ENGR 610 Applied Statistics Fall Week 7 Marshall University CITE Jack Smith

Overview for Today Review Hypothesis Testing, One-Sample Tests of the Mean Go over homework problem 9.2 Hypothesis Testing, 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 (Z c ) based on chosen level of significance,  Typically  = 0.05 (95% confidence), where Z c = 1.96 (area = 0.95/2 = 0.475)  = 0.01 (99%) and (99.9%) are also common, where Z c = 2.57 and 3.29 Null hypothesis ( = µ) rejected if Z > Z c or < -Z c, where

Z Test (  known) - One-tailed Critical value (Z c ) based on chosen level of significance,  Typically  = 0.05 (95% confidence), but where Z c = (area = = 0.45) Null hypothesis ( ≤ µ) rejected if Z > Z c, where

t Test (  unknown) - Two-tailed Critical value (t c ) based on chosen level of significance, , and degrees of freedom, n-1 Typically  = 0.05 (95% confidence), where, for example t c = (upper area = 0.05/2 = 0.025), for n-1 = 29 Null hypothesis rejected if t > t c or < -t c, 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 t c 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

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 H 0 : µ 1 = µ 2

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 Satterthwaite approximation 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 F L ( ,n 1 -1,n 2 -1)  F  F U ( ,n 1 -1,n 2 -1), where F L ( ,n 1 -1,n 2 -1) = 1/F U ( ,n 2 -1,n 1 -1), and where F U 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 (D i ) Z Test for known population  D t Test for unknown  D (with df = n-1) H 0 :  D = 0

 2 Test for the Difference among Two or More Proportions Uses contingency table to compute (f e ) i = n i p or n i (1-p) are the expected frequencies, where p = X/n, and (f o ) i are the observed frequencies For more than 1 factor, (f e ) ij = n i p j, where p j = X j /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 Read Chapter 10 Design of Experiments: One Factor and Randomized Block Experiments