Hypothesis Testing

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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 ? gt 13
• HA ? ? 0

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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, cant reject H0. But if x is
  far away, then reject. But whats far away ??

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Hypothesis Testing Errors
State of Nature (Truth) State of Nature (Truth)
H0 True H0 False
Reject H0
Fail to Reject H0
Action
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Hypothesis Testing Errors
Drug Testing Example H0 Not using drugs
State of Nature (Truth) State of Nature (Truth)
H0 True H0 False
Reject H0 Conclude a drug user
Fail to Reject H0 Conclude clean
Action
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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)

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If H0 is true

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x has a t distribution with 50 degrees of freedom
?x 2,000

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When to Reject H0?

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x has a t distribution with 50 degrees of freedom
Rejection Region
P(rejection region) ?

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?x 2,000

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xL
xUP
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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)?

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If H0 is true

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x has a t distribution with 39 degrees of freedom
?x 30

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When to Reject H0?

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x has a t distribution with 39 degrees of freedom
P(rejection region) ?
Rejection Region

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?x 30

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xUP
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t Values for 39 d.f.
x P(tltx)
1.55 0.9354
1.56 0.9366
1.57 0.9378
1.58 0.9389
1.59 0.9400
1.60 0.9412
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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 doesnt fit with p-values
• Its too easy to get confused between one-tailed
  and two-tailed tests
• So dont follow Siegel, follow Budd

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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)?

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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, n47)

Test the hypothesis that the unemployment rate is
not related to the injury rate (use ?0.01)
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Excel Output
Regression Statistics Regression Statistics Regression Statistics Regression Statistics
R Squared R Squared 0.955
Adj. R Squared Adj. R Squared Adj. R Squared 0.949
Standard Error Standard Error Standard Error 108.052
Obs. Obs. 47
ANOVA ANOVA df SS MS F Significance
Regression Regression 6 9975694.933 1662615.822 142.406 0.000
Residual Residual 40 467007.875 11675.197
Total Total 46 10442702.809
Coeff. Std. Error t stat p value Lower 95 Upper 95
Intercept -168.510 -168.510 258.819 -0.651 0.519 -691.603 354.583
hours 1.244 1.244 0.186 6.565 0.000 0.001 0.002
tons 0.048 0.048 0.403 0.119 0.906 -0.001 0.001
unemp 19.618 19.618 5.660 3.466 0.001 8.178 31.058
WWII 159.851 159.851 78.218 2.044 0.048 1.766 317.935
Act1952 -9.839 -9.839 100.045 -0.098 0.922 -212.038 192.360
Act1969 -203.010 -203.010 111.535 -1.820 0.076 -428.431 22.411
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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

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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?

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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)

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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 trueso 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 truethe safe bet is that H0
  is false
• Need the t or z distribution to distinguish
  small from large via probability of occurrence

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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)

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More Exercises

• Predictor Coef StDev T P
• Constant 6.010 0.235 25.6 0.000
• age -0.006 0.003 -1.71 0.088
• seniorty 0.011 0.003 3.56 0.000
• cognitve -0.005 0.032 -0.17 0.867
• strucint 2.129 0.894 2.38 0.017
• manual -1.513 0.239 -6.33 0.000
• Manlage -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 Performance n3525 Use ?0.01
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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?

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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)

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