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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:

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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,

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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)

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<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−

= ≥

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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>

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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

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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= × = >− = × = >

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<Student t table>

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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−

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Sample size decision