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How to do T-test? Cindy Wu 1 Difference Between Two Means Population means, independent samples * σ1 and σ2 known Use a Z test statistic σ1 and σ2 unknown, assumed equal Use Sp to estimate unknown σ , use a t test statistic and pooled standard deviation σ1 and σ2 unknown, not assumed equal Use S1 and S2 to estimate unknown σ1 and σ2, use a separate-variance t test Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 2 When is T-test performed? • To compare the means between two groups • One-tail T-test – μ1 ≧ μ2 or μ1 ≤ μ2 – ex: Foreign students have higher grade than local students • Two-tail T-test – μ1 ≠ μ2 – ex: the performance between foreign and local students are different. 3 Hypothesis Testing for Mean Difference, σD Unknown Lower-tail test: Upper-tail test: Two-tail test: H0: μ1 – μ2 ≧ 0 H1: μ1 – μ2 < 0 H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0 H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0 a a -ta Reject H0 if t < -ta ta Reject H0 if t > ta Where t has n - 1 d.f. a/2 -ta/2 a/2 ta/2 Reject H0 if t < -ta/2 or t > ta/2 Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 4 Hypothesis • Hypothesis 0: Gender differences have no influence on the preference toward chocolate • Hypothesis 1: Gender differences have influence on the preference toward chocolate 5 T-test step1: The test of variance (F-test) • To test if the variances of the two groups are equal – If the p-value of F value (Pr > F ) >α=0.05, the variances are equal • Equal variance: pooled method – If the p-value of F value (Pr > F ) <α=0.05, the variances are unequal • Unequal variance: Satterthwaite method 6 T-test step2: T-test • To test if the hypothesis 1 is accepted. – If the p-value of T value (Pr > |t|) >α=0.05, H0 is accepted • Gender differences have no influence on the preference toward chocolate – If the p-value of T value (Pr > |t|) <α=0.05, H0 is rejected • Gender differences do have influence on the preference toward chocolate 7 Example (SAS) Please open Tim’s HW 8 Common Mistakes 1. Didn’t check if the variance is equal or not first 2. T-test can’t be used to find the reason • Why do people ride bike? For convenience? For health? 3. Use pair T-test to test two different groups 4. When using Excel, don’t know how to interpret the result 9 Types of Data Data Categorical Numerical Examples: Marital Status Political Party Eye Color (Defined categories) Discrete Examples: Number of Children Defects per hour (Counted items) Continuous Examples: Weight Grades (Measured characteristics) Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 10 Levels of Measurement and Measurement Scales EXAMPLES: Ratio Data Differences between measurements, true zero exists Height, Age, Weekly Food Spending Interval Data Differences between measurements but no true zero Temperature in Fahrenheit, Standardized exam score Ordinal Data Ordered Categories (rankings, order, or scaling) Nominal Data Categories (no ordering or direction) Service quality rating, Standard & Poor’s bond rating, Student letter grades Marital status, Type of car owned Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 11 Two-Sample Tests in EXCEL For independent samples: • Independent sample Z test with variances known: – Tools | data analysis | z-test: two sample for means • Pooled variance t test: – Tools | data analysis | t-test: two sample assuming equal variances • Separate-variance t test: – Tools | data analysis | t-test: two sample assuming unequal variances For paired samples (t test): – Tools | data analysis | t-test: paired two sample for means For variances: • F test for two variances: – Tools | data analysis | F-test: two sample for variances Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 12 Example (Excel) 13 Example (SPSS) 14 Hypothesis • Hypothesis 0: Gender differences have no influence on the preference toward chocolate • Hypothesis 1: Gender differences have influence on the preference toward chocolate 15 Table 1. Descriptive statistics Gender number mean 1 (boys) 2 (girls) 33 44 3.2424 4.1364 Standard deviation 0.16872 0.13243 16 Table 2. T-test results (SPSS) Equal variance of Equal mean of t-test Levene test Chocolate preference Significance T value Significance Equal variances -4.227 0.000 -4.168 0.000 Unequal variances 0.558 P-value <0.05 Ho is rejected. So the gender differences have influence on the preference toward chocolate 17 Thanks. 18 F distribution • Test for the Difference in 2 Independent Populations • Parametric Test Procedure • Assumptions – Both populations are normally distributed • Test is not robust to this violation – Samples are randomly and independently drawn Source: Mark L, Berenson, David M. Levine, Timonthy C. Levine, Basic Business Statistics– Concepts and Applications, 10rd Edition, 2005 19 Hypothesis Tests for Variances Tests for Two Population Variances F test statistic * H0: σ12 = σ22 H1: σ12 ≠ σ22 Two-tail test H0: σ12 ≧ σ22 H1: σ12 < σ22 Lower-tail test H0: σ12 ≤ σ22 H1: σ12 > σ22 Upper-tail test 20