GPCO 453: Quantitative Methods I Sec 09: More on Hypothesis Testing
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1 GPCO 453: Quantitative Methods I Sec 09: More on Hypothesis Testing Shane Xinyang Xuan 1 ShaneXuan.com November 20, Department of Political Science, UC San Diego, 9500 Gilman Drive #0521. ShaneXuan.com 1 / 7
2 Contact Information Shane Xinyang Xuan The teaching staff is a team! Professor Garg Tu (RBC 1303) Shane Xuan M (SSB 332) M (SSB 332) Joanna Valle-luna Tu (RBC 3131) Th (RBC 3131) Daniel Rust F (RBC 3213) ShaneXuan.com 2 / 7
3 Comparing Two Populations Standard error σ x1 x 2 = σˆp1 ˆp 2 = σ1 2 + σ2 2 n 1 n 2 (1) p 1 (1 p 1 ) + p 2(1 p 2 ) n 1 n 2 (2) ShaneXuan.com 3 / 7
4 Comparing Two Populations Standard error σ x1 x 2 = σˆp1 ˆp 2 = Confidence interval σ1 2 + σ2 2 n 1 n 2 (1) p 1 (1 p 1 ) + p 2(1 p 2 ) n 1 n 2 (2) ( x 1 x 2 ) ± z σ x1 x 2 (3) (ˆp 1 ˆp 2 ) ± z σˆp1 ˆp 2 (4) ShaneXuan.com 3 / 7
5 Some Tweaks What if σ is unknown? ShaneXuan.com 4 / 7
6 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score ShaneXuan.com 4 / 7
7 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score How to calculate t-score again? t = (x 1 x 2 ) D 0 σ x1 x 2 (5) ShaneXuan.com 4 / 7
8 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score How to calculate t-score again? t = (x 1 x 2 ) D 0 σ x1 x 2 (5) Degree of freedom is calculated by ( ) s n 1 + s2 2 n 2 df = ( ) 1 s 2 2 ( ) 1 n 1 1 n s 2 2 (6) 2 n 2 1 n 2 ShaneXuan.com 4 / 7
9 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score How to calculate t-score again? t = (x 1 x 2 ) D 0 σ x1 x 2 (5) Degree of freedom is calculated by ( ) s n 1 + s2 2 n 2 df = ( ) 1 s 2 2 ( ) 1 n 1 1 n s 2 2 (6) 2 n 2 1 n 2 If we do not know p, we use ˆp instead: ˆp 1 (1 ˆp 1 ) σˆp1 ˆp 2 = + ˆp 2(1 ˆp 2 ) (7) n 1 n 2 ShaneXuan.com 4 / 7
10 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score How to calculate t-score again? t = (x 1 x 2 ) D 0 σ x1 x 2 (5) Degree of freedom is calculated by ( ) s n 1 + s2 2 n 2 df = ( ) 1 s 2 2 ( ) 1 n 1 1 n s 2 2 (6) 2 n 2 1 n 2 If we do not know p, we use ˆp instead: ˆp 1 (1 ˆp 1 ) σˆp1 ˆp 2 = + ˆp 2(1 ˆp 2 ) (7) n 1 n 2 Two-tailed tests ShaneXuan.com 4 / 7
11 Comparing Two Means Test the hypothesis at α = 0.05 that the average temperature in San Diego is higher than the average temperature in San Francisco. A random sample of 33 days and 37 days is obtained from San Diego and San Francisco. Note that x SD = 72, s SD = 10, x SF = 65, s SF = 12. ShaneXuan.com 5 / 7
12 Comparing Two Means Test the hypothesis at α = 0.05 that the average temperature in San Diego is higher than the average temperature in San Francisco. A random sample of 33 days and 37 days is obtained from San Diego and San Francisco. Note that x SD = 72, s SD = 10, x SF = 65, s SF = 12. Right-tailed test H 0 : µ SD µ SF 0; H A : µ SD µ SF > 0 ShaneXuan.com 5 / 7
13 Comparing Two Means Test the hypothesis at α = 0.05 that the average temperature in San Diego is higher than the average temperature in San Francisco. A random sample of 33 days and 37 days is obtained from San Diego and San Francisco. Note that x SD = 72, s SD = 10, x SF = 65, s SF = 12. Right-tailed test H 0 : µ SD µ SF 0; H A : µ SD µ SF > 0 Calculate the t-statistic t = x 1 x 2 D 0 σ x1 x 2 (8) = = 2.66 (9) ShaneXuan.com 5 / 7
