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Quantitative Methods HYPOTHESIS TESTING Notation for one group X̄ = sample mean s = sample standard deviation p̂ = sample proportion µ = population mean σ = population standard deviation p = population proportion n = sample size α = significance level = probability of making a type I error GENERAL PROCEDURE 1. State the null (H0 ) and the alternative (Ha ) hypotheses. 2. Calculate the test statistic using the appropriate formula. 3. Find the p-value using the appropriate distribution: (a) find the area in the tail of the appropriate distribution (b) double this area in the case of a two-tail test (6=) 4. Reach a conclusion: (a) if p < α, reject the null hypothesis (b) if p > α, fail to reject the null hypothesis Hypothesis testing for a mean Hypotheses: H0 : µ = µ0 Ha : µ 6= µ0 (two-tail) or µ < µ0 (one-tail) Test statistic: t∗ = or µ > µ0 (one-tail) X̄ − µ0 √ s/ n Distribution: use the t-distribution with df = n − 1. Hypothesis testing for a proportion Hypotheses: Test statistic: H0 : p = p0 Ha : p 6= p0 (two-tail) or p < p0 (one-tail) p̂ − p0 z∗ = p p0 (1 − p0 )/n Distribution: use the normal distribution. or p > p0 (one-tail) Notation for comparing two groups n1 X̄1 s1 p̂1 = = = = size of first group sample mean in first group standard deviation in first group proportion in first group n2 X̄2 s2 p̂2 = = = = size of second group sample mean in second group standard deviation in second group proportion in second group Hypothesis testing for comparing two means Hypotheses: H0 : µ1 = µ2 Ha : µ1 6= µ2 (two-tail) Test statistic: or µ1 < µ2 (one-tail) or µ1 > µ2 (one-tail) X̄1 − X̄2 t∗ = s s21 s2 + 2 n1 n2 Distribution: use the t-distribution with df equal to the minimum of n1 − 1 and n2 − 1. Hypothesis testing for comparing two proportions Hypotheses: H0 : p1 = p2 Ha : p1 6= p2 (two-tail) or Test statistic: z∗ = s p1 < p2 (one-tail) or p1 > p2 (one-tail) p̂1 − p̂2 1 1 p̂(1 − p̂) + n1 n2 where p̂ is the pooled sample proportion, the proportion if we combine the two groups. Distribution: use the normal distribution. Notation for test of independence r = number of rows c = number of columns fo = observed frequencies fe = expected frequencies = row total × column total table total Hypothesis testing for the independence between variables Hypotheses: H0 : the two variables are independent (no relationship) Ha : the two variables are dependent (always one-tail) Test statistic: χ2 = X (fo − fe )2 fe Distribution: use the chi-square distribution with df = (r − 1) × (c − 1).