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Chapter 8

Inferences Based on a Single

Sample: Tests of Hypothesis

The Elements of a Test of

Hypothesis

7 elements

1. The Null hypothesis

2. The alternate, or research hypothesis

3. The test statistic

4. The rejection region

5. The assumptions

6. The Experiment and test statistic calculation

7. The Conclusion

The Elements of a Test of

Hypothesis

Does a manufacturer’s pipe meet building

code?

Null hypothesis – Pipe does not meet code

(H0): < 2400

Alternate hypothesis – Pipe meets

specifications

(Ha): > 2400

The Elements of a Test of

Hypothesis

Test statistic to be used

Rejection region Determined by Type I error, which is the probability of

rejecting the null hypothesis when it is true, which is .

Here, we set =.05

Region is z>1.645, from

z value table

n

xxz

x

24002400

The Elements of a Test of

Hypothesis

Assume that s is a good approximation of

Sample of 60 taken, , s=200

Test statistic is

Test statistic lies in rejection region,

therefore we reject H0 and accept Ha that the

pipe meets building code

12.228.28

60

50200

240024602400

ns

xz

2460x

The Elements of a Test of

Hypothesis

Type I vs Type II Error

Conclusions and Consequences for a Test of Hypothesis

True State of Nature

Conclusion H0 True Ha True

Accept H0

(Assume H0 True)

Correct decision Type II error

(probability )

Reject H0

(Assume Ha True)

Type I error

(probability )

Correct decision

The Elements of a Test of

Hypothesis

1. The Null hypothesis – the status quo. What

we will accept unless proven otherwise.

Stated as H0: parameter = value

2. The Alternative (research) hypothesis (Ha) –

theory that contradicts H0. Will be accepted if

there is evidence to establish its truth

3. Test Statistic – sample statistic used to

determine whether or not to reject Ho and

accept Ha

The Elements of a Test of

Hypothesis

4. The rejection region – the region that will lead to

H0 being rejected and Ha accepted. Set to

minimize the likelihood of a Type I error

5. The assumptions – clear statements about the

population being sampled

6. The Experiment and test statistic calculation –

performance of sampling and calculation of

value of test statistic

7. The Conclusion – decision to (not) reject H0,

based on a comparison of test statistic to

rejection region

Large-Sample Test of Hypothesis

about a Population Mean

Null hypothesis is the status quo, expressed in one

of three forms

H0: = 2400

H0: ≤ 2400

H0: ≥ 2400

It represents what must be accepted if the

alternative hypothesis is not accepted as a result

of the hypothesis test

Large-Sample Test of Hypothesis

about a Population Mean

Alternative hypothesis can take one of 3 forms:

One-tailed, upper tail Ha: <2400

One-tailed, upper tail Ha: >2400

Two-tailed Ha: 2400

Large-Sample Test of Hypothesis

about a Population Mean

Rejection Regions for Common Values of

Alternative Hypotheses

Lower-Tailed Upper-Tailed Two-Tailed

= .10 z < -1.28 z > 1.28 z < -1.645 or z > 1.645

= .05 z < -1.645 z > 1.645 Z < -1.96 or z > 1.96

= .01 z < -2.33 z > 2.33 Z < -2.575 or z > 2.575

Large-Sample Test of Hypothesis

about a Population Mean

If we have: n=100, = 11.85, s = .5, and we

want to test if 12 with a 99% confidence

level, our setup would be as follows:

H0: = 12

Ha: 12

Test statistic

Rejection region z < -2.575 or z > 2.575

(two-tailed)

x

x

xz

12

Large-Sample Test of Hypothesis

about a Population Mean

CLT applies, therefore no assumptions

about population are needed

Solve

Since z falls in the rejection region, we

conclude that at .01 level of significance the

observed mean differs significantly from 12

3.105.

15.

10

1285.11

100

1285.111212

sn

xxz

x

Observed Significance Levels: p-

Values

The p-value, or observed significance level,

is the smallest that can be set that will

result in the research hypothesis being

accepted.

