Key Formulas From Larson/Farber Elementary Statistics: Picturing the World, Second Edition 2002 Prentice Hall

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1 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 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

2 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 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

3 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 Tes for a Mean m : = x - m for s unknown, x s> n, normally disribued, and n 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 -

4 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 n =, 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 -

5 64_INS.qxd /6/0 :56 AM Page 5 Table 4 Sandard Normal Disribuion Area Criical Values Level of Confidence c c c = 0 c c

6 64_INS.qxd /6/0 :56 AM Page 6 Table 4 Sandard Normal Disribuion (coninued) Area

7 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 One ail, d.f. Two ails, q

8 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