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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 610 applied statistics fall 2007 week 7

ENGR 610Applied StatisticsFall 2007 - Week 7

Marshall University

CITE

Jack Smith

overview for today
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
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
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 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
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
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
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
t test unknown two tailed
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
Z Test on Proportion
  • Use normal approximation to binomial distribution, where
p value vs critical value
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
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
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

t test for the difference between two means equal variances
t Test for the Difference between Two Means (Equal Variances)
  • Random samples from independent groups with normal distributions, but with equal and unknown1 and 2
  • Using the pooled sample variance

H0: µ1 = µ2

t test for the difference between two means unequal variances
t Test for the Difference between Two Means (Unequal Variances)
  • 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
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
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
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
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
Homework
  • Work through rest of Appendix 9.1
  • Work and hand in Problems 9.69, 9.71, 9.74
  • Read Chapter 10
    • Design of Experiments: One Factor and Randomized Block Experiments