GPCO 453: Quantitative Methods I Review: Hypothesis Testing
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1 GPCO 453: Quantitative Methods I Review: Hypothesis Testing Shane Xinyang Xuan 1 ShaneXuan.com December 6, Department of Political Science, UC San Diego, 9500 Gilman Drive #0521. ShaneXuan.com 1 / 11
2 Thought Process One population or two populations? ShaneXuan.com 2 / 11
3 Thought Process One population or two populations? Mean, proportion, or variance? ShaneXuan.com 2 / 11
4 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? ShaneXuan.com 2 / 11
5 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? ShaneXuan.com 2 / 11
6 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes ShaneXuan.com 2 / 11
7 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n ShaneXuan.com 2 / 11
8 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ n ShaneXuan.com 2 / 11
9 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n ShaneXuan.com 2 / 11
10 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n One population variance χ 2 = (n 1)s2 ; critical value σ0 2 approach ShaneXuan.com 2 / 11
11 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n One population variance χ 2 = (n 1)s2 ; critical value σ0 2 approach Two population variances F = s2 1 ; critical value approach s 2 2 ShaneXuan.com 2 / 11
12 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n One population variance χ 2 = (n 1)s2 ; critical value σ0 2 approach Two population variances F = s2 1 ; critical value approach s 2 2 σ1 Two population means, known σ z; s.e. = 2 n 1 + σ2 2 n 2 ShaneXuan.com 2 / 11
13 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n One population variance χ 2 = (n 1)s2 ; critical value σ0 2 approach Two population variances F = s2 1 s 2 2 Two population means, known σ z; s.e. = Two population means, unknown σ t; s.e. = ; critical value approach σ 2 1 n 1 + σ2 2 n 2 s 2 1 n 1 + s2 2 n 2 ShaneXuan.com 2 / 11
14 Thought Process One population or two populations? Mean, proportion, or variance? Right, left, or two tails? Use z-score or t-score? Here is what we will cover in the next 30 minutes One population mean, known σ z; s.e. = σ n One population mean, unknown σ t; s.e. = σ One population proportion z; s.e. = n p 0(1 p 0) n One population variance χ 2 = (n 1)s2 ; critical value σ0 2 approach Two population variances F = s2 1 s 2 2 Two population means, known σ z; s.e. = Two population means, unknown σ t; s.e. = Two population proportions z; s.e. = ; critical value approach σ 2 1 n 1 + σ2 2 n 2 s 2 1 n 1 + s2 2 n 2 p 1(1 p 1) n 1 + p2(1 p2) n 2 ShaneXuan.com 2 / 11
15 What We Won t Cover Important! You should be able to read z table, t table, χ 2 table, and F table; and you should be able to compute degree of freedoms for all the above cases ShaneXuan.com 3 / 11
16 What We Won t Cover Important! You should be able to read z table, t table, χ 2 table, and F table; and you should be able to compute degree of freedoms for all the above cases For each case, you should do at least one example We suggest that you go to the respective section in the textbook, and follow the example to make sure that you get it ShaneXuan.com 3 / 11
17 One Population Mean, known σ ShaneXuan.com 4 / 11
18 One Population Mean, unknown σ ShaneXuan.com 5 / 11
19 One Population Proportion ShaneXuan.com 6 / 11
20 One Population Variance ShaneXuan.com 7 / 11
21 Two Population Variances ShaneXuan.com 8 / 11
22 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 9 / 11
23 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 9 / 11
24 Some Tweaks What if σ is unknown? ShaneXuan.com 10 / 11
25 Some Tweaks What if σ is unknown? Use s to replace σ, and calculate t-score instead of z-score ShaneXuan.com 10 / 11
26 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 10 / 11
27 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 10 / 11
28 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 10 / 11
29 Concluding Remarks It might sound really easy, but you need to practice ShaneXuan.com 11 / 11
30 Concluding Remarks It might sound really easy, but you need to practice If you have not done teaching evaluations for the TAs yet, please consider doing so before the deadline (Monday) ShaneXuan.com 11 / 11
31 Concluding Remarks It might sound really easy, but you need to practice If you have not done teaching evaluations for the TAs yet, please consider doing so before the deadline (Monday) All section materials available at ShaneXuan.com/teaching ShaneXuan.com 11 / 11
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