Key Formulas. From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition 2012 Prentice Hall. Standard Score: CHAPTER 3.

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1 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Fifh Ediion 01 Prenice Hall CHAPTER Class Widh = Range of daa Number of classes 1round up o nex convenien number 1Lower class limi + 1Upper class limi Midpoin = Class frequency Relaive Frequency = = f Sample size n Populaion Mean: m = gx N Sample Mean: x = gx n Weighed Mean: x = g1x # w gw Mean of a Frequency Disribuion: x = g1x # f n Range = 1Maximum enry - 1Minimum enry g1x - Populaion Variance: s m = N Populaion Sandard Deviaion: g1x - m s = s = C N g1x - Sample Variance: s x = n - 1 g1x - x Sample Sandard Deviaion: s = s = C n - 1 Empirical Rule (or Rule) For daa wih a (symmeric) bell-shaped disribuion: 1. 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. Chebychev s Theorem The porion of any daa se lying wihin k sandard deviaions 1k 7 1 of he mean is a leas 1-1 k. Sample Sandard Deviaion of a Frequency Disribuion: g1x - x f s = C n - 1 Sandard Score: CHAPTER 3 Classical (or Theoreical) Probabiliy: Number of oucomes in even E P1E = Toal number of oucomes in sample space Empirical (or Saisical) Probabiliy: Frequency of even E P1E = = f Toal frequency n Probabiliy of a Complemen: Probabiliy of occurrence of boh evens A and B: P1A and B = P1A # P1B ƒa P1A and B = P1A # P1B if A and B are independen Probabiliy of occurrence of eiher A or B or boh: P1A or B = P1A + P1B - P1A and B P1A or B = P1A + P1B if A and B are muually exclusive Permuaions of n objecs aken r a a ime: n! np r = where r n 1n - r!, Disinguishable Permuaions: n 1 alike, n alike, Á, alike: n k z = n! n 1! # n! # n! Á n k!, Value - Mean Sandard deviaion = x - m s where n 1 + n + n 3 + Á + n k = n P1E = 1 - P1E Combinaion of n objecs aken r a a ime: n! nc r = 1n - r!r!

2 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Fifh Ediion 01 Prenice Hall CHAPTER 4 Mean of a Discree Random Variable: m = gxp1x Variance of a Discree Random Variable: Sandard Deviaion of he Sampling Disribuion (Sandard Error): z-score = s x = Value - Mean Sandard Error = x - m x = x - m s x s> 1n s 1n s = g1x - m P1x Sandard Deviaion of a Discree Random Variable: s = s = g1x - m P1x Expeced Value: Binomial Probabiliy of x successes in n rials: P1x = n C x p x q n - x = Populaion Parameers of a Binomial Disribuion: Mean: Sandard Deviaion: Variance: Geomeric Disribuion: The probabiliy ha he firs success will occur on rial number x is P1x = p1q x - 1, where q = 1 - p. Poisson Disribuion: The probabiliy of exacly x occurrences in an inerval is P1x = mx e -m, where x! e L.7188 and m is he mean number of occurences per inerval uni. CHAPTER 5 Sandard Score, or z-score: z = m = np E1x = m = gxp1x n! 1n - x!x! px q n - x s = 1npq Value - Mean Sandard deviaion = x - m s Cenral Limi Theorem ( n Ú 30 or populaion is normally disribued): Mean of he Sampling Disribuion: s = npq Transforming a z-score o an x-value: x = m + zs m x = m CHAPTER 6 c-confidence Inerval for m: x - E 6 m 6 x + E, s where E = z c if s is known and he populaion is 1n s normally disribued or n Ú 30, or E = c if he 1n populaion is normally or approximaely normally disribued, s is unknown, and n 6 30 Minimum Sample Size o Esimae m: Poin Esimae for p, he populaion proporion of successes: c-confidence Inerval for Populaion Proporion p (when np Ú 5 and nq Ú 5: pn - E 6 p 6 pn + E, where pnqn E = z c B n Minimum Sample Size o Esimae p: c-confidence Inerval for Populaion Variance : 1n - 1s 1n - 6 s 1s 6 c-confidence Inerval for Populaion Sandard Deviaion s: C x R 1n - 1s x R pn = x n x L 6 s 6 C 1n - 1s x L n = a z cs E b n = pnqn a z c E b s Variance of he Sampling Disribuion: s x = s n

