1
CH9. Hypothesis Testing (One Population)
• Hypotheses are a pair of mutually exclusive, collectively exhaustive
statements about the world.
• One statement or the other must be true, but they cannot both be true.
• H0: Null Hypothesis
H1 (or Ha ): Alternative Hypothesis
• Decision will be made to reject H0 or fail to reject (not reject) H0.
• We can not accept H0, we can only fail to reject H0.
• If H0 is rejected, we tentatively conclude H1 to be accepted.
• Statements to be proved are located in H1.
• A statistical hypothesis is a statement about the value of a population
parameter ө (not statistic).
• A hypothesis test is a decision between two competing mutually exclusive
and collectively exhaustive hypotheses about the value of parameter using a
proper test statistic.
• θ is a parameter and 0θ is a specific value.
• One/ two-side of the test is indicated by H1:
Left-side test Right-side test Two-side test
H0 : 0θ θ≥ 0( )θ θ= H0 : 0θ θ≤ 0( )θ θ= H0 : 0θ θ=
H1 : 0θ θ< H1 : 0θ θ> H1 : 0θ θ≠
• Ex 1) A tire company B claims that their newly developed tires’ average life
expectancy (μ) is more than 7 yrs. Company B will build the hypotheses as
follows:
H0: vs. H1:
• Ex 2) A consumer association claims that the average life expectancy of the
newly developed tire from company B is significantly different from 7 yrs.
The association will build the hypotheses as follows:
H0: vs. H1:
• Ex 3) A tire company D claims that the average life expectancy of the newly
developed tire from company B is less than 7 yrs. Company D will build the
hypotheses as follows:
H0: vs. H1:
2
Types of error
• Type I error: Rejecting the null hypothesis when it is true. This occurs with
probability α .
• Type II error: Failure to reject the null hypothesis when it is false. This
occurs with probability β .
Decision If H0 is true If H0 is false
Reject H0 Type I error (α risk) Correct decision
Not reject H0 Correct decision Type II error ( β risk)
<Type I error>
• α , the probability of a Type I error, is the level of significance (i.e., the
probability that the test statistic falls in the rejection region even though H0
is true).
α = P (reject H0 | H0 is true)
• If we choose a = .05, we expect to commit a Type I error about 5 times in
100.
• A smaller a is more desirable, other things being equal.
<Type II error>
• β , the probability of a type II error, is the probability that the test statistic
falls in the not rejection region even though H0 is false.
β = P (fail to reject H0 | H0 is false)
• A smaller β is more desirable, other things being equal.
<Power>
• The power of a test is the probability that a false hypothesis will be rejected.
• Power = 1 – β
• A low β risk means high power.
Power = P(reject H0 | H0 is false) = 1 – β
• Larger samples lead to increased power.
<Relationship btw type I & type II errors>
• Both a small α and a small β are desirable.
• For a given type of test and fixed sample size, there is a trade-off between α
and β .
• The larger critical value needed to reduce a risk makes it harder to reject H0,
3
thereby increasing β risk.
• Both α andβ can be reduced simultaneously only by increasing the sample
size.
Test procedures
1. Rejection region approach: M1
We decide on rejecting H0 (or not) using a test procedure based on the data we
have.
• Specify the level of significance
• We calculate a test statistic (function of the data)
• We specify the rejection region (a set or range of test statistics values for
which H0 is rejected)
• The null will be rejected in favor of the alternative if and only if the
observed or computed test statistic value falls in the rejection region.
(Choice of confidence level)
• Chosen in advance, common choices for α are
0.10, 0.05, 0.025, 0.01 and 0.005 (i.e., 10%, 5%, 2.5%, 1% and .5%).
• It depends on the purpose and property of test.
• The α risk is the area under the tail(s) of the sampling distribution of test
statistic.
• In a two-sided test, the α risk is split with α /2 in each tail since there are
two ways to reject H0.
(Decision rule)
• The decision rule uses the known sampling distribution of the test statistic
to establish the critical value that divides the sampling distribution into two
regions (rejection/ not rejection).
• Reject H0 if the test statistic lies in the rejection region.
• Right tailed test: reject H0 if the test statistic > right-tail critical value.
• Left tailed test: reject H0 if the test statistic < left-tail critical value.
• Two tailed test: reject H0 if the test statistic < left-tail critical value or if
the test statistic > right-tail critical value.
Ex)
4
<Right tailed test> <Left tailed test> <Two tailed test>
2. p-value approach: M2
• The p-value is the probability of the sample result assuming that H0 is true.
• Using the p-value, we reject H0 at significance level a if p-value ≤ a.
• The p-value is a direct measure of the level of significance at which we
could reject H0 (The smallest significance level at which H0 would be
rejected).
• Therefore, the smaller the p-value, the more we want to reject H0.
