64_INS.qxd /6/0 :56 AM Page Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Second Ediion 00 Prenice Hall CHAPTER Class Widh = round up o nex convenien number Maximum daa enry - Minimum daa enry of Number of classes Lower class limi + Upper class limi Midpoin = Class frequency Relaive Frequency = = f Sample sie n Populaion Mean: Sample Mean: Weighed Mean: Mean for Grouped Daa: Range Populaion Variance: Populaion Sandard Deviaion: Sample Variance: Sample Sandard Deviaion: Sample Sandard Deviaion for Grouped Daa: s = C gx - x f n - m = gx N x = gx n x = gx # w gw s = s = C gx - m N s = s = x = gx # f n = Maximum enry - Minimum enry gx - m gx - x n - Empirical Rule (or 68-95-99.7 Rule) For daa wih a (symmeric) bell-shaped disribuion. Abou 68% of he daa lies beween m - s and m + s.. Abou 95% of he daa lies beween m - s and m + s. 3. Abou 99.7% of he daa lies beween m - 3s and m + 3s. N s = s = C gx - x n - Chebychev s Theorem The porion of any daa se lying wihin k sandard deviaions k 7 of he mean is a leas - k. Sandard Score: CHAPTER 3 Classical (or Theoreical) Probabiliy: PE = Empirical (or Saisical) Probabiliy: Frequency of even E PE = = f Toal frequency n Probabiliy of a Complemen: Probabiliy of occurrence of boh evens A and B: PA and B = PA # PBƒA PA and B = PA # PB if A and B are independen Probabiliy of occurrence of eiher A or B or boh: PA or B = PA + PB - PA and B PA or B = PA + PB if evens are muually exclusive. Permuaions of n objecs aken r a a ime: np r = Disinguishable Permuaions: n alike, n alike, Á, alike: n k n! n! # n! # n! Á n k! where n + n + n 3 + Á + n k = n. Combinaion of n objecs aken r a a ime: nc r = Number of oucomes in E Toal number of oucomes in sample space n! where r n. n - r!, n! n - r!r! = value - mean sandard deviaion = x - m s PE = - PE
64_INS.qxd /6/0 :56 AM Page Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Second Ediion 00 Prenice Hall CHAPTER 4 Mean of a Discree Random Variable: Variance of a Discree Random Variable: s = gx - m Px Sandard Deviaion: Expeced Value: Binomial Probabiliy of x successes in n rials: Px = n C x p x q n - x = Populaion Parameers of a Binomial Disribuion: Mean: m = np s = s Ex = m = gxpx n! n - x!x! px q n - x Variance: s = npq m = gxpx CHAPTER 6 c-confidence Inerval for m: x - E 6 m 6 x + E s where E = c if s is known and he populaion is n s normal or n Ú 30, and E = c if he populaion is n normal, s is unknown, and n 6 30. Minimum Sample Sie o Esimae : n = a cs m E b Poin Esimae for p, he populaion proporion of successes: pn = x n c-confidence Inerval for Populaion Proporion p (when np Ú 5 and nq Ú 5: pn - E 6 p 6 pn + E, where Sandard Deviaion: s = npq Geomeric Disribuion: The probabiliy ha he firs success will occur on rial number x is Px = pq x -, where q = - p. pnqn E = c B n. Minimum Sample Sie o Esimae p: n = pnqn a c E b Poisson Disribuion: The probabiliy of exacly x occurrences in an inerval is Px = mx e -m, where e L.788. x! CHAPTER 5 Sandard Score, or -Score: = value - mean sandard deviaion = x - m s c-confidence Inerval for Populaion Variance s : n - s x R n - 6 s s 6 x L c-confidence Inerval for Populaion Sandard Deviaion s: n - s n - s 6 s 6 C x R C x L Transforming a -Score o an x-value: x = m + s m x = m Cenral Limi Theorem (Mean of he Sample Means) s x = s n Cenral Limi Theorem (Sandard Error) = x - m x s x = x - m s> n Cenral Limi Theorem
