GPCO 453: Quantitative Methods I Sec 10: Hypothesis Testing, III - PowerPoint PPT Presentation

GPCO 453: Quantitative Methods I Sec 10: Hypothesis Testing, III Shane Xinyang Xuan 1 ShaneXuan.com November 30, 2017 1 Department of Political Science, UC San Diego, 9500 Gilman Drive #0521. 1 / 11 ShaneXuan.com

Contact Information Shane Xinyang Xuan xxuan@ucsd.edu My office hours for the rest of the quarter 12/4 M 1100-1230 (SSB 332) 12/4 M 1330-1400 (SSB 332) 12/12 Tu 1100-1130 (SSB 332) 2 / 11 ShaneXuan.com

Interval Estimate of a Population Variance ◮ χ 2 are based on n − 1 d.f. and (1 − α ) confidence level ( n − 1) s 2 ≤ σ 2 ≤ ( n − 1) s 2 (1) χ 2 χ 2 α/ 2 1 − α/ 2 3 / 11 ShaneXuan.com

Test Statistic for Hyp. Tests about a Population Variance ◮ χ 2 follows a chi-square distribution with n − 1 d.f. χ 2 = ( n − 1) s 2 (2) σ 2 4 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, One-tailed ◮ A company produces metal pipes and claims that the standard deviation of the length is at most 1 . 2 cm. One of its clients decides to test this claim by taking a sample of 25 pipes. They found that the standard deviation of the sample is 1.5 cm. 5 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, One-tailed ◮ A company produces metal pipes and claims that the standard deviation of the length is at most 1 . 2 cm. One of its clients decides to test this claim by taking a sample of 25 pipes. They found that the standard deviation of the sample is 1.5 cm. ◮ H 0 : σ 2 ≤ 1 . 2 ; H A : σ 2 > 1 . 2 5 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, One-tailed ◮ A company produces metal pipes and claims that the standard deviation of the length is at most 1 . 2 cm. One of its clients decides to test this claim by taking a sample of 25 pipes. They found that the standard deviation of the sample is 1.5 cm. ◮ H 0 : σ 2 ≤ 1 . 2 ; H A : σ 2 > 1 . 2 ◮ Calculate the test statistics χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(1 . 5) 2 = 37 . 5 (1 . 2) 2 5 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, One-tailed ◮ A company produces metal pipes and claims that the standard deviation of the length is at most 1 . 2 cm. One of its clients decides to test this claim by taking a sample of 25 pipes. They found that the standard deviation of the sample is 1.5 cm. ◮ H 0 : σ 2 ≤ 1 . 2 ; H A : σ 2 > 1 . 2 ◮ Calculate the test statistics χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(1 . 5) 2 = 37 . 5 (1 . 2) 2 χ 2 -table → χ 2 ◮ Find the critical value α = 36 . 415 5 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, One-tailed ◮ A company produces metal pipes and claims that the standard deviation of the length is at most 1 . 2 cm. One of its clients decides to test this claim by taking a sample of 25 pipes. They found that the standard deviation of the sample is 1.5 cm. ◮ H 0 : σ 2 ≤ 1 . 2 ; H A : σ 2 > 1 . 2 ◮ Calculate the test statistics χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(1 . 5) 2 = 37 . 5 (1 . 2) 2 χ 2 -table → χ 2 ◮ Find the critical value α = 36 . 415 ◮ Note that χ 2 > χ 2 α → We reject the null 5 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, Two-tailed ◮ Given n = 25 , s = 17 . 7 , test against σ 2 � = 225 6 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, Two-tailed ◮ Given n = 25 , s = 17 . 7 , test against σ 2 � = 225 ◮ H 0 : σ 2 = 225 ; H A : σ 2 � = 225 6 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, Two-tailed ◮ Given n = 25 , s = 17 . 7 , test against σ 2 � = 225 ◮ H 0 : σ 2 = 225 ; H A : σ 2 � = 225 ◮ Calculate the test statistic χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(17 . 7) 2 = 33 . 42 225 6 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, Two-tailed ◮ Given n = 25 , s = 17 . 7 , test against σ 2 � = 225 ◮ H 0 : σ 2 = 225 ; H A : σ 2 � = 225 ◮ Calculate the test statistic χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(17 . 7) 2 = 33 . 42 225 ◮ Find the critical values at 95% CI χ 2 -table χ 2 0 . 025 = 39 . 364 χ 2 0 . 975 = 12 . 401 6 / 11 ShaneXuan.com

Hypothesis Tests about a Population Variance, Two-tailed ◮ Given n = 25 , s = 17 . 7 , test against σ 2 � = 225 ◮ H 0 : σ 2 = 225 ; H A : σ 2 � = 225 ◮ Calculate the test statistic χ 2 = ( n − 1) s 2 σ 2 = (25 − 1)(17 . 7) 2 = 33 . 42 225 ◮ Find the critical values at 95% CI χ 2 -table χ 2 0 . 025 = 39 . 364 χ 2 0 . 975 = 12 . 401 0 . 975 < χ 2 < χ 2 ◮ Note that χ 2 0 . 025 → We fail to reject the null 6 / 11 ShaneXuan.com

Hypothesis Tests about Two Population Variances ◮ Denoting the population with the larger sample variance as population 1, the test statistic has an F -distribution with n 1 − 1 degrees of freedom for the numerator and n 2 − 1 degrees of freedom for the denominator. 7 / 11 ShaneXuan.com

Hypothesis Tests about Two Population Variances ◮ Denoting the population with the larger sample variance as population 1, the test statistic has an F -distribution with n 1 − 1 degrees of freedom for the numerator and n 2 − 1 degrees of freedom for the denominator. ◮ The test statistic (when σ 2 1 = σ 2 2 ) is F = s 2 1 (3) s 2 2 7 / 11 ShaneXuan.com

Example: Two Population Variances ◮ Suppose n 1 = 31 , and n 2 = 26 , and F = s 2 = 120 1 80 = 1 . 5 (4) s 2 2 We want to test at 95% confidence level: H 0 : σ 2 1 ≤ σ 2 2 H A : σ 2 1 > σ 2 2 8 / 11 ShaneXuan.com

Example: Two Population Variances ◮ Suppose n 1 = 31 , and n 2 = 26 , and F = s 2 = 120 1 80 = 1 . 5 (4) s 2 2 We want to test at 95% confidence level: H 0 : σ 2 1 ≤ σ 2 2 H A : σ 2 1 > σ 2 2 ◮ We find that F distribution with 30 numerator degrees of freedom and 25 denominator degrees of freedom has F . 05 = 1 . 92 F distribution 8 / 11 ShaneXuan.com

Example: Two Population Variances ◮ Suppose n 1 = 31 , and n 2 = 26 , and F = s 2 = 120 1 80 = 1 . 5 (4) s 2 2 We want to test at 95% confidence level: H 0 : σ 2 1 ≤ σ 2 2 H A : σ 2 1 > σ 2 2 ◮ We find that F distribution with 30 numerator degrees of freedom and 25 denominator degrees of freedom has F . 05 = 1 . 92 F distribution ◮ Since the test statistic F is less than the critical value F . 05 , we conclude that we fail to reject H 0 . 8 / 11 ShaneXuan.com

Appendix: χ 2 − table back 9 / 11 ShaneXuan.com

Appendix: χ 2 − table back 10 / 11 ShaneXuan.com

Appendix: F distribution back 11 / 11 ShaneXuan.com