QMM 250 FORMULA SHEET EXAM III DRAFT n Rules of summation: 1. For any constant c, c nc i 1 n 3. (x i 1 n n i 1 i 1 x i 1 Sample x N 2 (x i 1 i x )2 N Standard Deviation of x: Population Sample binomial proportion: i 1 i 1 n i N Variance of x: Population n y i ) xi y i i N Mean of x: Population x n cxi c xi 2. p 2 x i 1 i n n Sample s 2 (x i 1 Sample s s i x)2 n 1 2 x n Binomial Distribution Parameters: E ( X ) n Standard Normal Random Variable: Standard Error of the Sample Mean: z Var ( X ) n (1 ) x X X X X n X Confidence Interval for X , given known X : x z / 2 Confidence Interval for X , given unknown X : x t / 2,n1 n sX n (1 ) Sampling Distribution for Sample Binomial Proportion: p ~ N , , (sample size n requirements of 𝑛𝜋 ≥ 10 𝑎𝑛𝑑 𝑛(1 − 𝜋) ≥ 10). Confidence Interval for : p z / 2 p(1 p) n 2 Z Sample Size Necessary to Estimate 𝜋 to within 𝐸∗: n / 2 .25 E* Test Statistic for population mean, known: Z X 0 , distributed N(0,1) if H0 is true. / n X 0 , distributed t n 1 if H0 is true. s/ n p 0 , distributed N(0,1) if H0 is true. 0 (1 0 ) n Test Statistic for population mean, unknown: T Test Statistic for binomial proportion ( ): Z p Value P(test statistic is at least as extreme as the observed value of the test statistic | H0 is true) Sampling Distribution for difference in means: X 1 X 2 ~ N ( 1 2 , Test Statistic for 1 2 , 12 and 22 known: Z X1 X 2 12 n1 22 12 (n1 1) s12 (n2 1) s22 n1 n2 2 Sum of Squares Total: SST ( yij y ) 2 c nj j 1 i 1 c Sum of Squares for Factor A: SSA n j ( y j y ) 2 j 1 c Sum of Squared Errors: nj SSE ( yij y j ) 2 j 1 i 1 Mean Square Factor A: Mean Square Error: ANOVA Test Statistic: SSA c 1 SSE MSE nc MSA , distributed Fc1,nc if H0 is true. F MSE MSA Equation of the Population Regression Model: Equation of the fitted regression line: yi 0 1 xi i yˆ i b0 b1 xi n2 ) n2 distributed t n1 n2 2 if H0 is true. s 2p 22 , distributed N(0,1) if H0 is true. Test Statistic for 1 2 , equal but unknown population variance: T Pooled Estimator of 2 : n1 X1 X 2 1 1 s 2p n1 n2 , n Sum of Cross-Products: SS xy ( xi x )( yi y ) i 1 n SS xx ( xi x ) 2 Sum of Squared Deviations in x: i 1 n OLS Slope Coefficient: b1 (x i i 1 x )( yi y ) n (x i 1 OLS Intercept Coefficient: i x )2 SS xy SS xx b0 y b1 x n Total Sum of Squares: SST ( yi y ) 2 i 1 Sum of Squared Residuals: Mean Square Error: n n i 1 i 1 SSE ( yi yˆ i ) 2 ( yi b0 b1 xi ) 2 SSE MSE n2 n Regression Sum of Squares: SSR ( yˆ i y ) 2 i 1 R 2 (coefficient of determination): R2 SSR SSE 1 SST SST Standard Error of the Regression: s SSE MSE n2 Standard Deviation of Sample Slope Coefficient: sb1 s SS xx Confidence Interval for Population Slope Coefficient, 1 : b1 t / 2,n2 sb1 Test Statistic for Regression Slope Coefficient: T b1 , distributed t n2 if H0 is true. sb1