Download Chapter 4 Hypothesis Testing1

HYPOTHESIS TESTING
CHAPTER 4
BCT 2053 APPLIED STATISTICS
CONTENT
•
4.1 Introduction to Hypothesis Testing
•
4.2 Hypothesis Testing for Mean with known and unknown
Variance
•
4.3 Hypothesis Testing for Difference Means with known
and unknown Population Variance
•
4.4 Hypothesis Testing for Proportion
•
4.5 Hypothesis Testing for the Difference between Two
Proportions
•
4.6 Hypothesis Testing for Variances and Standard
Deviations
•
4.7 Hypothesis Testing for Two Variances and Standard
Deviations
2
4.1 Introduction to
Hypothesis Testing
OBJECTIVES : After completing this chapter, you should be able to
1. Describe the meaning of terms used in
hypothesis testing.
2. State the Null and Alternative Hypothesis
General Terms in Hypothesis Testing
• Hypothesis
– A statement that something TRUE
• Statistical Hypothesis
– A statement about the parameters of one or more
populations.
• Null Hypothesis (Ho)
– A hypothesis to be tested
• Alternative Hypothesis (H1)
– A hypothesis to be considered as an alternative to the
null hypothesis
4
How to make Null hypothesis
• Should have an ‘equal’ sign
• Generally,
H o :   o
two tailed test
H o :   o
right tailed test
H o :   o
left tailed test
parameter
A value
5
How to make Alternative hypothesis
• Should reflect the purpose of the
hypothesis test and different from the
null hypothesis
• Generally (3 types);
H1 :   o
two tailed test
H1 :   o
right tailed test
H1 :   o
left tailed test
6
Basic logic of Hypothesis Testing
• Accept null hypothesis
– if the sample data are consistent with the null
hypothesis
• Reject null hypothesis
– if the sample data are inconsistent with the null
hypothesis, so accept Alternative hypothesis
• Test statistics
– the statistics used as a basis for deciding whether
the null hypothesis should be rejected (z, t, Chi, F)
7
Basic logic of Hypothesis Testing
• Rejection (Critical) Region, (α)
– the set of values for the test statistics that leads to
rejection of the null hypothesis
• Nonrejection (NonCritical) Region,(1 – α)
– the set of values for the test statistics that leads to
nonrejection of the null hypothesis
• Critical values
– the values of the test statistics that separate the
rejection and nonrejection regions.
– A critical value is considered part of the rejection
region
– In general, Reject Ho if test statistics > critical value
8
Rejection Region
Reject Ho
Reject Ho
Accept Ho
1–α
Critical value
Critical value
Reject Ho
Accept Ho
1–α
Critical value
Reject Ho
Accept Ho
1–α
Critical value
9
Type I and II Error
Ho is
Do not reject Ho
Reject Ho
True
False
Correct decision
Type I Error
Type II Error
Correct Decision
Type I Error
- Rejecting the null hypothesis when it is in fact true
  P  Reject H o H o is true   significance level
Type II Error
-Not rejecting the null hypothesis when it is in fact false
  P  Accept H o H o is false 
10
Hypothesis Testing Common Phrase
H1
H1
H0
H0
H0
H1
11
EXERCISE 4.1
State the null and alternative hypotheses for each conjecture.
a. A researcher thinks that if expectant mothers use vitamin
pills, the birth weight of the babies will increase. The average
birth weight of the population is 8.6 pounds.
b. An engineer hypothesizes that the mean number of defects
can be decreased in a manufacturing process of compact
disks by using robots instead of humans for certain tasks.
The mean number of defective disks per 1000 is 18.
c. A psychologist feels that playing soft music during a test will
change the results of the test. The psychologist is not sure
whether the grades will be higher or lower. In the past, the
mean of the scores was 73.
12
Steps In Hypothesis Testing
1.
2.
3.
4.
5.
6.
7.
Define the parameter used
Define the null and alternative hypothesis
Define all the given information
Chose appropriate Test Statistics (z,t,chi,F)
Find Critical value
Test the hypothesis (rejection region)
Make conclusion – there is enough
evidence to reject/accept the claim at α
13
4.2 Hypothesis Testing for Mean
with known and unknown
Variance
OBJECTIVES : After completing this chapter, you should be able to
1. Test mean when σ ² is known.
2. Test mean when σ ² is unknown.
Hypothesis Testing for Mean μ
ztest 
X 
 n
ztest
Where:
X 

