hypothesis testing

1

Hypothesis Testing

• Make a tentative assumption about a parameter• Evaluate how likely we think this assumption is true

• Null Hypothesis• Default possibility• H0: = 13• H0: = 0

• Alternative (or Research) Hypothesis• Values of a parameter if your theory is correct• HA: > 13• HA: 0

2

Hypothesis Testing

• Test Statistic• Measure used to assess the validity of the null

hypothesis• Rejection Region

• A range of values such that if our test statistic falls into this range, we reject the null hypothesis

• H0: = 13

• If x is close to 13, can’t reject H0. But if x is far away, then reject. But what’s “far away” ??

_ _

3

Hypothesis Testing Errors

State of Nature (Truth)

H0 True H0 False

Reject H0

Fail to Reject H0

Act

ion

4

Hypothesis Testing Errors

State of Nature (Truth)

H0 True H0 False

Reject H0

Conclude “a drug user”

Fail to Reject H0 Conclude

“clean”

Act

ion

Drug Testing ExampleH 0: Not using drugs

5

Testing

• A human resources executive for a huge company wants to set-up a self-insured workers’ compensation plan based on a company-wide average of 2,000 person-days lost per plant. A survey of 51 plants in the company reveals that x = 1,800 and s = 500. Is there sufficient evidence to conclude that company-wide days lost differs from 2,000? (Use = 0.05)

_

6

If H0 is true…

x =2,000

x has a t distribution with 50 degrees of freedom

_

_

7

When to Reject H0?

x =2,000

xL

x has a t distribution with 50 degrees of freedom

xUP

Rejection Region

P(rejection region) =

_

__ _

8

Testing

• Suppose you are a human resources manager and are investigating health insurance costs for your employees. You know that five years ago, the average weekly premium was $30.00. You take a random sample of 40 of your employees and calculate that x = $31.25 and s = 5. • Have health care costs increased (use a 5%

significance level)?

_

9

If H0 is true…

x =30

x has a t distribution with 39 degrees of freedom

_

_

10

When to Reject H0?

x =30

x has a t distribution with 39 degrees of freedom

xUP

Rejection Region

P(rejection region) =

_

_ _

11

t Values for 39 d.f.

x P(t<x)

1.55 0.9354

1.56 0.9366

1.57 0.9378

1.58 0.9389

1.59 0.9400

1.60 0.9412

12

Important Note

• Siegel emphasizes confidence intervals to do hypothesis tests• I do NOT want you to do it this way

• It does not fit the logic that I will emphasize• It doesn’t fit with p-values• It’s too easy to get confused between one-tailed

and two-tailed tests • So don’t follow Siegel, follow Budd

13

Testing p

• An HR manager of a large corporation surveys 1,000 workers and asks “Are you satisfied with your job?” The results are

Responses Percentage

Satisfied 77%

Not Satisfied 23%• You want to examine whether dissatisfaction is

increasing. You know that the fraction of workers who were dissatisfied with their job five years ago was 20%. Has the fraction increased (at the 5% significance level)?

14

Regression

• Recall Coal Mining Safety Problem• Dependent Variable: annual fatal injuries

injury = -168.51 + 1.224 hours + 0.048 tons (258.82) (0.186) (0.403)

+ 19.618 unemp + 159.851 WWII (5.660) (78.218)

-9.839 Act1952 -203.010 Act1969 (100.045) (111.535)

(R2 = 0.9553, n=47)

Test the hypothesis that the unemployment rate is not related to the injury rate (use =0.01)