3 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Fifh Ediion 01 Prenice Hall CHAPTER 7 z- Tes for a Mean m : z = x - m for s known wih a s> 1n, normal populaion, or for n Ú 30 - Tes for a Mean m : = x - m for s unknown, s> 1n, populaion is normal or nearly normal, and n d.f. = n - 1 z-tes for a Proporion p (when np Ú 5 and nq Ú 5: z = p n - m p n s p n Chi-Square Tes for a Variance 1n - x 1s = s CHAPTER 8 = p n - p 1pq>n or Sandard Deviaion s: Two-Sample z-tes for he Difference Beween Means (Independen samples; n 1 and n Ú 30 or normally disribued populaions): s 1d.f. = n - 1 z = 1x 1 - x - 1m 1 - m s x1 - x, -Tes for he Difference Beween Means (Dependen samples): = d - m d s d > 1n, where and d.f. = n - 1 Two-Sample z-tes for he Difference Beween Proporions ( n 1 p, n 1 q, n p, and n q mus be a leas 5): z = 1pn 1 - pn - 1p 1 - p, where and q = 1 - p. CHAPTER 9 Correlaion Coefficien: r = -Tes for he Correlaion Coefficien: = B pqa 1 n n b r 1 - r B n - d = gd n, ngxy - 1gx1gy ngx - 1gx ngy - 1gy (d.f. = n - ) Equaion of a Regression Line: s d = A g 1d - d yn = mx + b, n - 1 p = x 1 + x n 1 + n where s x1 - x = C s 1 n 1 + s n where m = ngxy - 1gx1gy and ngx - 1gx Two-Sample -Tes for he Difference Beween Means (Independen samples from normally disribued populaions, or n 6 30): n 1 = 1x 1 - x - 1m 1 - m s x1 - x If populaion variances are equal, and s x1 - x = C 1n 1-1s 1 + 1n - 1s n 1 + n - If populaion variances are no equal, d.f. is he smaller of n 1-1 or n - 1 and d.f. = n 1 + n - # B n 1 n s x1 - x = C s 1 n 1 + s n. b = y - mx = gy n - m gx n Coefficien of Deerminaion: r Explained variaion = Toal variaion Sandard Error of Esimae: c-predicion Inerval for y: yn - E 6 y 6 yn + E, where E = c s e C n + n1x 0 - x ngx - 1gx = g1yn i - y g1y i - y s e = C g1y i - yn i n - 1d.f. = n -

4 Key Formulas From Larson/Farber Elemenary Saisics: Picuring he World, Fifh Ediion 01 Prenice Hall CHAPTER 10 Chi-Square: x = g 1O - E E Goodness-of-Fi Tes: d.f. = k - 1 Tes Saisic for he Kruskal-Wallis Tes: Given hree or more independen samples, he es saisic for he Kruskal-Wallis es is H = 1 N1N + 1 a R 1 + R + Á + R k b n 1 n n k Tes of Independence: - 31N d.f. = k - 1 d.f. = 1no. of rows - 11no. of columns - 1 Spearman Rank Correlaion Coefficien: Two-Sample Tes for Variances: F = s 1 F-, where s r s = 1-6gd n1n - 1 s 1 Ú s, d.f. N = n 1-1, and d.f. D = n - 1 One-Way Analysis of Variance Tes: F = MS B, where MS B = MS W and MS W = SS W N - k = g1n i - 1s N - k ( d.f. N = k - 1, d.f. D = N - k) CHAPTER 11 Tes Saisic for Sign Tes: SS B k - 1 = gn iax i - xb k - 1 i Tes Saisic for he Runs Tes: When n 1 0 and n 0, he es saisic is G, he number of runs. When n or n 7 0, he es saisic is z = G - m G, where G = number of runs, s G m G = n 1n n 1 + n + 1, and s G = B n 1 n 1n 1 n - n 1 - n 1n 1 + n 1n 1 + n - 1. When n 5, he es saisic is he smaller number of + or - signs. 1x n When n 7 5, z =, where x is he n smaller number of + or - signs and n is he oal number of + and - signs. Tes Saisic for Wilcoxon Rank Sum Tes: z = R - m R, where R = s R sum of he ranks for he smaller sample, m R = n 11n 1 + n + 1, n 1 n 1n 1 + n + 1 s R =, and n 1 n B 1

5 Table 4 Sandard Normal Disribuion Area z 0 z z Criical Values Level of Confidence c z c z c z = 0 c z c z

6 Table 4 Sandard Normal Disribuion (coninued) Area 0 z z z

7 Table 5 -Disribuion c-confidence inerval Lef-ailed es Righ-ailed es Two-ailed es 1 1 Level of confidence, c One ail, A d.f. Two ails, A q

