Stat 104 – Lecture 27 Two Independent Samples • Do males and females at I.S.U. spend the same amount of time, on average, at the Lied Recreation Athletic Center? 1 Populations random selection 2. Male Inference 1. Female Samples random selection 2 Time (minutes) 1. Females 63, 32, 86, 53, 49 73, 39, 56, 45, 67 49, 51, 65, 54, 56 2. Males 52, 75, 74, 68, 93 77, 41, 87, 72, 53 84, 65, 66, 69, 62 3 1 Stat 104 – Lecture 27 Time (minutes) Sex=F Mean Std Dev n Sex=M 55.87 Mean 13.527 Std Dev 15 n 69.20 13.790 15 4 Description • This sample of I.S.U. females spends, on average, 13.33 minutes less time at the Lied Recreation Athletic Center than this sample of I.S.U. males. 5 Confidence Interval: 1 y1 y2 t * 2 s12 s22 n1 n2 t * from Table T, df nasty formula 6 2 Stat 104 – Lecture 27 Finding t* • Use Table T. • Confidence Level in last row. • df = 27.99 or 28. 7 Table T 80% Confidence Level 90% 95% 98% 99% df 1 2 3 4 2.048 28 8 s12 s22 n1 n2 13.5272 13.7922 15 15 24.88 4.988 9 3 Stat 104 – Lecture 27 Confidence Interval: 1 y1 y2 t * 2 s12 s22 n1 n2 55.87 69.2 2.0484.988 13.33 10.22 23.55 to 3.11 10 Interpretation • We are 95% confident that I.S.U. females spend, on average, from 3.11 to 23.55 minutes less time at the Lied Recreation Athletic Center than I.S.U. males do. 11 Inference • The confidence interval is making a generalization to ALL students (males and females) at I.S.U. who use the Lead Recreation Athletic Center 12 4 Stat 104 – Lecture 27 Test of Hypothesis: 1 2 • Step 1: Hypotheses. H 0 : 1 2 or H 0 : 1 2 0 H A : 1 2 or H A : 1 2 0 13 Test of Hypothesis: 1 2 • Step 2: Conditions. –Quantitative response for two groups. –Independent random samples. –Approximately normal distribution for both groups. 14 Test of Hypothesis: 1 2 • Step 3: Test Statistic. t y1 y 2 0 s12 s 22 n1 n 2 15 5 Stat 104 – Lecture 27 Time (minutes) Sex=F Sex=M Mean 55.87 Mean Std Dev 69.20 13.527 Std Dev 13.790 n 15 n 15 16 s12 s22 n1 n2 13.5272 13.7922 15 15 24.88 4.988 17 Test of Hypothesis: 1 2 • Step 3: Test Statistic. t y1 y2 0 55.87 69.20 s12 s22 n1 n2 4.988 t 2.672 18 6 Stat 104 – Lecture 27 Table T 0.100 Right-Tail probability 0.050 0.025 0.010 P-value 0.005 df 1 2 3 4 28 1.313 1.701 2.048 2.467 2.672 2.763 19 Test of Hypothesis: 1 2 • Step 4: Probability value –The P-value is between 2(0.005) = 0.010 and 2(0.010) = 0.020. 20 Test of Hypothesis: 1 2 • Step 5: Results. – Reject the null hypothesis because the P-value is smaller than 0.05 – The difference in mean times is not zero. 21 7 Stat 104 – Lecture 27 Conclusion in Context • Therefore, on average, females and males at I.S.U. spend different amounts of time at the Lied Recreation Athletic Center. 22 Comment • This conclusion agrees with the results of the confidence interval. • Zero is not contained in the 95% confidence interval (–23.55 mins to –3.11 mins), therefore the difference in population mean times is not zero. 23 JMP • Data in two columns. –Response variable: • Numeric – Continuous –Explanatory variable: • Character – Nominal 24 8 Stat 104 – Lecture 27 Time 1 2 3 ⁞ 28 29 30 Sex 52 75 74 ⁞ 65 54 56 M M M ⁞ F F F 25 Analyze – Fit Y by X • Y, Response: Time • X, Factor: Sex Click on OK – Means and Std Dev – t Test 26 Oneway Analysis of Time By Sex 100 90 Time 80 70 60 50 40 30 F M Sex Means and Std Deviations Level F M Number 15 15 Mean 55.8667 69.2000 Std Dev 13.5270 13.7903 Std Err Mean Lower 95% 3.4927 48.376 3.5606 61.563 Upper 95% 63.358 76.837 t Test M-F Assuming unequal variances Difference 13.3333 t Ratio Std Err Dif 4.9877 DF Upper CL Dif 23.5503 Prob > |t| Lower CL Dif 3.1164 Prob > t Confidence 0.95 Prob < t 2.67326 27.98961 0.0124 * 0.0062 * 0.9938 -15 -10 -5 0 5 10 15 27 9