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Economics 105: Statistics • Go over GH 11 & 12 • GH 13 & 14 due Thursday Hypothesis Testing for Using t • Pharmaceutical manufacturer is concerned about impurity concentration in pills, not wanting it to be different than 3%. A random sample of 16 pills was drawn and found to have a mean impurity level of 3.07% and a standard deviation (s) of .6%. • Test the following hypothesis at the 1% level on the test statistic scale. H0 : m = 3 H1 : m ¹ 3 • Perform the test on the sample statistic scale. • What is the p-value for this test? • Calculate the 99% confidence interval. When to use z or t-test for H0: = 0 Xi~N n ≥ 30 Xi not ~ N s n<30 n ≥ 30 n<30 n ≥ 30 s n<30 n ≥ 30 n<30 Nonparametric versus Parametric Hypothesis Testing Parametric Tests • • • • require the estimation of one or more unknown parameters (e.g., population mean or variance). often, unrealistic assumptions are made about the normality of the underlying population. large sample sizes are often required to invoke the Central Limit Theorem typically more powerful if normality can be assumed Nonparametric versus Parametric Hypothesis Testing Nonparametric Tests (“distribution-free”) • • • • • • usually focus on the sign or rank of the data rather than the exact numerical value do not specify the shape of the parent population can often be used in smaller samples can be used for ordinal data usually more powerful if normality can’t be assumed require special tables of critical values if small n Nonparametric Counterparts Source: Doane and Seward (2009), Applied Statistics in Business & Economics, 2e; McGraw-Hill One-Sample Runs Test Wald-Wolfowitz Runs Test • • • • The one-sample runs test (Wald-Wolfowitz test) detects non-randomness. Ask – Is each observation in a sequence of binary events independent of its predecessor? A nonrandom pattern suggests that the observations are not independent. The hypotheses are H0: Events follow a random pattern H1: Events do not follow a random pattern One-Sample Runs Test Wald-Wolfowitz Runs Test • • To test the hypothesis, first count the number of outcomes of each type. n1 = number of outcomes of the first type n2 = number of outcomes of the second type n = total sample size = n1 + n2 A run is a series of consecutive outcomes of the same type, surrounded by a sequence of outcomes of the other type. One-Sample Runs Test Wald-Wolfowitz Runs Test • For example, consider the following series representing 44 defective (D) or acceptable (A) computer chips: DAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDAAA AAAAAAA • The grouped sequences are: • A run can be a single outcome if it is preceded and followed by outcomes of the other type. One-Sample Runs Test Wald-Wolfowitz Runs Test • • There are 8 runs (R = 8). n1 = number of defective chips (D) = 11 n2 = number of acceptable chips (A) = 33 n = total sample size = n1 + n2 = 11 + 33 = 44 The hypotheses are: H0: Defects follow a random sequence H1: Defects follow a nonrandom sequence One-Sample Runs Test Wald-Wolfowitz Runs Test • When n1 > 10 and n2 > 10, then the number of runs R may be assumed ~ N(m ,s 2 ) R calc R One-Sample Runs Test Wald-Wolfowitz Runs Test • Decision rule for large-sample runs tests at .01 level • Critical values on test statistic scale = +/- 2.576 • Test statistic mR = 17.5, s R = 2.438 8 -17.5 Z= = -3.90 2.438 • Conclusion? Nonparametric Counterparts Source: Doane and Seward (2009), Applied Statistics in Business & Economics, 2e; McGraw-Hill Wilcoxon Signed-Rank Test • • Compares a single sample median with a benchmark using only ranks of the data instead of the original observations. Can also be used to compare paired observations • That’s a two-sample test • • Advantages are - freedom from the normality assumption - robustness to outliers - applicability to ordinal data The population should be roughly symmetric Wilcoxon Signed-Rank Test • To compare the sample median (M) with a benchmark median (M0), the hypotheses are: Wilcoxon Signed-Rank Test • • • Calculate the difference between each observation and the hypothesized median. Rank the differences from smallest to largest by absolute value. Same rank only if same sign before abs value. Add the ranks of the positive differences to obtain the rank sum W. Wilcoxon Signed-Rank Test • • For small samples, a special table is required to obtain critical values. For large samples (n > 20), the test statistic is approximately normal. H 0 : M specialty = 20.2 H1 : M specialty ¹ 20.2 • • Use Excel to get a p-value Reject H0 if p-value < a Wilcoxon Signed-Rank Test pvalue = 2*(1- normsdist(.4344)) = .6639 Hypothesis Testing for Using z • A marketing company claims that it receives an 8% response rate from its mailings to potential customers. To test this claim, a random sample of 500 potential customers were surveyed. 25 responded. • a =.05 • Calculate power and graph a “power curve” • Reminder: CI for uses p in standard error, not ! – because CI does not assume H0 is true