Important Formulas, Lecture notes of Calculus

Download Important Formulas and more Lecture notes Calculus in PDF only on Docsity! Chapter 3 Data Description Mean for individual data: Mean for grouped data: Standard deviation for a sample: or (Shortcut formula) Standard deviation for grouped data: Range rule of thumb: Chapter 4 Probability and Counting Rules Addition rule 1 (mutually exclusive events): P(A or B)  P(A)  P(B) Addition rule 2 (events not mutually exclusive): P(A or B)  P(A)  P(B)  P(A and B) Multiplication rule 1 (independent events): P(A and B)  P(A)  P(B) Multiplication rule 2 (dependent events): P(A and B)  P(A)  P(B  A) Conditional probability: Complementary events: P( )  1  P(E) Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: k1  k 2  k 3    kn Permutation rule: Number of permutations of n objects taking r at a time is Combination rule: Number of combinations of r objects selected from n objects is nCr  n! n  r!r! n Pr  n! n  r! E  PB  A  PA and B PA s  range 4 s  n f • X 2 m   f • Xm 2 nn  1 s  nX 2  X2 nn  1 s  X  X 2 n  1 X    f • Xm n X   X n Chapter 5 Discrete Probability Distributions Mean for a probability distribution: m  [X  P(X)] Variance and standard deviation for a probability distribution: s2  [X2  P(X)]  m2 Expectation: E(X)  [X  P(X)] Binomial probability: Mean for binomial distribution: m  n  p Variance and standard deviation for the binomial distribution: s2  n  p  q s  Multinomial probability: Poisson probability: P(X; l)  where X  0, 1, 2, . . . Hypergeometric probability: Chapter 6 The Normal Distribution Standard score Mean of sample means: mX  m Standard error of the mean: sX Central limit theorem formula: Chapter 7 Confidence Intervals and Sample Size z confidence interval for means: t confidence interval for means: Sample size for means: where E is the maximum error of estimate Confidence interval for a proportion: p̂  z 2 p̂ q̂ n p p̂  z 2  p̂ q̂ n n  z 2 • E  2 X   t  2  s n  X   t 2  s n  X   z  2  n  X   z 2  n  z  X   n  n z  X  or z  X  X  s PX  aCX • bCnX abCn e X X! PX  n! X1!X2!X3! . . . Xk! • pX1 1 • pX2 2 • pX3 3 • • • pXk k n • p • q PX  n! n  X!X! • pX • q nX s [X 2 • PX]  m2 Important Formulas blu38582_IF_1-8.qxd 9/27/10 9:19 PM Page 1 Sample size for a proportion: where Confidence interval for variance: Confidence interval for standard deviation: Chapter 8 Hypothesis Testing z test: for any value n. If n 30, population must be normally distributed. t test: (d.f.  n  1) z test for proportions: Chi-square test for a single variance: (d.f.  n  1) Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances z test for comparing two means (independent samples): Formula for the confidence interval for difference of two means (large samples): t test for comparing two means (independent samples, variances not equal): (d.f.  the smaller of n1  1 or n2  1) t  X  1  X  2   1  2  s2 1 n1  s2 2 n2 X  1  X  2  z 2  2 1 n1  2 2 n2 X  1  X  2  z 2  2 1 n1  2 2 n2 1  2 z  X  1  X  2   1  2  1 2 n1  2 2 n2  2  n  1s2 2 z  p̂  p pq n t  X   s n z  X   n n  1s2  2 right n  1s2  2 left n  1s2  2 right 2 n  1s2  2 left p̂  X n and q̂  1  p̂ n  p̂q̂ z 2 E  2 Formula for the confidence interval for difference of two means (small independent samples, variance unequal): (d.f.  smaller of n1  1 and n2  1) t test for comparing two means for dependent samples: Formula for confidence interval for the mean of the difference for dependent samples: (d.f.  n  1) z test for comparing two proportions: where Formula for the confidence interval for the difference of two proportions: F test for comparing two variances: where is the larger variance and d.f.N.  n1  1, d.f.D.  n2  1 s2 1F  s2 1 s2 2  p̂1  p̂2  z 2  p̂1q̂1 n1  p̂2q̂2 n2  p̂1  p̂2  z 2  p̂1q̂1 n1  p̂2q̂2 n2 p1  p2 q _  1  p _ p̂2  X2 n2 p _  X1  X2 n1  n2 p̂1  X1 n1 z   p̂1  p̂2   p1  p2 p _ q _  1 n1  1 n2  D   t 2 SD n D D   t 2 SD n sD  nD 2  D2 nn  1 d.f.  n  1 t  D   D sD n where D   D n and X  1  X  2  t 2  s2 1 n1  s2 2 n2 X  1  X  2  t 2  s2 1 n1  s2 2 n2 1  2 blu38582_IF_1-8.qxd 9/27/10 9:19 PM Page 2