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One and Two-Sample Tests of
Hypotheses
IEEM 320
1
Statistical Hypotheses

Statistical Hypothesis: an assertion or conjecture
concerning (parameters of) one or more populations
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IEEM
e.g., the population mean is equal to a particular
value: m = m0
Hypothesis testing: accept or reject a hypothesis based
on the sample information
Notes 20, Page 2
Null and Alternative Hypotheses
 H0,
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null hypothesis: the hypothesis subject to
testing
H1, alternative hypothesis: H1 is rejected if H0 is
accepted, and vice versa
condition to reject H0: if H0 is true, it is highly
unlikely to get the given set of sample values
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the rejection provides a firmer, clearer assertion
tend to set the desirable conclusion as H1
Notes 20, Page 3
Null and Alternative Hypotheses
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the form of H1 affects the procedure of the test
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IEEM
two-tailed test: H0: m = m0 vs. H1: m  m0
one-tailed test: H0: m = m0 vs. H1: m > m0
one-tailed test: H0: m = m0 vs. H1: m < m0
Notes 20, Page 4
Null and Alternative Hypotheses

Null Hypothesis: the hypothesis we wish to test and is
denoted by H0 .
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Alternative Hypothesis: the rejection of the null
hypothesis implies the acceptance of an Alternative
hypothesis denoted by H1 .
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e.g.,
Acceptance region
(accept H0 if x here)
H0 : m =m0
H1 : m≠m0
IEEM
1– 
/2
Critical values
Critical regions
(reject H0 if x here)
m  z / 2

n
m0
/2
m  z / 2

x
n
(1-)100%
confidence interval
Notes 20, Page 5
Type I and Type II Error
H0 is true
H0 is false
Accept H0
Correct decision
Type II error
Reject H0
Type I error
Correct decision

Rejection of the null hypothesis when it is true is
called a type I error.
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Acceptance of the null hypothesis when it is false is
called a type II error.
Probability of committing
a type II error if m=m1
/2
m1
IEEM
b
Probability of committing
a type I error.
1– 
/2
m0
x
Notes 20, Page 6
Important Properties

IEEM
Relationships among , b and sample size
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type I error  type II error; type I error  type
II error
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type I error changes with the critical value(s)
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n   and b
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b if the difference between the true value and the
hypothesized value increases
Notes 20, Page 7
The Power of A Test
IEEM

The power of a test is the probability of rejecting H0
given that a specific alternative is true.
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The power of a test = 1 – b.
Notes 20, Page 8
One- and two-Tailed Tests
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One-tailed test:
H0: m =m0
H1: m >m0
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Two-tailed test:
or
H0: m =m0
H1: m <m0
H0: m =m0
H1: m m0
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e.g., a one-tailed test:
H0: m =68
H1: m >68
IEEM
Notes 20, Page 9
One- and two-Tailed Tests
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One-tailed test:
H0: m =m0
H1: m >m0
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Two-tailed test:
or
H0: m =m0
H1: m <m0
H0: m =m0
H1: m m0
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e.g., a one-tailed test:
H0: m =68
H1: m >68
IEEM
Notes 20, Page 10
Two-Tailed Test on Mean
H0: m = m0, H1: m  m0
 X1, …, Xn ~ i.i.d. normal with variance 2

X  m0
~ normal(0, 1)
/ n
P(
X m 0
/ n
  z / 2 or
X m 0
/ n
 z / 2 )  
if the true mean is m0,
it is unlikely for
X m0
/ n
 z / 2 ;
X  m 0  z / 2
it is unlikely for
or m0  z / 2
IEEM

n
X

n
Notes 20, Page 11
Two-Tailed Test on Mean
Acceptance region
(accept H0 if x here)
Critical regions
(reject H0 if x here)
1– 
/2
/2
m  z/2

n
m0
m  z/2

x
n
Critical values
(1-)100% confidence
interval
IEEM
Notes 20, Page 12
One-Tailed Test on Mean
H0: m = m0, H1: m > m0
 X1, …, Xn ~ i.i.d. normal with variance 2

