ieem 3201 one and two-sample tests of hypotheses
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IEEM 320 1
One and Two-Sample Tests of
Hypotheses
IEEM 320IEEM151
Notes 20, Page 2
Statistical Hypotheses
Statistical Hypothesis: an assertion or conjecture concerning (parameters of) one or more populations
? e.g., the population mean is equal to a particular value: = 0
? Hypothesis testing: accept or reject a hypothesis based on the sample information
IEEM 320IEEM151
Notes 20, Page 3
Null and Alternative Hypotheses
H0, 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 ? the rejection provides a firmer, clearer assertion
? tend to set the desirable conclusion as H1
IEEM 320IEEM151
Notes 20, Page 4
Null and Alternative Hypotheses
? the form of H1 affects the procedure of the test
? two-tailed test: H0: = 0 vs. H1: 0
? one-tailed test: H0: = 0 vs. H1: > 0
? one-tailed test: H0: = 0 vs. H1: < 0
IEEM 320IEEM151
Notes 20, Page 5
Null and Alternative Hypotheses
Null Hypothesis: the hypothesis we wish to test and is denoted by H0 .
? Alternative Hypothesis: the rejection of the null hypothesis implies the acceptance of an Alternative hypothesis denoted by H1 .
/ 2zn
1– /2/2
(1-)100% confidence interval
Critical values
Critical regions(reject H0 if here)
Acceptance region(accept H0 if here)
0
/ 2zn
x xx
? e.g.,
H0 : =0
H1 : ≠0
IEEM 320IEEM151
Notes 20, Page 6
Type I and Type II Error
Rejection of the null hypothesis when it is true is called a type I error.
? Acceptance of the null hypothesis when it is false is called a type II error.
Correct decisionType I errorReject H0
Type II errorCorrect decisionAccept H0
H0 is falseH0 is true
1– /2/2
1
Probability of committing a type I error.
Probability of committing a type II error if =1
0
x
IEEM 320IEEM151
Notes 20, Page 7
Important Properties
Relationships among , and sample size
? type I error type II error; type I error type II error
? type I error changes with the critical value(s)
? n and ? if the difference between the true value and the
hypothesized value increases
IEEM 320IEEM151
Notes 20, Page 8
The Power of A Test
The power of a test is the probability of rejecting H0 given that a specific alternative is true.
? The power of a test = 1 – .
IEEM 320IEEM151
Notes 20, Page 9
One- and two-Tailed Tests
? One-tailed test:
H0: =0 H0: =0
H1: >0 or H1: <0
? Two-tailed test: H0: =0
H1: 0
? e.g., a one-tailed test:
H0: =68
H1: >68
IEEM 320IEEM151
Notes 20, Page 10
One- and two-Tailed Tests
? One-tailed test:
H0: =0 H0: =0
H1: >0 or H1: <0
? Two-tailed test: H0: =0
H1: 0
? e.g., a one-tailed test:
H0: =68
H1: >68
IEEM 320IEEM151
Notes 20, Page 11
Two-Tailed Test on Mean
H0: = 0, H1: 0
X1, …, Xn ~ i.i.d. normal with variance 2
)1 ,0(normal~/
0
n
X
)or ( 2//2//00 zzPn
X
n
X
if the true mean is 0,
;2//0
zn
Xit is unlikely for it is unlikely for
nzX 2/0
Xzn
2/0or
IEEM 320IEEM151
Notes 20, Page 12
Two-Tailed Test on Mean
/2zn
1–
/2/2
0
/2zn
x
Critical regions(reject H0 if here)x
Acceptance region(accept H0 if here)x
(1-)100% confidence interval
Critical values
IEEM 320IEEM151
Notes 20, Page 13
One-Tailed Test on Mean
H0: = 0, H1: > 0
X1, …, Xn ~ i.i.d. normal with variance 2
)1 ,0(normal~/
0
n
X
)(/
0 zPn
X
if the true mean is 0,
;/
0
zn
Xit is unlikely for it is unlikely for
nzX 0
IEEM 320IEEM151
Notes 20, Page 14
One-Tailed Test on Mean
1–
0 x
nz
Critical regions(reject H0 if here)x
Acceptance region(accept H0 if here)x
Critical values
IEEM 320IEEM151
Notes 20, Page 15
One-Tailed Test on Mean
H0: = 0, H1: < 0
X1, …, Xn ~ i.i.d. normal with variance 2
)1 ,0(normal~/
0
n
X
)(
/0 zPn
X
if the true mean is 0,
;/
0
zn
Xit is unlikely for it is unlikely for
nzX 0
IEEM 320IEEM151
Notes 20, Page 16
One-Tailed Test on Mean
1–
0 xn
z
Critical regions(reject H0 if here)xAcceptance region
(accept H0 if here)x
Critical values
IEEM 320IEEM151
Notes 20, Page 17
Type I and Type II Error
type I error: Rejecting H0 when it is true
? type II error: Accepting H0 when it is false
Correct decisionType I errorReject H0
Type II errorCorrect decisionAccept H0
H0 is falseH0 is true
1
x
1– /2/2
0
x
Probability of committing a type I error.
Probability of committing a type II error if =1
Type II error: change with
a given 1
IEEM 320IEEM151
Notes 20, Page 18
Effect of Sample Size on Type I Error
Solution:
follows a normal distribution with = 68 and =3.6/6 = 0.6
x 0/ 2 / 2( ) 1
/
XP z z
n
Example: Find the type 1 error. H0: = 68, H1: 68.
given = 3.6, n = 36; critical regions: 69or 67 xx
67.1 and 67.1 6.06869
26.06867
1 zz
Hence 095.0)67.1()67.1( zPzP
IEEM 320IEEM151
Notes 20, Page 19
Effect of Sample Size on Type I
Solution:
follows a normal distribution with = 68 and =3.6/8 = 0.45
x 0/ 2 / 2( ) 1
/
XP z z
n
1 2
67 68 69 682.22 and 2.22
0.45 0.45z z
Hence ( 2.22) ( 2.22) 0.0264P z P z
Example: Find the type 1 error. H0: = 68, H1: 68.
given = 3.6, n = 64; critical regions: 69or 67 xx
IEEM 320IEEM151
Notes 20, Page 20
p-Value
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.
IEEM 320IEEM151
Notes 20, Page 21
Type II Error
accepting when H0 is false
type II error: a function of the true value of parameter
Find type II error. H0: = 68, H1: 68.
= 3.6; n = 64, critical regions:
the true = 70
69or 67 xx
Example on page 292
IEEM 320IEEM151
Notes 20, Page 22
Important Properties
Relationships among , and sample size
? type I error type II error; type I error type II error
? type I error changes with the critical value(s)
? n and ? if the difference between the true value and the
hypothesized value increases
IEEM 320IEEM151
Notes 20, Page 23
Examples
? 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.
IEEM 320IEEM151
Notes 20, Page 24
Examples
? 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.
IEEM 320IEEM151
Notes 20, Page 25
Examples
? 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.
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