Formulas and Tables for AP Statistics

Formulas and Tables for AP Statistics I. Descriptive Statistics x = ∑ xi 1 ∑ xi = n n sx = y = a + bx ŷ = a + bx r = II. 2 1 ∑ ( x − x ) 2 = ∑ ( xi − x ) i n− 1 n− 1 x − x   yi − y  1 ∑ i n − 1  sx   sy  b= r sy sx Probability and Distributions P (A | B ) = P ( A ∪ B ) = P ( A ) + P (B ) − P ( A ∩ B ) Probability Distribution Mean Discrete random variable, X µ X = E ( X ) = ∑ xi P ( xi ) If 𝑋𝑋 has a binomial distribution with parameters n and p, then:  n P( X = x) =   p x ( 1 − p) n − x  x P (A ∩ B ) P (B ) Standard Deviation σ X = ∑ ( xi − µX µX = np )2 P( xi ) σ X = np (1 − p ) where x = 0, 1, 2, 3,  , n If 𝑋𝑋 has a geometric distribution with parameter p, then: P( X = x ) = (1 − p ) x −1 p µX = 1 p σX = 1− p p where x = 1, 2, 3,  III. Sampling Distributions and Inferential Statistics Standardized test statistic: statistic − parameter standard error of the statistic Confidence interval: statistic ± ( critical value)( standard error of statistic) Chi-square statistic: χ 2 = ∑ ( observed − expected )2 expected III. Sampling Distributions and Inferential Statistics (continued) Sampling distributions for proportions: Random Variable For one population: p̂ For two populations: ˆp − pˆ 1 2 Standard Error* of Sample Statistic Parameters of Sampling Distribution µ p̂ = p µ ˆp ˆ2 1−p = p1 − p 2 p(1 − p) n σ p̂ = σ ˆp 1 − pˆ 2 s p̂ = p ( 1 − p2 ) p1 (1 − p1) + 2 n2 n1 = p̂1 (1 − pˆ1 ) pˆ2 (1 − p̂2 ) + n1 n2 s ˆp − ˆp = 1 p̂ (1 − p̂ ) n 2 When p1 = p2 is assumed: s ˆp ˆ2 1 −p   pˆ c (1 − pˆ c )  1 + 1  n2   n1 = where p̂ c = X1 + X 2 n1 + n2 Sampling distributions for means: Random Variable For one population: X For two populations: X1 − X 2 Standard Error* of Sample Statistic Parameters of Sampling Distribution σ σX = µX = µ µX 1− X 2 = µ1 − µ 2 σX 1 − X2 = σ 12 n1 = s n s X n + σ 22 s X 1− X 2 n2 = s12 s2 + 2 n1 n2 Sampling distributions for simple linear regression: Random Variable For slope: Standard Error* of Sample Statistic Parameters of Sampling Distribution µ =β b σb = b where σ x = σ σx n sb = , ∑ ( xi − x ) n 2 where s = and sx = s , sx n − 1 ∑( yi − ˆyi ) n− 2 ∑ ( xi − x ) n −1 2 2 *Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error. Table entry for p and C is the point t* with probability p lying above it and probability C lying between t * and t*. Probability p t* Table B t distribution critical values Tail probability p df .25 .20 .15 .10 .05 .025 .02 .01 .005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 80 100 1000 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .679 .678 .677 .675 .674 1.376 1.061 .978 .941 .920 .906 .896 .889 .883 .879 .876 .873 .870 .868 .866 .865 .863 .862 .861 .860 .859 .858 .858 .857 .856 .856 .855 .855 .854 .854 .851 .849 .848 .846 .845 .842 .841 1.963 1.386 1.250 1.190 1.156 1.134 1.119 1.108 1.100 1.093 1.088 1.083 1.079 1.076 1.074 1.071 1.069 1.067 1.066 1.064 1.063 1.061 1.060 1.059 1.058 1.058 1.057 1.056 1.055 1.055 1.050 1.047 1.045 1.043 1.042 1.037 1.036 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.299 1.296 1.292 1.290 1.282 1.282 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.676 1.671 1.664 1.660 1.646 1.645 12.71 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.009 2.000 1.990 1.984 1.962 1.960 15.89 4.849 3.482 2.999 2.757 2.612 2.517 2.449 2.398 2.359 2.328 2.303 2.282 2.264 2.249 2.235 2.224 2.214 2.205 2.197 