X  m0
~ normal(0, 1)
/ n
P(
X m 0
/ n
 z )  
if the true mean is m0,
it is unlikely for
it is unlikely for
IEEM
X m 0
/ n
 z ;
X  m 0  z

n
Notes 20, Page 13
One-Tailed Test on Mean
Acceptance region
(accept H0 if x here)
Critical regions
(reject H0 if x here)
1– 

m0
m  z

n
x
Critical values
IEEM
Notes 20, Page 14
One-Tailed Test on Mean
H0: m = m0, H1: m < m0
 X1, …, Xn ~ i.i.d. normal with variance 2

X  m0
~ normal(0, 1)
/ n
X m 0
P(
/ n
  z )  
if the true mean is m0,
it is unlikely for
it is unlikely for
IEEM
X m0
/ n
  z ;
X  m 0  z

n
Notes 20, Page 15
One-Tailed Test on Mean
Critical regions
(reject H0 if x here)
Acceptance region
(accept H0 if x here)
1– 

m  z
m0

n
x
Critical values
IEEM
Notes 20, Page 16
Type I  and Type II Error b

type I error: Rejecting H0 when it is true
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type II error: Accepting H0 when it is false
H0 is true
H0 is false
Accept H0
Correct decision
Type II error
Reject H0
Type I error
Correct decision
Probability of committing
a type II error if m=m1
Type II error:
change with a
given m1
IEEM
/2
m1
b
Probability of committing
a type I error.
1– 
/2
m0
x
Notes 20, Page 17
Effect of Sample Size on Type I Error
Example: Find the type 1 error. H0: m = 68, H1: m  68.
given  = 3.6, n = 36; critical regions: x  67 or x  69
Solution:
follows a normal distribution with m = 68 and  =3.6/6 = 0.6
X  m0
xP (  z
 z / 2 )  1  
 /2 
/ n
z1  670.668  1.67 and z2  690.668  1.67
Hen
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IEEM
  P( z  1.67)  P( z  1.67)  0.095
Notes 20, Page 18
Effect of Sample Size on Type I
Example: Find the type 1 error. H0: m = 68, H1: m  68.
given  = 3.6, n = 64; critical regions: x  67 or x  69
Solution:
follows a normal distribution with m = 68 and  =3.6/8 = 0.45
X  m0
xP (  z
 z / 2 )  1  
 /2 
/ n
z1 
Hen
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IEEM
67  68
69  68
 2.22 and z2 
 2.22
0.45
0.45
  P( z  2.22)  P( z  2.22)  0.0264
Notes 20, Page 19
p-Value
IEEM

A p-value is the lowest level (of significance) at
which the observed value of the test statistic is
significant.

Calculate the p-value and compare it with a preset
significance level . If the p-value is smaller than
, we reject the null hypothesis.
Notes 20, Page 20
Type II Error

accepting when H0 is false

type II error: a function of the true value of parameter
 Find
type II error. H0: m = 68, H1: m  68.
 = 3.6; n = 64, critical regions: x  67 or x  69
the true m = 70
Example on
page 292
IEEM
Notes 20, Page 21
Important Properties

IEEM
Relationships among , b and sample size
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type I error  type II error; type I error  type
II error
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type I error changes with the critical value(s)
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n   and b
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b if the difference between the true value and the
hypothesized value increases
Notes 20, Page 22
Examples
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IEEM
A random sample of 100 recorded deaths in U.S>
during the past year showed an average life span of
71.8 years. Assuming a population standard
deviation of 8.9 years, does this seem to indicate
that the mean life span today is greater than 70
years? Use a 0.05 level of significance.
Notes 20, Page 23
Examples
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IEEM
A manufacturer of sports equipment has developed
a new fishing line that claims has a mean breaking
strength of 8 kilograms with s standard deviation of
0.5 kilogram. Test the hypothesis that u=8
kilograms again H1 that u is not equal to 8 if a
random of sample of 50 lines is tested and found to
have a mean breaking strength of 7.8 kilograms.
Use a 0.01 level significance.
Notes 20, Page 24
Examples
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IEEM
Some company has published figures on the annual
number of kilowatt-hours expended by various
home appliances. It is claimed that a vacuum
cleaner expends an average of 46 kilowatt-hours
per hour. If a random sample of 12 homes included
in a planned study indicates that vacuum cleaners
expended an average of 42 kilowatt-hours per year
with a standard deviation of 11.9 kilowatt-hours,
does this suggest at the 0.05 level of significance
that vacuum cleaners expend, on average, less than
46 kilowatt-hours annually? Assume the population
of kilowatt-hours to be normal.
Notes 20, Page 25