8 Table 6 Chi-Square Disribuion 1 1 χ χ χ L χ R χ Righ ail Two ails Degrees of A freedom

9 Apple Correlaion Apple Concep Illusraed Descripor Random numbers This apple simulaes selecing a random sample from a populaion by firs assigning a unique ineger o each experimenal uni and hen using he random numbers generaed o deermine he experimenal unis ha will be included in he sample. This apple generaes random numbers from a range of inegers specified by he user. Apple Aciviy 1.3 Mean versus median The mean and he median of a daa se respond differenly o changes in he daa. This apple invesigaes how skewedness and ouliers affec measures of cenral endency. This apple allows he user o visualize he relaionship beween he mean and median of a daa se. The user may easily add and delee daa poins. The apple auomaically updaes he mean and median for each change in he daa..3 Sandard deviaion Sandard deviaion measures he spread of a daa se. Use his apple o invesigae how he shape and spread of a disribuion affec he sandard deviaion. This apple allows he user o visualize he relaionship beween he mean and sandard deviaion of a daa se. The user may easily add and delee daa poins. The apple auomaically updaes he mean and sandard deviaion for each change in he daa..4 Simulaing he sock marke Theoreical probabiliies are long run experimenal probabiliies. This apple simulaes flucuaion in he sock marke, where on any given day going up is equally likely as going down. The user specifies he number of days and he apple repors wheher he sock marke goes up or down each day and creaes a bar graph for he oucomes. I also calculaes and plos he proporion of days ha he sock marke goes up during he simulaion. 3.1 Simulaing he probabiliy of rolling a 3 or 4 Theoreical probabiliies are long run experimenal probabiliies. Use his apple o invesigae he relaionship beween he heoreical and experimenal probabiliies of rolling a 3 or a 4 as he number of imes he die is rolled increases. This apple simulaes rolling a fair die. The user specifies he number of rolls and he apple repors he oucome of each roll and creaes a frequency hisogram for he oucomes. I also calculaes and plos he proporion of 3s and 4s rolled during he simulaion. 3.3 Binomial disribuion As he number of samples increases, he esimaed probabiliy ges closer o he rue value. This apple simulaes values from a binomial disribuion. The user specifies he parameers for he binomial disribuion (n and p) and he number of values o be simulaed (N). The apple plos N values from he specified binomial disribuion in a bar graph and repors he frequency of each oucome. 4. Sampling disribuions The mean and sandard deviaion of he disribuion of sample means are unbiased esimaors of he mean and sandard deviaion of he populaion disribuion. This apple compares he means and sandard deviaions of he disribuions and assesses he effec of sample size. This apple simulaes repeaedly choosing samples of a fixed size n from a populaion. The user specifies he size of he sample, he number of samples o be chosen, and he shape of he populaion disribuion. The apple repors he means, medians, and sandard deviaions of boh he sample means and he sample medians and creaes plos for boh. 5.4 (coninued on nex page)

10 Apple Concep Illusraed Descripor Confidence inervals for a mean (he impac of no knowing he sandard deviaion) Confidence inervals obained using he sample sandard deviaion are differen from hose obained using he populaion sandard deviaion. This apple invesigaes he effec of no knowing he populaion sandard deviaion. This apple generaes confidence inervals for a populaion mean. The user specifies he sample size, he shape of he disribuion, he populaion mean, and he populaion sandard deviaion. The apple simulaes selecing 100 random samples from he populaion and finds he 95% z-inerval and 95% -inerval for each sample. The confidence inervals are ploed and he number and proporion conaining he rue mean are repored. Apple Aciviy 6. Confidence inervals for a proporion No all confidence inervals conain he populaion mean. This apple invesigaes he meaning of 95% and 99% confidence. This apple generaes confidence inervals for a populaion proporion. The user specifies he populaion proporion and he sample size. The apple simulaes selecing 100 random samples from he populaion and finds he 95% and 99% confidence inervals for each sample. The confidence inervals are ploed and he number and proporion conaining he rue proporion are repored. 6.3 Hypohesis ess for a mean No all ess of hypoheses lead correcly o eiher rejecing or failing o rejec he null hypohesis. This apple invesigaes he relaionship beween he level of confidence and he probabiliies of making Type I and Type II errors. This apple performs hypoheses ess for a populaion mean. The user specifies he shape of he populaion disribuion, he populaion mean and sandard deviaion, he sample size, and he null and alernaive hypoheses. The apple simulaes selecing 100 random samples from he populaion and calculaes and plos he saisic and P-value for each sample. The apple repors he number and proporion of imes he null hypohesis is rejeced a boh he 0.05 level and he 0.01 level. 7. Hypohesis ess for a proporion No all ess of hypoheses lead correcly o eiher rejecing or failing o rejec he null hypohesis. This apple invesigaes he relaionship beween he level of confidence and he probabiliies of making Type I and Type II errors. This apple performs hypoheses ess for a populaion proporion. The user specifies he populaion proporion, he sample size, and he null and alernaive hypoheses. The apple simulaes selecing 100 random samples from he populaion and calculaes and plos he z saisic and P-value for each sample. The apple repors he number and proporion of imes he null hypohesis is rejeced a boh he 0.05 level and he 0.01 level. 7.4 Correlaion by eye The correlaion coefficien measures he srengh of a linear relaionship beween wo variables. This apple eaches he user how o assess he srengh of a linear relaionship from a scaer plo. This apple compues he correlaion coefficien r for a se of bivariae daa ploed on a scaer plo. The user can easily add or delee poins and guess he value of r. The apple hen compares he guess o is calculaed value. 9.1 Regression by eye The leas squares regression line has a smaller SSE han any oher line ha migh approximae a se of bivariae daa. This apple eaches he user how o approximae he locaion of a regression line on a scaer plo. This apple compues he leas squares regression line for a se of bivariae daa ploed on a scaer plo. The user can easily add or delee poins and guess he locaion of he regression line by manipulaing a line provided on he scaer plo. The apple will hen plo he leas squares line. I displays he equaions and he SSEs for boh lines. 9.