s n
ttest
X 

s n
o  population mean
NOTE: Ztest and ttest are test statistics
15
Example 1: Hypothesis testing for mean
μ with known σ²
• A lecturer state that the IQ score for IPT students must be more
higher than other people IQ’s which is known to be normally
distributed with mean 110 and standard deviation 10. To prove
his hypothesis, 25 IPT students were chosen and were given an
IQ test. The result shows that the mean IQ score for 25 IPT
students is 114. Can we accept his hypothesis at significance
level, α = 0.05?
16
Example 2: Hypothesis testing for mean
μ with unknown σ² and n ≥ 30
• UMP students said that they have no enough time to
sleep. A sample of 36 students give that, the mean of
sleep time is 6 hours and the standard deviation is 0.9
hours. It is known that the mean sleep time for adult is
6.5 hours. Can we accept their hypothesis at
significance level, α = 0.01?
17
Example 3: Hypothesis testing for mean
μ with unknown σ² and n < 30
• In a wood cutting process to produce rulers, the mean of rulers
height is set to be equal 100 cm at all times. If the mean height of
rulers is not equal to 100 cm, the process will stop immediately.
The height for a sample of 10 rulers produces by the process
shows below:
100.13 100.11 100.02 99.99 99.98
100.14 100.03 100.10 99.97 100.21
Can we stop the process at significance level, α = 0.05?
18
4.3 Hypothesis Testing for Difference
Means with known and unknown
Population Variance
OBJECTIVES : After completing this chapter, you should be able to
1.
Test the difference between two means when σ ’s
are known.
2.
Test the difference between two means when σ ’s
are unknown and equal.
3.
Test the difference between two means when σ ’s
are unknown and not equal.
Hypothesis Testing for Different Mean
with known and unknown Variance
ztest 
X 1  X 2  o
 12
n1
ztest
X  X 2  o
 1
1 1
sp

n1 n2
ttest
X  X 2  o
 1
1 1
sp

n1 n2
ttest 
ztest 
X 1  X 2  o
s12 s22

n1 n2
v

 22
n2
X 1  X 2  o
s12 s22

n1 n2
 s12 s 22 
  
 n1 n 2 
2
2
2
 s12 
 s 22 
 
 
20
 n1    n 2 
n1  1 n 2  1
Example 4: Hypothesis testing for μ – μ
with known σ ² and σ ²
1
1
2
2
• The mean lifetime for 30 battery type A is 5.3 hours while the
mean lifetime for 35 battery type B is 4.8 hours. If the lifetime
standard deviation for the battery type A is 1 and the lifetime
standard deviation for the battery type B is 0.7 hours, can we
conclude that the lifetime for both batteries type A and type B
are same at significance level, α = 0.05?
21
Example 5: Hypothesis testing for μ – μ with
unknown σ ² & σ ², σ ² ≠ σ ², n ≥ 30 & n ≥ 30
1
1
•
2
1
2
1
2
2
The mean price of 30 acre of land in Cerok before a highway is build is
RM20,000 per acre with standard deviation RM 3000 per acre. The mean price
of 36 acre of land in Cerok after the highway is build is RM50,000 per acre
with standard deviation RM 4000 per acre. Test a hypothesis that the new
highway will increases the land price in Cerok among RM35,000 at significance
level, α = 0.05. Assume that the variances of land price in Cerok are not same
before and after the highway is build.
22
Example 6: Hypothesis testing for μ – μ with
unknown σ ² & σ ², σ ² ≠ σ ², n < 30 & n < 30
1
1
2
1
2
1
2
2
• The average size of a farm in Indiana is 191 acres. The average
size of a farm in Greene is 199 acres. Assume the data were
obtained from two samples with standard deviations of 38 and 12
acres, respectively, and sample sizes of 8 and 10, respectively.
Can we conclude that the at α = 0.05, the average size of the farms
in Indiana is less than Greene? Assume that the variance for both
23
countries are different.
Example 7: Hypothesis testing for μ – μ with
unknown σ ² & σ ², σ ² = σ ² , n ≥ 30 & n ≥ 30
1
1
•
2
1
2
1
2
2
Many studies have been conducted to test the effects of marijuana use on
mental abilities. In a study, groups of light and heavy users of marijuana
were tested for memory recall, with the result below.
– Item sort correctly by light marijuana users:
– Item sort correctly by heavy marijuana users:
n1  64, x1  53.3, s1  3.6
n2  65, x2  51.3, s2  4.5
Use 0.01 significance level to test the claim that the population of heavy
marijuana users has a lower mean than the light users if the variance
population for both users are same.
24
Example 8: Hypothesis testing for μ – μ with
unknown σ ² & σ ², σ ² = σ ² , n < 30 & n < 30
1
1
•
2
1
2
1
2
2
Two catalyst are being analyzed to determine the mean yield of a chemical
process. A test is run in the pilot plant and results are shown below.
catalyst 1: 91.50 94.18 92.18 95.39 91.79 89.07 94.72 89.21
catalyst 2: 89.19 90.95 90.46 93.21 97.19 97.04 91.07 92.75
Is there any different between the mean yield? Use α = 0.05 and assume the
variances population are equal.
25
4.4 Hypothesis Testing for
Proportion
OBJECTIVES : After completing this chapter, you should be able to
1. Test proportions using z-test.
Hypothesis testing for proportion p
ztest 
Where:
x
pˆ  ,
n
pˆ  po
po 1  po  n
po  population proportion
27
Example 9: Hypothesis testing
for proportion p
• An attorney claims that more than 25% of all lawyers
advertise. A sample of 200 lawyers in a certain city showed
that 63 had used some form of advertising. At α = 0.01, is
there enough evidence to support the attorney’s claim?
28
Example 10: Hypothesis testing
for proportion p
• A group of scientist believes that their new medicine can heal
40% of patients. The current medicine in market can only
heal 30% of patients. A research is done to test the
hypothesis made by the scientists. The new medicine is given
to the 100 patients and it shows that only 26 patients are
recovered. Can we accept their hypothesis at significance
level α = 0.05?
29
4.5 Hypothesis Testing for the
Difference between two
Proportions
OBJECTIVES : After completing this chapter, you should be able to
1. Test the difference between two
proportions.
Hypothesis testing for the difference
between two proportions p – p
1
z test 
Where:
x
pˆ  ,
n
2
pˆ 1  pˆ 2  p o
pˆ 1 1  pˆ 1  pˆ 2 1  pˆ 2 