Observed Significance Levels: p-

Values

Steps:

Determine value of test statistic z

The p-value is the area to the right of z if Ha

is one-tailed, upper tailed

The p-value is the area to the left of z if Ha is

one-tailed, lower tailed

The p-valued is twice the tail area beyond z

if Ha is two-tailed.

Observed Significance Levels: p-

Values

When p-values are used, results are

reported by setting the maximum you are

willing to tolerate, and comparing p-value to

that to reject or not reject H0

Small-Sample Test of Hypothesis

about a Population Mean

When sample size is small (<30) we use a

different sampling distribution for determining the

rejection region and we calculate a different test

statistic

The t-statistic and t distribution are used in cases

of a small sample test of hypothesis about

All steps of the test are the same, and an

assumption about the population distribution is

now necessary, since CLT does not apply

Small-Sample Test of Hypothesis

about a Population Mean

where t and t/2 are based on (n-1) degrees of freedom

Rejection region:Rejection region:

(or when Ha:

Test Statistic:Test Statistic:

Ha:Ha: (or Ha: )

H0:H0:

Two-Tailed TestOne-Tailed Test

Small-Sample Test of Hypothesis about

0

0

ns

xt

0

0

0

ns

xt

0

tt

tt

2tt

0

0

Large-Sample Test of Hypothesis

about a Population Proportion

Rejection region:Rejection region:

(or when

where, according to H0, and

Test Statistic:Test Statistic:

Ha:Ha: (or Ha: )

H0:H0:

Two-Tailed TestOne-Tailed Test

Large-Sample Test of Hypothesis about

0pp

p

0pp

p

ppz

ˆ

0ˆ

0pp

0pp

p

ppz

0ˆ

zz

zz

2zz

0pp

0pp

nqpp 00ˆ

001 pq

Large-Sample Test of Hypothesis

about a Population Proportion

Assumptions needed for a Valid Large-Sample

Test of Hypothesis for p

•A random sample is selected from a binomial

population

•The sample size n is large (condition satisfied if

falls between 0 and 1 pp

ˆ03

Calculating Type II Error

Probabilities: More about

Type II error is associated with , which is

the probability that we will accept H0 when

Ha is true

Calculating a value for can only be done if

we assume a true value for

There is a different value of for every value

of

Calculating Type II Error

Probabilities: More about

Steps for calculating for a Large-Sample Test about

1. Calculate the value(s) of corresponding to the

borders of the rejection region using one of the

following:

Upper-tailed test:

Lower-tailed test:

Two-tailed test:

x

n

szzx

x

000

n

szzx

x

000

n

szzx

xL

000

n

szzx

xU

000

Calculating Type II Error

Probabilities: More about

2. Specify the value of in Ha for

which is to be calculated.

3. Convert border values of to

z values using the mean , and

the formula

4. Sketch the alternate distribution,

shade the area in the acceptance

region and use the z statistics and

table to find the shaded area,

a

x

ax

z

0

0x

a

Calculating Type II Error

Probabilities: More about

The Power of a test – the probability that the

test will correctly lead to the rejection of H0

for a particular value of in Ha. Power is

calculated as 1- .

Tests of Hypothesis about a

Population Variance

Hypotheses about the variance use the Chi-

Square distribution and statistic

The quantity has a sampling

distribution that follows the

chi-square distribution

assuming the population the

sample is drawn from is

normally distributed.

2

21

sn

Tests of Hypothesis about a

Population Variance

where is the hypothesized variance and the distribution of is based

on (n-1) degrees of freedom

Rejection region:

Or

Rejection region:

(or when Ha:

Test Statistic:Test Statistic:

Ha:Ha: (or Ha: )

H0:H0:

Two-Tailed TestOne-Tailed Test

Small-Sample Test of Hypothesis about

2

0

2

2

2

0

2

2

0

2

2 1

sn

1

22

2

0

2

2

0

2

2

0

2

2

0

2

2 1

sn

22

21

22

2

22

2

0

2

22

0