64_INS.qxd /6/0 :56 AM Page 3 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Second Ediion 00 Prenice Hall CHAPTER 7 - Tes for a Mean m : = x - m for s known wih a s> n, normal populaion, or for n Ú 30. - Tes for a Mean m : = x - m for s unknown, x s> n, normally disribued, and n 6 30. -Tes for a Proporion p (when np Ú 5 and nq Ú 5: = Chi-Square Tes for a Variance or Sandard Deviaion: CHAPTER 8 Two-Sample -Tes for he Difference Beween Means: (Independen samples; n and n Ú 30 or normally disribued populaions) where Two-Sample -Tes for he Difference Beween Means: (Independen samples from normally disribued populaions, or n 6 30) If populaion variances are equal, d.f. = n + n - and pn - p pq>n n - x s = s n = x - x - m - m s x - x If populaion variances are no equal, d.f. is he smaller of n - or n - and d.f. = n - = x - x - m - m s x - x, s x - x = C n - s + n - s n + n - s x - x = B s n + s n. -Tes for he Difference Beween Means: (Dependen samples) d.f. = n - s x - x # = B s n + s n +. B n n = d - m d s d > n, where and d.f. = n -. Two-Sample -Tes for he Difference Beween Proporions: Noe: n p, n q, n p, and n q mus be a leas 5. = pn - pn - p - p, where CHAPTER 9 Correlaion Coefficien: r = -Tes for he Correlaion Coefficien: = B p qa + b n n r - r B n - (d.f. = n - ) Equaion of a Regression Line: ngxy - gxgy where m = and ngx - gx b = y - mx = gy n d = gd n, ngxy - gxgy ngx - gx ngy - gy - m gx n Coefficien of Deerminaion: explained variaion r = oal variaion Sandard Error of Esimae: = gyn i - y gy i - y c-predicion Inerval for y is yn - E 6 y 6 yn + E, where E = c s e C + n + nx 0 - x ngx - gx. ngd - gd s d =, C nn - p = x + x n + n. yn = mx + b s e = C gy i - yn i n - d.f. = n -
64_INS.qxd /6/0 :56 AM Page 4 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Second Ediion 00 Prenice Hall CHAPTER 0 Chi-Square: Goodness-of-Fi Tes: Tes of Independence: d.f. N = n -, and d.f. D = n - One-Way Analysis of Variance Tes: F = MS B MS W, O - E x = g E where MS B = and MS W = SS W N - k = gn i - s i. N - k ( d.f. N = k -, d.f. D = N - k) d.f. = k - d.f. = no. of rows - no. of columns - Two-Sample F-Tes for Variances: s Ú s SS B k - = gn i ax i - gx N b k - F = s s CHAPTER Tes Saisic for Sign Tes: When n 7 5, x + 0.5-0.5n =, where x is he smaller number of 0.5n + or - signs and n is he oal number of + and - signs. When n 5, he es saisic is he smaller number of + or - signs. Tes Saisic for Wilcoxon Rank Sum Tes: = R - m R, s R where R = sum of he ranks for he smaller sample, m R = n n + n +, n n. Tes Saisic for he Kruskal-Wallis Tes: Given hree or more independen samples, he es saisic for he Kruskal-Wallis es is H = n n n + n + s R =, B NN + a R + R + Á + R k b - 3N +. n n n k and d.f. = k - The Spearman Rank Correlaion Coefficien: r s = - 6gd nn -
