FORMULAS / STRATEGY FOR STATISTICS Essay Example For Students

FORMULAS / STRATEGY FOR STATISTICS Essay Probability Complement Law P(A) = 1 P(A) Laws Of Addition -P(A B) = P(A) + P(B) P(A B), if A and B not mutually exclusive P(A B) = P(A) + P(B), if A and B are mutually exclusive Conditional Probability P(A|B) = P(A B) P(B) Independent Condition If A and B are independent, P(A B) = P(A) x P(B) Laws Of Multiplication If A and B are dependent, P(A B) = P(A) x P(B|A) or P(A B) = P(B) x P(A|B) Descriptive Statistics Population Mean, m= all values N Sample Mean, x = all values n Population Variance, s2 = (X m)2 N Sample Variance, S2 = (x x)2 n-1 Standard Deviation = square root of s2 or S2 Probability Distribution Expected Value, E(x) = all x P(xi = x) = m Properties of E(x), E(a) = a E(ax) = aE(x) E(ax b) = aE(x) b E(x1 x2) = E(x1) E(x2) E(x2) = all x2 P(xi = x) Variance, Var(x) = E(x m)2 or Var(x) = E(x2) n(x)2 Properties of Var(x), Var(a) = 0 Var(ax) = a2Var(x) Var(ax b) = a2E(x) Var(x1 x2) = Var(x1) + Var(x2) E(x2) = all x2 P(xi = x) Standard Deviation = square root of var(x) Binomial Distribution x ~ Bin (n , p) Characteristics, Experiment consist of a number of trials Results of trials are only either success or failure Probability of each test between trials are the same E(x) = np Var(x) = npq Continuous Distribution x ~ N(m , s2) Standardising, z = x m s Normal Approximation to Binomial Distribution x ~ N(np , npq) Conditions, Number of trials n > 50 Must use continuity correction Joint Probability Conditional Mean E(x | y=y1) = all x P(xi | y) E(XY) = all x all y P(xi = x and yi = y) When x and y are independent, E(XY) = E(X) E(Y) Covariance of 2 random variables, sxy Cov(XY) = E(XY) E(X)E(Y) When X and Y are independent, Cov(XY) = 0, since E(XY) = E(X)E(Y) Correlation Coefficient, r = Cov(XY),-1 r 1 Var(x) Var(y) Formula for Variance of linear combinations of 2 dependent variables Var(X Y) = Var(X) + Var (Y) 2Cov(XY) Var(aX bY) = a2Var(X) + b2Var (Y) 2abCov(XY) Distribution Of Sample Mean Sample Proportion Let X denote the population variable. m the population mean and s2 the population variance. then, x ~ N(m,s2/n) Let P denote the population proportion with proportion P with n, the number of samples, then P ~ N { p , p (1-p)/n } if P is unknown, P ~ N { P , P (1-P)/n } approx. where P is the sample proportion with the use of continuity correction x (1/2n) Theory Of Estimation Mean Square Error MSE = E(V q)2 where V is the value of the estimator from the true value q Best estimator of the true value is the one that yields the lowest MSE Confidence Interval The interval of which the true value is probable to be included. 3 Cases Of Formula For Confidence Interval For population mean where m, s2 given,-m = x (s2/n)1/2 Zsig level m given but s2 unknown, samples size n > 50-m = x (S2/n)1/2 Zsig level m given but s2 unknown, samples size n < 50-m = x (S2/n)1/2 tsig level For difference in population means mx my where m, s2 given,- mD = (x y) (sx2/nx + sy2/ny)1/2 Zsig level m given but s2 unknown, samples size n > 50- mD = (x y) (Sx2/nx + Sy2/ny)1/2 Zsig level m given but s2 unknown, samples size n < 50- mD = (x y) (Sp2/nx + Sp2/ny)1/2 tsig level where pooled variance, Sp2 = S(x-x)2 + S(y-y)2 nx + ny - 2 Sp2 = Sx2(nx-1) + Sy2(ny-1) nx + ny - 2 Paired Samples- mD = D (SD2/nD)1/2 tsig level where D is the difference between the paired samples. For Population Proportion, p ~ N { p, p(1-p)/n } p not given, then it is estimated with variance P(1-P)/n, in the confidence interval of p = P (P(1-P)/n)1/2 Zsig level Hypothesis Testing Procedure: State Null and Alternate hypothesis Determine one or two sided test Find Ztest or ttest and compare the result with Zcritical and Tcritical respectively Decision Rule, |Ztest| < Zcritical or |ttest| < Tcritical then null hypothesis is true Conclude in relation to hypothesis / question e.g., Ztest = x - m s/n P-value - Decision Rule Reject H0 if p-value < level of significance Accept H0 if p-value level of significance .u39460ac9855cae49de3f21ea69457c87 , .u39460ac9855cae49de3f21ea69457c87 .postImageUrl , .u39460ac9855cae49de3f21ea69457c87 .centered-text-area { min-height: 80px; position: relative; } .u39460ac9855cae49de3f21ea69457c87 , .u39460ac9855cae49de3f21ea69457c87:hover , .u39460ac9855cae49de3f21ea69457c87:visited , .u39460ac9855cae49de3f21ea69457c87:active { border:0!important; } .u39460ac9855cae49de3f21ea69457c87 .clearfix:after { content: ""; display: table; clear: both; } .u39460ac9855cae49de3f21ea69457c87 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u39460ac9855cae49de3f21ea69457c87:active , .u39460ac9855cae49de3f21ea69457c87:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u39460ac9855cae49de3f21ea69457c87 .centered-text-area { width: 100%; position: relative ; } .u39460ac9855cae49de3f21ea69457c87 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u39460ac9855cae49de3f21ea69457c87 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u39460ac9855cae49de3f21ea69457c87 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u39460ac9855cae49de3f21ea69457c87:hover .ctaButton { background-color: #34495E!important; } .u39460ac9855cae49de3f21ea69457c87 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u39460ac9855cae49de3f21ea69457c87 .u39460ac9855cae49de3f21ea69457c87-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u39460ac9855cae49de3f21ea69457c87:after { content: ""; display: block; clear: both; } READ: Banning Child Labour in Developing Countries Essay Type I Error - The error of rejecting H0 when H0 is true P(type I error) = the level of significance Type II Error, b .