14 Comparing Two Means Test the hypothesis at α = 0.05 that the average temperature in San Diego is higher than the average temperature in San Francisco. A random sample of 33 days and 37 days is obtained from San Diego and San Francisco. Note that x SD = 72, s SD = 10, x SF = 65, s SF = 12. Right-tailed test H 0 : µ SD µ SF 0; H A : µ SD µ SF > 0 Calculate the t-statistic t = x 1 x 2 D 0 σ x1 x 2 (8) = = 2.66 (9) Find degree of freedom (d.f.=68) ShaneXuan.com 5 / 7
15 Comparing Two Means: Cont. x SD = 72, s SD = 10, x SF = 65, s SF = 12, t = 2.66, d.f.=68 ShaneXuan.com 6 / 7
16 Comparing Two Means: Cont. x SD = 72, s SD = 10, x SF = 65, s SF = 12, t = 2.66, d.f.=68 Find the critical vaule Degree of Freedom... Area in Upper Tail = ShaneXuan.com 6 / 7
17 Comparing Two Means: Cont. x SD = 72, s SD = 10, x SF = 65, s SF = 12, t = 2.66, d.f.=68 Find the critical vaule Degree of Freedom... Area in Upper Tail = Compare t-statistic to the critical value: 2.66 > (10) ShaneXuan.com 6 / 7
18 Comparing Two Means: Cont. x SD = 72, s SD = 10, x SF = 65, s SF = 12, t = 2.66, d.f.=68 Find the critical vaule Degree of Freedom... Area in Upper Tail = Compare t-statistic to the critical value: 2.66 > (10) We conclude that we can reject the null hypothesis and the average SD temperature is higher than the average SF temperature ShaneXuan.com 6 / 7
19 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. ShaneXuan.com 7 / 7
20 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. Right-tailed test H 0 : p SD p NY 0; H A : p SD p NY > 0 ShaneXuan.com 7 / 7
21 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. Right-tailed test H 0 : p SD p NY 0; H A : p SD p NY > 0 Note that ˆp 1 = 6 35 and ˆp 2 = 5 46 ShaneXuan.com 7 / 7
22 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. Right-tailed test H 0 : p SD p NY 0; H A : p SD p NY > 0 Note that ˆp 1 = 6 35 and ˆp 2 = 5 46 Calculate the z-statistic z = ˆp 1 ˆp 2 D 0 σˆp1 ˆp 2 (11) = ( )(1 35) ( 46)(1 5 46) 5 46 = (12) ShaneXuan.com 7 / 7
23 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. Right-tailed test H 0 : p SD p NY 0; H A : p SD p NY > 0 Note that ˆp 1 = 6 35 and ˆp 2 = 5 46 Calculate the z-statistic z = ˆp 1 ˆp 2 D 0 σˆp1 ˆp 2 (11) = ( )(1 35) ( 46)(1 5 46) 5 46 = (12) Find p-value: Pr(z > 0.799) = = > 0.01 ShaneXuan.com 7 / 7
24 Comparing Two Proportions Test the hypothesis at α = 0.01 that the unemployment rate in San Diego is higher than the unemployment rate in New York. A random sample of 35 people is obtained in San Diego and 6 of them are unemployed. A random sample of 46 people is obtained in New York and 5 of them are unemployed. Right-tailed test H 0 : p SD p NY 0; H A : p SD p NY > 0 Note that ˆp 1 = 6 35 and ˆp 2 = 5 46 Calculate the z-statistic z = ˆp 1 ˆp 2 D 0 σˆp1 ˆp 2 (11) = ( )(1 35) ( 46)(1 5 46) 5 46 = (12) Find p-value: Pr(z > 0.799) = = > 0.01 We fail to reject the null hypothesis and we do not have evidence that the SD unemployment rate is higher than the NY unemployment rate. ShaneXuan.com 7 / 7
25 Conclusion One population or two populations ShaneXuan.com 8 / 7
26 Conclusion One population or two populations Mean or proportion ShaneXuan.com 8 / 7
27 Conclusion One population or two populations Mean or proportion One-tailed or two-tailed tests ShaneXuan.com 8 / 7
28 Conclusion One population or two populations Mean or proportion One-tailed or two-tailed tests Whether we know σ or not ShaneXuan.com 8 / 7
29 Conclusion One population or two populations Mean or proportion One-tailed or two-tailed tests Whether we know σ or not Whether we should compute a z-statistic or t-statistic ShaneXuan.com 8 / 7
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