15

Excel OutputRegression Statistics

R Squared 0.955

Adj. R Squared 0.949

Standard Error 108.052

Obs. 47

ANOVA df SS MS F Significance

Regression 6 9975694.933 1662615.822 142.406 0.000

Residual 40 467007.875 11675.197

Total 46 10442702.809

Coeff. Std. Error t stat p value Lower 95% Upper 95%

Intercept -168.510 258.819 -0.651 0.519 -691.603 354.583

hours 1.244 0.186 6.565 0.000 0.001 0.002

tons 0.048 0.403 0.119 0.906 -0.001 0.001

unemp 19.618 5.660 3.466 0.001 8.178 31.058

WWII 159.851 78.218 2.044 0.048 1.766 317.935

Act1952 -9.839 100.045 -0.098 0.922 -212.038 192.360

Act1969 -203.010 111.535 -1.820 0.076 -428.431 22.411

16

Minitab Output

Predictor Coef StDev T P

Constant -168.5 258.8 -0.65 0.519

hours 1.2235 0.186 6.56 0.000

tons 0.0478 0.403 0.12 0.906

unemp 19.618 5.660 3.47 0.001

WWII 159.85 78.22 2.04 0.048

1952Act -9.8 100.0 -0.10 0.922

1969Act -203.0 111.5 -1.82 0.076

S = 108.1 R-Sq = 95.5% R-Sq(adj) = 94.9%

17

Testing 1- 2

• To compare wages in two large industries, we draw a random sample of 46 hourly wage earners from each industry and find x1 = $7.50 and x2 = $7.90 with s1 = 2.00 and s2 = 1.80.

• Is there sufficient evidence to conclude (using = 0.01) that the average hourly wage in industry 2 is greater than the average in industry 1?

_ _

18

Testing p1- p2

• In a random survey of 850 companies in 1995, 73% of the companies reported that there were no difficulties with employee acceptance of job transfers. In a random survey of 850 companies in 1990, the analogous proportion was 67%. Do these data provide sufficient evidence to conclude that the proportion of companies with no difficulties with employee acceptance of job transfers has changed between 1990 and 1995? (Use = 0.05)

_

19

Many Cases, Same Logic

• If you get a “small” test statistic, then there is a decent probability that you could have drawn this sample with H0 true—so not enough evidence to reject H0

• If you get a “large” test statistic, then there is a low probability that you could have drawn this sample with H0 true—the safe bet is that H0 is false

• Need the t or z distribution to distinguish “small” from “large” via probability of occurrence

statistic

0

error.std

)trueH|parameter(statisticzort

20

More Exercises

• A personnel department has developed an aptitude test for a type of semiskilled worker. The test scores are normally distributed. The developer of the test claims that the mean score is 100. You give the test to 36 semiskilled workers and find that x = 98 and s = 5. Do you agree that µ = 100 at the 5% level?

• Have 50% of all Cyberland Enterprises employees completed a training program? Recall that for the Cyberland Enterprises sample, 29 of the 50 employees sampled completed a training program. (Use = 0.05)

_

21

More Exercises

Predictor Coef StDev T PConstant 6.010 0.235 25.6 0.000age -0.006 0.003 -1.71 0.088seniorty 0.011 0.003 3.56 0.000cognitve -0.005 0.032 -0.17 0.867strucint 2.129 0.894 2.38 0.017manual -1.513 0.239 -6.33 0.000Manl*age -0.042 0.004 -10.4 0.000

• On average, is performance related to seniority?• Do those with structured interviews have higher average

performance levels than those without?• Do those with structured interviews have higher average

performance levels at least two units greater than those without?• Does the relationship between age and performance differ

between manual and non-manual jobs?

Dep. Var: Job Performancen=3525Use =0.01

22

More Exercises

• A large company is analyzing the use of its Employee Assistance Program (EAP). In a random sample of 500 employees, it finds:

Single Employees Married Employees

number of employees 200 300

number using the EAP 75 90

• Using =0.01, is there sufficient evidence to conclude that single and married employees differ in the usage rate of the EAP?

23

More Exercises

• Independent random samples of male and female hourly wage employees yield the following summary statistics:

Male Employees Female Employees

n1 = 45 n2 = 32

x1 = 9.25 x2 = 8.70

s1 = 1.00 s2 = 0.80

• Is there sufficient evidence to conclude that, on average, women earn less than men? (Use = 0.10)

__