Summary of level α hypothesis tests (one population case)
1. Test for µ (population mean):
1) Population is a normal distribution. Population variance 2σ is known.
a) 0 0:H µ µ= vs. 1 0:H µ µ>
Reject 0H if 0
/x Z
n αµ
σ−
≥
or if _p value α≤ , where 0_ ( )/
xp value P Znµ
σ−
= ≥
b) 0 0:H µ µ= vs. 1 0:H µ µ<
Reject 0H if 0
/x Z
n αµ
σ−
≤ −
or if _p value α≤ , where 0_ ( )/
xp value P Znµ
σ−
= ≤
c) 0 0:H µ µ= vs. 1 0:H µ µ≠
Reject 0H if 0/ 2| |
/x Z
n αµ
σ−
≥
or if _p value α≤ , where 0_ 2 ( | |)/
xp value P Znµ
σ−
= ≥
5
2) Population is a normal distribution. Population variance 2σ is unknown.
a) 0 0:H µ µ= vs. 1 0:H µ µ>
Reject 0H if 0( 1),/ n
x ts n α
µ−
−≥
or if _p value α≤ , where 0( 1)_ ( )
/nxp value P ts n
µ−
−= ≥
b) 0 0:H µ µ= vs. 1 0:H µ µ<
Reject 0H if 0( 1),/ n
x ts n α
µ−
−≤ −
or if _p value α≤ , where 0( 1)_ ( )
/nxp value P ts n
µ−
−= ≤
c) 0 0:H µ µ= vs. 1 0:H µ µ≠
Reject 0H if 0( 1), / 2| |
/ nx ts n α
µ−
−≥
or if _p value α≤ , where 0( 1)_ 2 ( | |)
/nxp value p ts n
µ−
−= ≥ .
Note that ‘s’ is a sample standard deviation (2 2
1( ) /( 1)
n
ii
s x x n=
= − −∑ ).
3) Population distribution is not known, but large samples.
(i) Population variance 2σ is known: same as 1)
(ii) Population variance 2σ is unknown
a) 0 0:H µ µ= vs. 1 0:H µ µ>
Reject 0H if 0
/x Zs n α
µ−≥
b) 0 0:H µ µ= vs. 1 0:H µ µ<
Reject 0H if 0
/x Zs n α
µ−≤ −
c) 0 0:H µ µ= vs. 1 0:H µ µ≠
Reject 0H if 0/ 2| |
/x Zs n α
µ−≥
2. Test for π (population proportion):
By central limit theorem;
a) 0 0:H π π= vs. 1 0:H π π>
6
Reject 0H if 0
0 0(1 ) /p Z
n απ
π π−
≥−
or if _p value α≤ , where 0
0 0
_ ( )(1 ) /pp value P Z
nπ
π π−
= ≥−
b) 0 0:H π π= vs. 1 0:H π π<
Reject 0H if 0
0 0(1 ) /p Z
n απ
π π−
≤ −−
or if _p value α≤ , where 0
0 0
_ ( )(1 ) /pp value P Z
nπ
π π−
= ≤−
c) 0 0:H π π= vs. 1 0:H π π≠
Reject 0H if 0/ 2
0 0
| |(1 ) /p Z
n απ
π π−
≥−
or if _p value α≤ , where 0
0 0
_ 2 ( | |)(1 ) /pp value P Z
nπ
π π−
= ≥−
Note that p is the sample proportion.
3. Test for 2σ (population variance):
Population is a normal distribution.
a) 2 2
0 0:H σ σ= vs. 2 2
1 0:H σ σ>
Reject 0H if 2
2( 1),2
0
( 1)n
n sαχ
σ −
−≥
b) 2 2
0 0:H σ σ= vs. 2 2
1 0:H σ σ<
Reject 0H if 2
2( 1),12
0
( 1)n
n sαχ
σ − −
−≤
c) 2 2
0 0:H σ σ= vs. 2 2
1 0:H σ σ≠
Reject 0H if (2
2( 1), / 22
0
( 1)n
n sαχ
σ −
−≥ or
22( 1),1 / 22
0
( 1)n
n sαχ
σ − −
−≤ )
Analogy to confidence interval
• A two-tailed hypothesis test at the 5% level of significance (α = .05) is
exactly equivalent to asking whether the 95% confidence interval for the
mean includes the hypothesized mean.
• If the confidence interval includes the hypothesized mean, then we cannot
7
reject the null hypothesis.
Practice problems
8
9
10
0
1
: 0.6: 0.6
HH
ππ=>
11
0
0 0
0.738 0.6(1 ) / 0.6 0.4 /160pZ
nπ
π π− −
= =− ×
0
0 0
0.25 0.2(1 ) / 0.8 0.2 / 60pZ
nπ
π π− −
= =− ×
0
0
160 0.6 96 10(1 ) 160 0.4 64 10
nnπ
π= × = >− = × = >
0
1
: 0.2: 0.2
HH
ππ=>
0
0
60 0.2 12 10(1 ) 60 0.8 48 10
nnπ
π= × = >− = × = >
12
<Student t table>
13
Power curve for a mean
• Power depends on how far the true value of the parameter ( 1µ in H1) is
from the null hypothesis value ( 0µ in H0).
• The further away the true population value is from the assumed value, the
easier it is for your hypothesis test to detect and the more power it has.
• Remember that β = P (fail to reject H0 | H0 is false)
Power = P (reject H0 | H0 is false) = 1 – β
• We want power to be as close to 1 as possible.
• The values of β and power will vary, depending on the difference between
the true mean 1µ and the hypothesized mean 0µ , the standard deviation, the
sample size n and the level of significance a
Power = f ( , σ , n, a)
Compute Type II error & Power
1 0| |µ µ−
14
Sample size decision
