Chapter 8

Inferences Based on a Single Sample: Tests of Hypothesis

2

The Elements of a Test of Hypothesis

7 elements1. 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

3

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

4

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

5

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

6

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

7

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

8

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

9

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

10

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

11

Large-Sample Test of Hypothesis about a Population Mean

12

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

13

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

14

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.

15

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.

16

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

17

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

18

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

19

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 00 1 pq

20

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 ˆ0 3

21

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

22

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

23

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

axz

0

0xa

24

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- .

25

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

26

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

20

2

2

20

2

20

22 1

sn

1

22

20

2 20

2 20

2

20

22 1

sn

22 21

22

2

22 2

02

220