64_INS.qxd /6/0 :56 AM Page 5 Table 4 Sandard Normal Disribuion Area 0.09.08.07.06.05.04.03.0.0.00 3.4.000.0003.0003.0003.0003.0003.0003.0003.0003.0003 3.3.0003.0004.0004.0004.0004.0004.0004.0005.0005.0005 3..0005.0005.0005.0006.0006.0006.0006.0006.0007.0007 3..0007.0007.0008.0008.0008.0008.0009.0009.0009.000 3.0.000.000.00.00.00.00.00.003.003.003.9.004.004.005.005.006.006.007.007.008.009.8.009.000.00.00.00.003.003.004.005.006.7.006.007.008.009.0030.003.003.0033.0034.0035.6.0036.0037.0038.0039.0040.004.0043.0044.0045.0047.5.0048.0049.005.005.0054.0055.0057.0059.0060.006.4.0064.0066.0068.0069.007.0073.0075.0078.0080.008.3.0084.0087.0089.009.0094.0096.0099.00.004.007..00.03.06.09.0.05.09.03.036.039..043.046.050.054.058.06.066.070.074.079.0.083.088.09.097.00.007.0.07.0.08.9.033.039.044.050.056.06.068.074.08.087.8.094.030.0307.034.03.039.0336.0344.035.0359.7.0367.0375.0384.039.040.0409.048.047.0436.0446.6.0455.0465.0475.0485.0495.0505.056.056.0537.0548.5.0559.057.058.0594.0606.068.0630.0643.0655.0668.4.068.0694.0708.07.0735.0749.0764.0778.0793.0808.3.083.0838.0853.0869.0885.090.098.0934.095.0968..0985.003.00.038.056.075.093..3.5..70.90.0.30.5.7.9.34.335.357.0.379.40.43.446.469.49.55.539.56.587 0.9.6.635.660.685.7.736.76.788.84.84 0.8.867.894.9.949.977.005.033.06.090.9 0.7.48.77.06.36.66.96.37.358.389.40 0.6.45.483.54.546.578.6.643.676.709.743 0.5.776.80.843.877.9.946.98.305.3050.3085 0.4.3.356.39.38.364.3300.3336.337.3409.3446 0.3.3483.350.3557.3594.363.3669.3707.3745.3783.38 0..3859.3897.3936.3974.403.405.4090.49.468.407 0..447.486.435.4364.4404.4443.4483.45.456.460 0.0.464.468.47.476.480.4840.4880.490.4960.5000 Criical Values Level of Confidence c c 0.80.8 0.90.645 0.95.96 0.99.575 c = 0 c c
64_INS.qxd /6/0 :56 AM Page 6 Table 4 Sandard Normal Disribuion (coninued) Area 0.00.0.0.03.04.05.06.07.08.09 0.0.5000.5040.5080.50.560.599.539.579.539.5359 0..5398.5438.5478.557.5557.5596.5636.5675.574.5753 0..5793.583.587.590.5948.5987.606.6064.603.64 0.3.679.67.655.693.633.6368.6406.6443.6480.657 0.4.6554.659.668.6664.6700.6736.677.6808.6844.6879 0.5.695.6950.6985.709.7054.7088.73.757.790.74 0.6.757.79.734.7357.7389.74.7454.7486.757.7549 0.7.7580.76.764.7673.7704.7734.7764.7794.783.785 0.8.788.790.7939.7967.7995.803.805.8078.806.833 0.9.859.886.8.838.864.889.835.8340.8365.8389.0.843.8438.846.8485.8508.853.8554.8577.8599.86..8643.8665.8686.8708.879.8749.8770.8790.880.8830..8849.8869.8888.8907.895.8944.896.8980.8997.905.3.903.9049.9066.908.9099.95.93.947.96.977.4.99.907.9.936.95.965.978.99.9306.939.5.933.9345.9357.9370.938.9394.9406.948.949.944.6.945.9463.9474.9484.9495.9505.955.955.9535.9545.7.9554.9564.9573.958.959.9599.9608.966.965.9633.8.964.9649.9656.9664.967.9678.9686.9693.9699.9706.9.973.979.976.973.9738.9744.9750.9756.976.9767.0.977.9778.9783.9788.9793.9798.9803.9808.98.987..98.986.9830.9834.9838.984.9846.9850.9854.9857..986.9864.9868.987.9875.9878.988.9884.9887.9890.3.9893.9896.9898.990.9904.9906.9909.99.993.996.4.998.990.99.995.997.999.993.993.9934.9936.5.9938.9940.994.9943.9945.9946.9948.9949.995.995.6.9953.9955.9956.9957.9959.9960.996.996.9963.9964.7.9965.9966.9967.9968.9969.9970.997.997.9973.9974.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.998.9.998.998.998.9983.9984.9984.9985.9985.9986.9986 3.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 3..9990.999.999.999.999.999.999.999.9993.9993 3..9993.9993.9994.9994.9994.9994.9994.9995.9995.9995 3.3.9995.9995.9995.9996.9996.9996.9996.9996.9996.9997 3.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998