n1
n2
po  population proportion
31
Example 11: Hypothesis testing for
difference proportion p – p
1
2
• In a sample of 200 surgeons, 15% thought the
government should control health care. In a sample of
200 practitioners, 21% felt the same way. At α = 0.01, is
there a difference in the proportions between surgeons
32
and practitioners?
Example 12: Hypothesis testing for
difference proportion p – p
1
2
• Random samples of 747 Malaysian men and 434 Malaysian women
were taken. Of those sampled, 276 men and 195 women said that they
sometimes ordered dish without meat or fish when they eat out. Do the
data provide sufficient evidence to conclude that, in Malaysia, the
percentage of men who sometimes order a dish without meat or fish is
smaller than the percentage of women who sometimes order a dish
33
without meat or fish at significance level α = 0.05?
4.6 Hypothesis Testing for
Variances and Standard
Deviations
OBJECTIVES : After completing this chapter, you should be able to
1. Test single variance and standard
deviation
Hypothesis testing for variance σ²

Where:
2
test

n  1s

2
 o2
 o  population variance
35
Example 13: Hypothesis Testing for σ²
• In a wood cutting process to produce rulers, the variance of
ruler’s height is set to be equal 2 cm² at all times. If the
variance of ruler’s height is not equal to 2 cm², the process will
stop immediately. The height for a sample of 10 rulers produces
by the process shows below:
100.23 100.11 100.42 99.66
100.14 100.33 100.10 99.50
99.68
100.21
Can we stop the process at significance level α = 0.05?
36
Example 14: Hypothesis Testing for σ²
• A hospital administrator believes that the standard deviation of
the number of people using outpatient surgery per day is
greater than 8. A random sample of 15 days is selected and the
standard deviation is 11.2. At α = 0.05, is there enough
evidence to support the administrator’s claim?
37
4.7 Hypothesis Testing for Two
Variances and Standard
Deviations
OBJECTIVES : After completing this chapter, you should be able to
1. Test the difference between two
variances.
Hypothesis testing for variance ratio
σ1²/ σ2²
Ftest
s12
 2
s2
39
Example 15: Hypothesis Testing for
difference proportions σ1²/ σ2²
• Before service, a machine can packed 10 packets of sugar with
variance weight 64 g² while after service the variance weight for 5
packets of sugar are 25 g². Do the services improve the packaging
process at significance level, α = 0.05?
40
Example 16: Hypothesis Testing for
difference proportions σ1²/ σ2²
• A medical researcher whishes to see whether the variance of the
heart rates (in beats per minute) of smokers is different from the
variance of heart rates of people do not smoke. Two samples are
selected, and the data are as shown below. Using α = 0.1, is there
enough evidence to support the claim?
– Smokers:
n1  26, s12  36
Nonsmokers:
n2  18, s2 2  10
41
Summary
• 0.01, 0.05 and 0.1 significance levels are
usually used in testing a hypothesis.
• Hypothesis test are closely related to
confidence interval. Whenever a
confidence interval can be computed, a
hypothesis test can also be performed, and
vice versa.
The End
42