OAD30763Statistics in Business and Economics

Week 5Dr. Jenne Meyer

Article Reviews

Article review

Hypothesis Testing

5-Step Hypothesis Testing Procedure

Step 1: Set up the null and alternative hypotheses.Step 2: Pick the level of significance (value of )

and find the rejection region.Step 3: Calculate the test statistics.Step 4: Decide whether or not to reject the null

hypothesis.Step 5: Interpret the statistical decision in terms of

the stated problem.

Two-tail testH0: = 750HA: 750

Lower-tail test Upper-tail testH0: 700 H0: 800HA: < 700 HA: >800

Hypothesis Testing

Review

One-tailed Two-tailed

/2 /2

The Rejection Region is the range of values of the test statistics that will lead you to reject the null hypothesis.

Review

For a large sample:

n

XZ

For a small sample:

ns

Xt

**At least 30 units

Apply hypothesis testing to different populations and samples in business research situations

Test of single population, small sample size

Test of a single proportion

Test of two populations, large sample size

Test of two populations, small sample size

Test for difference in two population proportions

The 5-Step Hypothesis Testing Procedure is the same for all

these processes.

Hypothesis Testing

Test for Single Population, Small n

For a small sample: Small sample and unknown σ Calculations are identical to those for

z Becomes identical to z for n > 30 Uses degrees of freedom: df = n - 1

ns

Xt

Review t-table Common Values of a and df and the Corresponding t-Values

Upper Tail Area

Degrees of Freedom (df) 0.25 0.10 0.05 0.01 0.005

20 1.185 1.725 2.086 2.845 3.15321 1.183 1.721 2.080 2.831 3.13522 1.182 1.717 2.074 2.819 3.11923 1.180 1.714 2.069 2.807 3.10424 1.179 1.711 2.064 2.797 3.09125 1.178 1.708 2.060 2.787 3.07826 1.177 1.706 2.056 2.779 3.067

1000000 1.150 1.645 1.960 2.576 2.807

Test for Single Population, Small n

ExampleA State Highway Patrol periodically samples vehicle

speeds at various locations on a particular roadway. The sample of vehicle speeds is used to test the hypothesis

H0: m < 65mph

The locations where H0 is rejected (average speed exceeds 65mph) are the best for radar traps.At Location X, a sample of 16 vehicles shows a mean speed of 68.2 mph with a standard deviation of 3.8 mph. Use an level of significance=.05 to test the hypothesis.

Test for Single Population, Small n

Example, cont.

a=.05

d.f.=16-1=15, ta = 1.753

n = 16

= 68.2 mph

s = 3.8 mph

x

37.316/8.3

652.68

/0

ns

xt

Rejection Region

t 1.753

Since 3.37 > 1.753, we reject H0.

Conclusion: We are 95% confident that the mean speed of vehicles at Location X is greater than 65 mph. Location X is a good candidate for a radar trap.

H0: m < 65mphHA: m > 65mph

Test for Single Population, Small n

Use if concerned with a proportion of the population, p, that have a particular characteristic

Can be used with nominal data Use the same 5-Step Hypothesis Testing Procedures Test Statistic calculated

Z = p -

(1- ) / n√

Test for Single Proportion

Test for Single Proportion Example For a Christmas and New Year’s

week, the National Safety Council estimated that 500 people would be killed and 25,000 injured on the nation’s roads. The NSC claimed that 50% of the accidents would be caused by drunk driving.

A sample of 120 accidents showed that 67 were caused by drunk driving. Use these data to test the NSC’s claim with a = 0.05.

Example, cont.H0: p = .5

HA: p .5

(67/120) - .5

= 1.278

.5(1 - .5)

120

a=.05

Rejection Region

Rejection Region

p = 67/120

n = 120

Z = Since –1.96 < 1.278 < 1.96, we do not reject H0.

Conclusion: There is insufficient evidence to suggest that the population proportion of accidents caused by drunk driving is different from 50%

Test for Single Proportion

Example, cont.

Rejection Region

Z 1.645

250

)1036.1(1036.

300

)1036.1(1036.

0880.1167.

-+

-

-=

Since 1.099 < 1.645, we do not reject H0.

