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ENGR 610
Applied Statistics
Fall 2007 - Week 6
Marshall University
CITE
Jack Smith
Overview for Today
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Go over exam #1
Hypothesis Testing (Ch 9, sections 1-3)
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Introduction and Basic Concepts
One-Sample Tests for the Mean
Homework assignment
Hypothesis Testing
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Introduction and Basic Concepts
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Null hypothesis
Critical value and regions of rejection,
acceptance (non-rejection)
Type I and Type II errors
Level of significance
Power of a test
Hypothesis Testing
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One-Sample Tests for the Mean
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Z Test ( known)
t Test ( unknown)
Two-tailed and one-tailed tests
p-value
Connection with Confidence Interval
Z Test for the proportion
Null hypothesis
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A suspect “no difference” claim about a population
parameter
Tested by sample statistics and either rejected or
accepted based on critical value
Rejection implies that an alternative (opposite)
hypothesis is more probable
Analogous to ‘proof by contradiction’ or ‘innocent until
proven guilty’
Only the null hypothesis involves an equality, while
the alternative hypothesis deals only with inequalities
Critical Regions
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Critical value (Z, t, F, 2,…)
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Rejection (alternative hypothesis) region
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Based on desired level of significance
One-tailed or two-tailed
Acceptance (non-rejection) region
Type I and Type II errors
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Type I error - false negative
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Null hypothesis rejected when in fact true
Occurs with probability 
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Type II error - false positive
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Null hypothesis accepted when in fact false
Occurs with probability 
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 = level of significance - chosen!
(1- ) = confidence coefficient
 = consumer’s risk
(1- ) = power of test
Depends on , difference between hypothesized and actual
parameter value, and sample size
Seek proper balance between Type I and II errors
Z Test ( known) - Two-tailed
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Critical value (Zc) based on chosen level of
significance, 
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Typically  = 0.05 (95% confidence), where
Zc = 1.96 (area = 0.95/2 = 0.475)
 = 0.01 (99%) and 0.001 (99.9%) are also common, where
Zc = 2.57 and 3.29
Null hypothesis rejected if sample Z > Zc or < -Zc,
where
Z
X 

n
Z Test ( known) - One-tailed
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Critical value (Zc) based on chosen level of
significance, 
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Typically  = 0.05 (95% confidence), but where
Zc = 1.645 (area = 0.95 - 0.50 = 0.45)
Null hypothesis rejected if sample Z > Zc, where
Z
X 

n
t Test ( unknown) - Two-tailed
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Critical value (tc) based on chosen level of
significance, , and degrees of freedom, n-1
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Typically  = 0.05 (95% confidence), where, for example
tc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29
Null hypothesis rejected if sample t > tc or < -tc, where
X 
t
s
n
t
Z Test on Proportion
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Using normal approximation to binomial
distribution
p
Z
n (1  )
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p-value
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Use probabilities corresponding to Z
values
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If the p-value  , accept null hypothesis
If the p-value < , reject null hypothesis
More direct, does not necessarily
assume distribution is normal
Connection with Confidence Interval
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Compute the Confidence Interval for the
statistic (e.g., the mean) as in Ch 8
If the computed statistic is within the
interval, accept the null hypothesis,
otherwise reject it
Homework
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Read Chapter 9 (all sections)
Work through Appendix 9.1 (Z test and t test)
Work and hand in Problem 9.2