2.189 2.183 2.177 2.172 2.167 2.162 2.158 2.154 2.150 2.147 2.123 2.109 2.099 2.088 2.081 2.056 2.054 31.82 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.403 2.390 2.374 2.364 2.330 2.326 63.66 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.678 2.660 2.639 2.626 2.581 2.576 50% 60% 70% 80% 90% 95% 96% 98% 99% • Confidence level C .0025 .001 .0005 127.3 14.09 7.453 5.598 4.773 4.317 4.029 3.833 3.690 3.581 3.497 3.428 3.372 3.326 3.286 3.252 3.222 3.197 3.174 3.153 3.135 3.119 3.104 3.091 3.078 3.067 3.057 3.047 3.038 3.030 2.971 2.937 2.915 2.887 2.871 2.813 2.807 318.3 22.33 10.21 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.646 3.611 3.579 3.552 3.527 3.505 3.485 3.467 3.450 3.435 3.421 3.408 3.396 3.385 3.307 3.261 3.232 3.195 3.174 3.098 3.091 636.6 31.60 12.92 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.496 3.460 3.416 3.390 3.300 3.291 99.5% 99.8% 99.9% Probability p Table entry for p is the point (  2 ) with probability p lying above it. (χ2 ) Table C c 2 critical values Tail probability p df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 80 100 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.22 11.39 12.55 13.70 14.85 15.98 17.12 18.25 19.37 20.49 21.60 22.72 23.83 24.93 26.04 27.14 28.24 29.34 30.43 31.53 32.62 33.71 34.80 45.62 56.33 66.98 88.13 109.1 1.64 3.22 4.64 5.99 7.29 8.56 9.80 11.03 12.24 13.44 14.63 15.81 16.98 18.15 19.31 20.47 21.61 22.76 23.90 25.04 26.17 27.30 28.43 29.55 30.68 31.79 32.91 34.03 35.14 36.25 47.27 58.16 68.97 90.41 111.7 2.07 3.79 5.32 6.74 8.12 9.45 10.75 12.03 13.29 14.53 15.77 16.99 18.20 19.41 20.60 21.79 22.98 24.16 25.33 26.50 27.66 28.82 29.98 31.13 32.28 33.43 34.57 35.71 36.85 37.99 49.24 60.35 71.34 93.11 114.7 2.71 4.61 6.25 7.78 9.24 10.64 12.02 13.36 14.68 15.99 17.28 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 29.62 30.81 32.01 33.20 34.38 35.56 36.74 37.92 39.09 40.26 51.81 63.17 74.40 96.58 118.5 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.89 40.11 41.34 42.56 43.77 55.76 67.50 79.08 101.9 124.3 5.02 7.38 9.35 11.14 12.83 14.45 16.01 17.53 19.02 20.48 21.92 23.34 24.74 26.12 27.49 28.85 30.19 31.53 32.85 34.17 35.48 36.78 38.08 39.36 40.65 41.92 43.19 44.46 45.72 46.98 59.34 71.42 83.30 106.6 129.6 5.41 7.82 9.84 11.67 13.39 15.03 16.62 18.17 19.68 21.16 22.62 24.05 25.47 26.87 28.26 29.63 31.00 32.35 33.69 35.02 36.34 37.66 38.97 40.27 41.57 42.86 44.14 45.42 46.69 47.96 60.44 72.61 84.58 108.1 131.1 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89 63.69 76.15 88.38 112.3 135.8 7.88 10.60 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 26.76 28.30 29.82 31.32 32.80 34.27 35.72 37.16 38.58 40.00 41.40 42.80 44.18 45.56 46.93 48.29 49.64 50.99 52.34 53.67 66.77 79.49 91.95 116.3 140.2 9.14 11.98 14.32 16.42 18.39 20.25 22.04 23.77 25.46 27.11 28.73 30.32 31.88 33.43 34.95 36.46 37.95 39.42 40.88 42.34 43.78 45.20 46.62 48.03 49.44 50.83 52.22 53.59 54.97 56.33 69.70 82.66 95.34 120.1 144.3 10.83 13.82 16.27 18.47 20.51 22.46 24.32 26.12 27.88 29.59 31.26 32.91 34.53 36.12 37.70 39.25 40.79 42.31 43.82 45.31 46.80 48.27 49.73 51.18 52.62 54.05 55.48 56.89 58.30 59.70 73.40 86.66 99.61 124.8 149.4 12.12 15.20 17.73 20.00 22.11 24.10 26.02 27.87 29.67 31.42 33.14 34.82 36.48 38.11 39.72 41.31 42.88 44.43 45.97 47.50 49.01 50.51 52.00 53.48 54.95 56.41 57.86 59.30 60.73 62.16 76.09 89.56 102.7 128.3 153.2