Conclusion: There is insufficient evidence to suggest that there is an difference between the proportion of unmarried workers missing more than 5 days of work than the proportion of married ones

= 1.099

21

21

)1()1(

n

pp

n

pp

ppz

cccc

Test of Two Population Proportions

Often we are interested in comparing two different, independent populations

Sample 1

Population 1Population 2

Sample 2

Figure 13.1 Two Populations and Two Samples

Test of Two Populations

When comparing two different, independent populations the Null Hypothesis takes on the form:

H0: s- p = 0 H0: s = p

H0: s ≤ p

Test of Two Populations

When comparing two different, independent populations the with large n, the test statistic looks like

2

22

1

21

21 )(

nn

XXz

If population std. dev. are unknown, use s1 and s2 instead of σ’s

Test of Two Populations, large n

Example

A study was conducted to compare the mean years of service for those retiring in 1979 with those retiring last year at Acme Manufacturing Co. At the .01 significance level can we conclude that the workers retiring last year gave more service based on the following sample data? Note: Let pop #1= “last year”

Population #1 Population #2

"Last Year" 1979

Sample Mean 30.4 25.6

Sample Standard Deviation 3.6 2.9

Sample Size 45 40

Test of Two Populations, large n

Example, cont.

H0: LY < 1979

HA: LY > 1979

Rejection Region

Z 2.326

a=.01

Test of Two Populations, large n

Example, cont.

Population #1 Population #2

"Last Year" 1979

Sample Mean 30.4 25.6

Sample Standard Deviation 3.6 2.9

Sample Size 45 40

Z 2.326

z

30 4 25 6

3 645

2 940

6 802 2

. .

. ..

Since 6.80 > 2.326, we reject H0.

Conclusion: There is sufficient evidence at the 99% confidence level to suggest that the mean years of service of those retiring last year is greater than the mean years of service of those retiring in 1979.

Test of Two Populations, large n

When comparing two different, independent populations with unknown variances that are assumed equal) with small n, the test statistic looks like

sn s n s

n np2 1 1

22 2

2

1 2

1 1

2

( ) ( )

(Pooled Sample Variance)

df = n1 + n2 – 2t( X X

sn np

=-

+æ

èç

ö

ø÷

1 2

2

1 2

1 1

- D)

Test of Two Populations, small n

Example

To determine whether there is a difference in the time involved in using two versions of software, the new version of the software is compared to the original. Samples are taken from two independent groups using the software (data below). At the .01 significance level, is there a difference in the mean amount of time required to use two versions of software?

Version 1 Version 2

6 58 96 89 7

10 76

Test of Two Populations, small n

Example, cont.Version 1 Version 2

6 58 96 89 7

10 76

S12 = variance 1 S2

2 = variance 2

3.2 2.0

X1 bar = mean 1 X2 bar = mean 2

7.8 7.0

H0 : 1- 2 = 0

HA : 1- 2 = 0/

a =.01 Because we have a two tailed test, there is a/2 = .005 in each tail

df = n1 + n2 – 2 = 5 + 6 – 2 = 9

From t-table, critical cutoffs for two-tail, alpha/2=.005, df=9 is 3.25

Test of Two Populations, small n

Example, cont.

53.2265

)0.2)(5()2.3)(4(

2

))(1())(1(

21

222

2112

nn

snsnsp

83.963.

80.

)6/15/1(53.2

0.78.7 ==+

-=

)/1/1(21

22

1+

-=

nnS

XXt

p

Since .83 < 3.25, we do not reject H0.

Conclusion: There is insufficient evidence to suggest that there is a difference between the mean time to use the two versions of software

Test of Two Populations, small n

When comparing two different population proportions, the Null Hypothesis takes on the form:

The test statistic looks like:

H0: p1- p2 = 0 H0: p1 = p2

21

21

)1()1(

n

pp

n

pp

ppz

cccc

where pTotal number of successes

Total number in samples

X X

n nc

1 2

1 2

= (the weighted mean of the two sample proportions)

Test of Two Population Proportions

Test of Two Population Proportions

Are unmarried workers more likely to be absent from work than married workers? A sample of 250 married workers showed 22 missed more than 5 days last year, while a sample of 300 unmarried workers showed 35 missed more than five days. Use a .05 significance level. Note: let pop #1= unmarried workers.

Example, cont.

H0 : pu = pm

HA : pu > pm

a = .05

Rejection Region

Z 1.645

pu = Unmarried Workers = X1/n1 = 35/300 = .1167

pm = Married Workers = X2/n2 = 22/250 = .0880

1036.250300

2235=

+

+=cp

Test of Two Population Proportions

Assignments

Chapter 9: problems 10, 15, 17, 18, 20, 40

Chapter 10: problems 12, 14 (two sample), 32, 33 (proportions)