Hypothesis Testing
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Transcript of 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: > 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, can’t reject H0. But if x is far away, then reject. But what’s “far away” ??
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Hypothesis Testing Errors
State of Nature (Truth)
H0 True H0 False
Reject H0
Fail to Reject H0
Act
ion
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Hypothesis Testing Errors
State of Nature (Truth)
H0 True H0 False
Reject H0
Conclude “a drug user”
Fail to Reject H0 Conclude
“clean”
Act
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Drug Testing ExampleH 0: Not using drugs
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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…
x =2,000
x has a t distribution with 50 degrees of freedom
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When to Reject H0?
x =2,000
xL
x has a t distribution with 50 degrees of freedom
xUP
Rejection Region
P(rejection region) =
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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…
x =30
x has a t distribution with 39 degrees of freedom
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When to Reject H0?
x =30
x has a t distribution with 39 degrees of freedom
xUP
Rejection Region
P(rejection region) =
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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
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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 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
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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, n=47)
Test the hypothesis that the unemployment rate is not related to the injury rate (use =0.01)
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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
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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 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
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error.std
)trueH|parameter(statisticzort
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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 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
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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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