Download Hypothesis Testing & Confidence Intervals for Two Population Means - Prof. Mark Woychick and more Study notes Introduction to Business Management in PDF only on Docsity! 11/10/2010 1 Inference Overview: Chapters 8-10 Popu- lation Method Parameter Distribution Chap- ter One Required sample size n/a z 8 Confidence interval Mean z for sigma known t for sigma unknown 8 Proportion z 8 Hypothesis test Mean z for sigma known t for sigma unknown 9 Proportion z 9 Two Confidence interval Mean z for sigma known t for sigma unknown 10 Hypothesis test Mean z for sigma known t for sigma unknown 10 Hypothesis testing – two populations • http://blog.asmartbear.com/easy-statistics-for-adwords-ab-testing-and-hamsters.html Hypothesis testing recap • The Null: − Is the statement about the population parameter that will be tested − Always includes an equality (= <=, >=) − The “benefit of the doubt” goes to the null hypothesis − The status quo goes into the null hypothesis • The Alternative − Is the opposite of the null hypothesis − Challenges the status quo − Never contains the “=” , “≤” or “” sign − Is generally the hypothesis that is believed (or needs to be supported) by the researcher – a research hypothesis Process of Hypothesis Testing • 1. Specify population parameter of interest • 2. Formulate the null and alternative hypotheses • 3. Specify the desired significance level, α • 4. Define the rejection region • 5. Take a random sample and determine whether or not the sample result is in the rejection region • 6. Reach a decision and draw a conclusion Working with two populations • Form interval estimates • Test hypotheses • For two independent population means • Standard deviations known • Standard deviations unknown Assumptions • When σ1 and σ2 are known − Samples are independent − Sample size >= 30 • When σ1 and σ2 are unknown: − Populations are normally distributed − Populations have equal variances − Samples are independent 11/10/2010 2 Confidence Interval Estimate Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval Point Estimate (Critical Value)(Standard Error) Point estimate and standard error • Point Estimate (Critical Value)(Standard Error) • Point estimate for the difference is x1 – x2 • Critical value is z for known sigma; t for unknown • Standard error formulas: σ1 and σ2 are known: σ1 and σ2 are unknown : 2 2 2 1 2 1 xx n σ n σ σ 21 2nn s1ns1n s 21 2 22 2 11 p Confidence intervals: μ1 – μ2 2 2 2 1 2 1 21 n σ n σ xx z 21 p21 n 1 n 1 sxx t n s tx n σ zx For two populations, formulas are similar: σ1 and σ2 are known: σ1 and σ2 are unknown : σ known: σ unknown : Single population Hypothesis Tests for the Difference Between Two Means •Testing Hypotheses about μ1 – μ2 • Use the same situations discussed already: −Standard deviations known −Standard deviations unknown Hypothesis Tests for Two Populations Lower tail test: H0: μ1 μ2 HA: μ1 < μ2 i.e., H0: μ1 – μ2 0 HA: μ1 – μ2 < 0 Upper tail test: H0: μ1 ≤ μ2 HA: μ1 > μ2 i.e., H0: μ1 – μ2 ≤ 0 HA: μ1 – μ2 > 0 Two-tailed test: H0: μ1 = μ2 HA: μ1 ≠ μ2 i.e., H0: μ1 – μ2 = 0 HA: μ1 – μ2 ≠ 0 Two Population Means, Independent Samples Two Population Means, Independent Samples Lower tail test: H0: μ1 – μ2 0 HA: μ1 – μ2 < 0 Upper tail test: H0: μ1 – μ2 ≤ 0 HA: μ1 – μ2 > 0 Two-tailed test: H0: μ1 – μ2 = 0 HA: μ1 – μ2 ≠ 0 a a/2 a/2 a -za -za/2 za za/2 Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -za/2 or z > za/2 Hypothesis tests for μ1 – μ2 Example: σ1 and σ2 known:
