ENGR 610 Applied Statistics Fall 2007 - Week 7

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# ENGR 610 Applied Statistics Fall 2007 - Week 7 - PowerPoint PPT Presentation

ENGR 610 Applied Statistics Fall 2007 - Week 7. Marshall University CITE Jack Smith. Overview for Today. Review Hypothesis Testing , 9.1-9.3 One-Sample Tests of the Mean Go over homework problem 9.2 Hypothesis Testing , 9.4-9.7 Testing for the Difference between Two Means

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### ENGR 610Applied StatisticsFall 2007 - Week 7

Marshall University

CITE

Jack Smith

Overview for Today
• Review Hypothesis Testing, 9.1-9.3
• One-Sample Tests of the Mean
• Go over homework problem 9.2
• Hypothesis Testing, 9.4-9.7
• Testing for the Difference between Two Means
• Testing for the Difference between Two Variances
• Testing for Paired Data or Repeated Measures
• Testing for the Difference among Proportions
• Homework assignment
Hypothesis Testing
• One-Sample Tests for the Mean
• 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
• A “no difference” claim about a population parameter under suspicion based on a sample
• Tested by sample statistics and either rejected or accepted based on critical test (Z, t, F, 2) value
• Rejection implies that an alternative (the opposite) hypothesis is more probable
• Analogous to a mathematical ‘proof by contradiction’ or the legal notion of ‘innocent until proven guilty’
• Only the null hypothesis involves an equality, while the alternative hypothesis deals only with inequalities
Critical Regions
• Critical value of test statistic (Z, t, F, 2,…)
• Based on desired level of significance
• Acceptance (null hypothesis) region, and a
• Rejection (alternative hypothesis) region
• One-tailed or two-tailed
Type I and Type II errors
• Seek proper balance between Type I and II errors
• Type I error - false negative
• Null hypothesis rejected when in fact it is true
• Occurs with probability 
•  = level of significance - chosen!
• (1- ) = confidence coefficient
• Type II error - false positive
• Null hypothesis accepted when in fact it is false
• Occurs with probability 
•  = consumer’s risk
• (1- ) = power of test
• Depends on , difference between hypothesized and actual parameter value, and sample size
Z Test ( known) - Two-tailed
• Critical value (Zc) based on chosen level of significance, 
• 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 (<X> = µ) rejected if Z > Zc or < -Zc, where
Z Test ( known) - One-tailed
• Critical value (Zc) based on chosen level of significance, 
• Typically  = 0.05 (95% confidence), but where Zc = 1.645 (area = 0.95 - 0.50 = 0.45)
• Null hypothesis (<X> ≤µ) rejected if Z > Zc, where
tTest ( unknown) - Two-tailed
• Critical value (tc) based on chosen level of significance, , and degrees of freedom, n-1
• Typically  = 0.05 (95% confidence), where, for exampletc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29
• Null hypothesis rejected if t > tc or < -tc, where

t

Z Test on Proportion
• Use normal approximation to binomial distribution, where
p-value vs critical value
• Use probabilities corresponding to values of test statistic (Z, t,…)
• If the p-value  , accept null hypothesis
• If the p-value < , reject null hypothesis
• E.g., compare p to α instead of t to tc
• More direct
• Does not necessarily assume distribution is normal
Connection with Confidence Interval
• Compute the Confidence Interval for the sample statistic (e.g., the mean) as in Ch 8
• If the hypothesized population parameter is within the interval, accept the null hypothesis, otherwise reject it
• Equivalent to a two-tailed test
• Double α for half-interval (one-tail) test
Z Test for the Difference between Two Means
• Random samples from independent groups with normal distributions and known1 and 2
• Any linear combination (e.g., the difference) of normal distributions (k, k) is also normal

CLT:

Populations 1 & 2 the same

• Random samples from independent groups with normal distributions, but with equal and unknown1 and 2
• Using the pooled sample variance

H0: µ1 = µ2

• Random samples from independent groups with normal distributions, with unequal and unknown1 and 2
• Using the Satterthwaiteapproximation to the degrees of freedom (df)
• Use Excel Data Analysis tool!
F test for the Difference between Two Variances
• Based on F Distribution - a ratio of 2 distributions, assuming normal distributions
• FL(,n1-1,n2-1)  F  FU(,n1-1,n2-1), whereFL(,n1-1,n2-1) = 1/FU(,n2-1,n1-1), and whereFU is given in Table A.7 (using nearest df)
Mean Test for Paired Data or Repeated Measures
• Based on a one-sample test of the corresponding differences (Di)
• Z Test for known population D
• t Test for unknown D (with df = n-1)

H0: D = 0

2 Test for the Difference among Two or More Proportions
• Uses contingency table to compute
• (fe)i = nip or ni(1-p) are the expected frequencies, where p = X/n, and (fo)i are the observed frequencies
• For more than 1 factor, (fe)ij = nipj, where pj = Xj/n
• Uses the upper-tail critical 2 value, with the df = number of groups – 1
• For more than 1 factor, df = (factors -1)*(groups-1)

Sum over all cells

Other Tests
• 2 Test for the Difference between Variances
• Follows directly from the 2 confidence interval for the variance (standard deviation) in Ch 8.
• Very sensitive to non-Normal distributions, so not a robust test.
• Wilcoxon Rank Sum Test between Two Medians
Homework
• Work through rest of Appendix 9.1
• Work and hand in Problems 9.69, 9.71, 9.74