Testing Hypotheses about a Population Mean: Z and t Tests - Prof. Mihails Levins, Study notes of Data Analysis & Statistical Methods

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Lecture 15: Tests about a Population Mean

Devore: Section 8.

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A Normal Population with known

σ

basic principles of test procedure designThis case is not common in practice. We will use it to illustrate

Let

X

1 ,... , X

n

be a sample size

n

from the normal

population. The null value of the mean is usually denoted

μ

0

μ > μ and we consider testing either of the three possible alternatives

0 ,

μ < μ

0

and

μ

μ

0

The test statistic that we will use is

Z

X

μ

0

σ/

n

It measures the distance of

X

from

μ

0

in standard deviation

units.

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Now consider the case of

H

a

μ

μ

0

. The rejection region

here consists of

z

c

and

z

c

.

For simplicity, consider the case

α

. Then,

P

Z

c

or

Z

c | Z

N

c

) + 1

c ) = 2[

c

)]

Therefore, we select

c

such that

c ) =

P

Z

c

) = 0

; it is

z

0 . 025

. This test

is called a

two-tailed test

.

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Summary

Let

H 0 : μ = μ 0

; define the test statistic

Z

¯

X

−

μ

0

σ/

√

n

.

H

a

μ > μ

0

has the rejection region

z

z

α

and is called

an

upper-tailed test

H

a

μ < μ

0

has the rejection region

z

z

α

and is

called an

lower-tailed test

H

a

μ

μ

0

has the rejection region

z

z

α/

2

or

z

z

α/

2

and is called a

two-tailed test

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Example

Consider Ex. 8.6 in Devore.

Parameter of interest is

μ

true average activation

temperature.

H

0

μ

; H a : μ 

.

Test statistic is

Z

¯x

μ

0

σ/

n

¯x

1. 5 / √ n

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Rejection region is

z

z

α/

2

or

z

z

α/

2

. If

α

, we

have

z

z

0 . 005

and

z

z

0 . 005

With

n

and

¯x

,

z

Since

is outside the rejection region, we fail to reject

H

0

at

significance level

.

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1. can be summarized as follows:probabilities for a lower-tailed test and a two-tailed test. ResultsSimilar derivations can help us to derive Type II error

H

a

μ > μ

0

has the probability of Type II Error

Φ ( z α + μ 0 − μ ′

σ/

√

n

H

a

μ < μ

0

has the probability of Type II Error

1 − Φ ( − z α + μ 0 − μ ′

σ/

√

n

H

a

μ

μ

0

has the probability of Type II Error

z

α/

2 + μ 0 − μ ′

σ/

√ n ) − Φ ( − z

α/

2 + μ 0 − μ ′

σ/

√

n

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Sample Size Determination

specific valueSometimes, we want to bound the value of Type II error for a

μ

′ .

Consider Ex. 8.6 again, fix

α

and specify

β

for such an

alternative value. For

μ

′

= 132

we may want to require

β

in addition to

α

.

The sample size required for that purpose is such that

Φ ( z α + μ 0 − μ ′

σ/

n

β

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Large Sample Tests

normal population distribution or a knownmodified to yield valid test procedures without requiring either aWhen the sample size is large, the z tests described earlier are

σ

.

Let us assume

n >

. Then, the test statistic

Z

X

μ

0

S/

n

is approximately standard normal

procedure for which the significance level is approximatelyThe use of the same rejection regions as before results in a test

α

.

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A Normal Population Distribution

When

n

is small, we can no longer invoke CLT as a justification

for the large sample test

X Remember that for a normally distributed random sample

1 ,... , X

n

, the statistic

T

X

μ

S/

n

has a

t

distribution with

n

df

Therefore, we have the test with

H 0 : μ = μ 0

and a test

statistic value

t

¯x −

μ 0

s/

√

n

(^).

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Example

The

Edison Electric Institute

publishes figures on the annual

number of kilowatt hours expended by various home appliances.

It is claimed that a vacuum cleaner expends an average of

kilowatt hours per year.

Suppose a planned study includes a random sample of

homes and it indicates that VC’s expend an average of

kilowatt hours per year with

s

kilowatt hours.

Assuming the population normality, design a

level test to

see whether VC’s spend less than

kilowatt hours annually

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H

0

μ

kilowatt hours and

H

a

μ <

kilowatt hours

Assuming

α

, we have a critical region

t <

where

t

¯x

μ

0

s/

n

with

df

The value of the statistic is

t

Since

t

is not in the rejection region, we fail to reject

H

0 .

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Figure 1:

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Calculating

β

First, we select

μ

′

and the estimated value for unknown

σ

.

Then, we find an estimated value of

d = | μ 0 − μ ′ |

/σ

. Finally,

the value of

β

is the height of the

n

df curve above the

value of

d

If

n

is not the value for which the corresponding curve

appears visual interpolation is necessary

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