Estimation and Hypothesis Testing - Business Statistics - Lecture Slides, Slides of Business Statistics

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Business Statistics Estimation & Hypothesis Testing for Two Population Parameters Business Statistics Topic Index Probability Distribution Sampling Distribution Point & Interval Estimates Hypothesis Testing – Single Mean & Single Proportion Hypothesis Testing – Difference of two means, two proportions Hypothesis Testing – One way ANOVA Hypothesis Testing – Non parametric Tests Bivariate Correlation, Linear Regression Index Numbers & Time Series Independent Samples • Different data sources – Unrelated – Independent • Sample selected from one population has no effect on the sample selected from the other population • Use the difference between 2 sample means • Use z test or pooled variance t test 5 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 * σ1 and σ2 known 6 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 Assumptions:  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are ≥ 30  Population standard deviations are known * σ1 and σ2 known 7 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 …and the standard error of x1 – x2 is When σ1 and σ2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z-value… 2 2 2 1 2 1 xx n σ n σσ 21 +=− * σ1 and σ2 unknown, large samples 10 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 Forming interval estimates:  use sample standard deviation s to estimate σ  the test statistic is a z value * σ1 and σ2 unknown, large samples 11 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 ( ) 2 2 2 1 2 1 /221 n s n szxx +±− α The confidence interval for μ1 – μ2 is: * σ1 and σ2 unknown, small samples 12 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 Assumptions:  populations are normally distributed  the populations have equal variances  samples are independent * σ1 and σ2 unknown, small samples 15 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 ( ) 21 p/221 n 1 n 1stxx +±− α The confidence interval for μ1 – μ2 is: Where tα/2 has (n1 + n2 – 2) d.f., and ( ) ( ) 2nn s1ns1ns 21 2 22 2 11 p −+ −+− =* Paired Samples Tests Means of 2 Related Populations – Paired or matched samples – Repeated measures (before/after) – Use difference between paired values: • Eliminates Variation Among Subjects • Assumptions: – Both Populations Are Normally Distributed – Or, if Not Normal, use large samples 16 Paired samples d = x1 - x2 Paired Differences The ith paired difference is di , where 17 Paired samples di = x1i - x2i The point estimate for the population mean paired difference is d : 1n )d(d s n 1i 2 i d − − = ∑ = n d d n 1i i∑ == The sample standard deviation is n is the number of pairs in the paired sample Hypothesis Tests for Two Population Proportions 20 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 Hypothesis tests for μ1 – μ2 21 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 Use a z test statistic Use s to estimate unknown σ , approximate with a z test statistic Use s to estimate unknown σ , use a t test statistic and pooled standard deviation σ1 and σ2 known 22 Population means, independent samples σ1 and σ2 known σ1 and σ2 unknown, n1 and n2 ≥ 30 σ1 and σ2 unknown, n1 or n2 < 30 ( ) ( ) 2 2 2 1 2 1 2121 n σ n σ μμxxz + −−− = The test statistic for μ1 – μ2 is: * Hypothesis tests for μ1 – μ2 25 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 α α/2 α/2 α -zα -zα/2 zα zα/2 Reject H0 if z < -zα Reject H0 if z > zα Reject H0 if z < -zα/2 or z > zα/2 Pooled sp t Test: Example You’re a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming equal variances, is there a difference in average yield (α = 0.05)? 26 Calculating the Test Statistic 27 ( ) ( ) ( ) ( ) 1.2256 22521 1.161251.30121 2nn s1ns1ns 22 21 2 22 2 11 p =−+ −+− = −+ −+− = ( ) ( ) ( ) 2.040 25 1 21 11.2256 02.533.27 n 1 n 1s μμxxz 21 p 2121 = + −− = + −−− = The test statistic is: Hypothesis Testing for Paired Samples 30 Lower tail test: H0: μd ≥ 0 HA: μd < 0 Upper tail test: H0: μd ≤ 0 HA: μd > 0 Two-tailed test: H0: μd = 0 HA: μd ≠ 0 Paired Samples α α/2 α/2 α -tα -tα/2 tα tα/2 Reject H0 if t < -tα Reject H0 if t > tα Reject H0 if t < -tα/2 or t > tα/2 Where t has n - 1 d.f. • Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: 31 Paired Samples Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, di C.B. 6 4 - 2 T.F. 20 6 -14 M.H. 3 2 - 1 R.K. 0 0 0 M.O. 4 0 - 4 -21 d = Σ di n 5.67 1n )d(d s 2 i d = − − = ∑ = -4.2  Has the training made a difference in the number of complaints (at the 0.01 level)? 32 - 4.2 d = 1.66 55.67/ 04.2 n/s μdt d d −= −− = − = H0: μd = 0 HA: μd ≠ 0 Test Statistic: Critical Value = ± 4.604 d.f. = n - 1 = 4 Reject α/2 - 4.604 4.604 Decision: Do not reject H0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints. Paired Samples: Solution Reject α/2 - 1.66 α = .01 Hypothesis Tests for Two Population Proportions 35 Population proportions Lower tail test: H0: p1 ≥ p2 HA: p1 < p2 i.e., H0: p1 – p2 ≥ 0 HA: p1 – p2 < 0 Upper tail test: H0: p1 ≤ p2 HA: p1 > p2 i.e., H0: p1 – p2 ≤ 0 HA: p1 – p2 > 0 Two-tailed test: H0: p1 = p2 HA: p1 ≠ p2 i.e., H0: p1 – p2 = 0 HA: p1 – p2 ≠ 0 Two Population Proportions 36 Population proportions 21 21 21 2211 nn xx nn pnpnp + + = + + = The pooled estimate for the overall proportion is: where x1 and x2 are the numbers from samples 1 and 2 with the characteristic of interest Since we begin by assuming the null hypothesis is true, we assume p1 = p2 and pool the two p estimates Two Population Proportions 37 Population proportions ( ) ( )       +− −−− = 21 2121 n 1 n 1)p1(p ppppz The test statistic for p1 – p2 is: • The hypothesis test is: 40 H0: p1 – p2 = 0 (the two proportions are equal) HA: p1 – p2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p1 = 36/72 = .50 Women: p2 = 31/50 = .62 .549 122 67 5072 3136 nn xxp 21 21 == + + = + + =  The pooled estimate for the overall proportion is: Example: Two population Proportions Example: Two population Proportions 41 The test statistic for p1 – p2 is: .025 -1.96 1.96 .025 -1.31 Decision: Do not reject H0 Conclusion: There is not significant evidence of a difference in proportions who will vote yes between men and women. ( ) ( ) ( ) ( ) 1.31 50 1 72 1.549)(1.549 0.62.50 n 1 n 1p)(1p ppppz 21 2121 −=       +− −− =       +− −−− = Reject H0 Reject H0 Critical Values = ±1.96 For α = .05