- Define critical region.
Answer:
The region where we reject our null hypothesis, when it is true is
Called critical region.
2-When we commit type-I error?
Answer:
We commit type-I error when we reject null hypothesis ,if it is true.
3-Write down the types of hypotheses.
Answer:
i)Null hypothesis.
ii)Alternate hypothesis.
4 -When we commit type-II error?
Answer:
We commit type-II error when we accept null hypothesis, if it is false.
Q-2 choose the correct one.
1- There are -------- types of hypotheses.
a) Three.
b) Five.
c) Two.
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2- When µ > µ0 ,we use,
a) Right tailed test.
b) Left tailed test.
c) Non of these.
3- When 𝜎 known n >30,we use,
a) Z- table
b) t- table
c) F table
4- Usually we use significance level.
a) 1%
b) 5%
c) Both of these.
5- Accepting null hypothesis, if it is false, is called
a) Power of the test.
b) Type-I error.
c) Type-II error.
Q-3 Write true or false for the following.
1- Rejecting null hypothesis, if it is false is called type-I error.
2- When µ < µo , it is two tailed test.
3- When 𝜎 unknown, n>30 we use z-table.
4- We have three types of alternate hypotheses.
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