64_INS.qxd /6/0 :56 AM Page 7 Table 5 -Disribuion c-confidence inerval Lef-ailed es Righ-ailed es Two-ailed es Level of confidence, c 0.50 0.80 0.90 0.95 0.98 0.99 One ail, 0.5 0.0 0.05 0.05 0.0 0.005 d.f. Two ails, 0.50 0.0 0.0 0.05 0.0 0.0.000 3.078 6.34.706 3.8 63.657.86.886.90 4.303 6.965 9.95 3.765.638.353 3.8 4.54 5.84 4.74.533.3.776 3.747 4.604 5.77.476.05.57 3.365 4.03 6.78.440.943.447 3.43 3.707 7.7.45.895.365.998 3.499 8.706.397.860.306.896 3.355 9.703.383.833.6.8 3.50 0.700.37.8.8.764 3.69.697.363.796.0.78 3.06.695.356.78.79.68 3.055 3.694.350.77.60.650 3.0 4.69.345.76.45.64.977 5.69.34.753.3.60.947 6.690.337.746.0.583.9 7.689.333.740.0.567.898 8.688.330.734.0.55.878 9.688.38.79.093.539.86 0.687.35.75.086.58.845.686.33.7.080.58.83.686.3.77.074.508.89 3.685.39.74.069.500.807 4.685.38.7.064.49.797 5.684.36.708.060.485.787 6.684.35.706.056.479.779 7.684.34.703.05.473.77 8.683.33.70.048.467.763 9.683.3.699.045.46.756 q.674.8.645.960.36.576
64_INS.qxd /6/0 :56 AM Page 8 Table 6 Chi-Square Disribuion χ χ χ L χ R χ Righ ail Two ails Degrees of freedom 0.995 0.99 0.975 0.95 0.90 0.0 0.05 0.05 0.0 0.005 0.00 0.004 0.06.706 3.84 5.04 6.635 7.879 0.00 0.00 0.05 0.03 0. 4.605 5.99 7.378 9.0 0.597 3 0.07 0.5 0.6 0.35 0.584 6.5 7.85 9.348.345.838 4 0.07 0.97 0.484 0.7.064 7.779 9.488.43 3.77 4.860 5 0.4 0.554 0.83.45.60 9.36.07.833 5.086 6.750 6 0.676 0.87.37.635.04 0.645.59 4.449 6.8 8.548 7 0.989.39.690.67.833.07 4.067 6.03 8.475 0.78 8.344.646.80.733 3.490 3.36 5.507 7.535 0.090.955 9.735.088.700 3.35 4.68 4.684 6.99 9.03.666 3.589 0.56.558 3.47 3.940 4.865 5.987 8.307 0.483 3.09 5.88.603 3.053 3.86 4.575 5.578 7.75 9.675.90 4.75 6.757 3.074 3.57 4.404 5.6 6.304 8.549.06 3.337 6.7 8.99 3 3.565 4.07 5.009 5.89 7.04 9.8.36 4.736 7.688 9.89 4 4.075 4.660 5.69 6.57 7.790.064 3.685 6.9 9.4 3.39 5 4.60 5.9 6.6 7.6 8.547.307 4.996 7.488 30.578 3.80 6 5.4 5.8 6.908 7.96 9.3 3.54 6.96 8.845 3.000 34.67 7 5.697 6.408 7.564 8.67 0.085 4.769 7.587 30.9 33.409 35.78 8 6.65 7.05 8.3 9.390 0.865 5.989 8.869 3.56 34.805 37.56 9 6.844 7.633 8.907 0.7.65 7.04 30.44 3.85 36.9 38.58 0 7.434 8.60 9.59 0.85.443 8.4 3.40 34.70 37.566 39.997 8.034 8.897 0.83.59 3.40 9.65 3.67 35.479 38.93 4.40 8.643 9.54 0.98.338 4.04 30.83 33.94 36.78 40.89 4.796 3 9.6 0.96.689 3.09 4.848 3.007 35.7 38.076 4.638 44.8 4 9.886 0.856.40 3.848 5.659 33.96 36.45 39.364 4.980 45.559 5 0.50.54 3.0 4.6 6.473 34.38 37.65 40.646 44.34 46.98 6.60.98 3.844 5.379 7.9 35.563 38.885 4.93 45.64 48.90 7.808.879 4.573 6.5 8.4 36.74 40.3 43.94 46.963 49.645 8.46 3.565 5.308 6.98 8.939 37.96 4.337 44.46 48.78 50.993 9 3. 4.57 6.047 7.708 9.768 39.087 4.557 45.7 49.588 5.336 30 3.787 4.954 6.79 8.493 0.599 40.56 43.773 46.979 50.89 53.67 40 0.707.64 4.433 6.509 9.05 5.805 55.758 59.34 63.69 66.766 50 7.99 9.707 3.357 34.764 37.689 63.67 67.505 7.40 76.54 79.490 60 35.534 37.485 40.48 43.88 46.459 74.397 79.08 83.98 88.379 9.95 70 43.75 45.44 48.758 5.739 55.39 85.57 90.53 95.03 00.45 04.5 80 5.7 53.540 57.53 60.39 64.78 96.578 0.879 06.69.39 6.3 90 59.96 6.754 65.647 69.6 73.9 07.565 3.45 8.36 4.6 8.99 00 67.38 70.065 74. 77.99 8.358 8.498 4.34 9.56 35.807 40.69