A34 APPENDIX D Key Formulas CHAPTER 8 Two-Sample z-Test for the Difference Between Means (s1 and s2 are known, the samples are random and independent, and either the populations are normally distributed or both n1 Ú 30 and n2 Ú 30): z = (x1 - x2) - (m1 - m2) sx 1 -x2 , where sx 1 -x2 = As 2 1 n1 + s 2 2 n2 Two-Sample t-Test for the Difference Between Means (s1 and s2 are unknown, the samples are random and independent, and either the populations are normally distributed or both n1 Ú 30 and n2 Ú 30): t = (x1 - x2) - (m1 - m2) sx 1 -x2 If population variances are equal, d.f. = n1 + n2 - 2 and sx 1 -x2 = A(n1 - 1)s 2 1 + (n2 - 1)s 2 2 n1 + n2 - 2 # A1 n1 + 1 n2 . If population variances are not equal, d.f. is the smaller of n1 - 1 or n2 - 1 and sx 1 -x2 = As2 1 n1 + s2 2 n2 . t-Test for the Difference Between Means (the samples are random and dependent, and either the populations are normally distributed or n Ú 30): t = d - md sd/2n , where d = gd n , sd = Ag(d - d)2 n - 1 , and d.f. = n - 1. Two-Sample z-Test for the Difference Between Proportions (the samples are random and independent, and n1p, n1q, n2p, and n2q are at least 5): z = (p1 n - p 2 n ) - (p 1 - p2) A pqa 1 n1 + 1 n2b , where p = x1 + x2 n1 + n2 and q = 1 - p. CHAPTER 9 Correlation Coefficient: r = ngxy - (gx)(gy) 2 ngx2 - (gx)22ngy2 - (gy)2 t-Test for the Correlation Coefficient: t = r A1 - r2 n - 2 (d.f. = n - 2) Equation of a Regression Line: yn = mx + b, where m = ngxy - (gx)(gy) ngx2 - (gx)2 and b = y - mx = gy n - m gx n . Coefficient of Determination: r2 = Explained variation Total variation = g(yn i - y) 2 g(yi - y) 2 Standard Error of Estimate: se = Ag(yi - y n i) 2 n - 2 c-Prediction Interval for y: yn - E 6 y 6 yn + E, where E = tcseA1 + 1 n + n(x0 - x) 2 ngx2 - (gx)2 (d.f. = n - 2) CHAPTER 10 Chi-Square: x 2 = g (O - E)2 E Goodness-of-Fit Test: d.f. = k - 1 Independence Test: d.f. = (no. of rows - 1)(no. of columns - 1) Expected frequency Er, c = 1Sum of row r2 # 1Sum of column c2 Sample size . Two-Sample F-Test for Variances: F = s2 1 s2 2, where s2 1 Ú s 2 2, d.f.N = n1 - 1, and d.f.D = n2 - 1. One-Way Analysis of Variance Test: F = MSB MSW , where MSB = SSB d.f.N = gni(xi - x) 2 k - 1 and MSW = SSW d.f.D = g(ni - 1)s 2 i N - k . (d.f.N = k - 1, d.f.D = N - k)
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