Equation Sheet On Elementary Statistics for Final Exam | MATH 220, Study notes of Statistics

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Equation Sheet Denton Asdourian Deviation- ( x− x̄ ) IQR: Q3 – Q1 Standard Deviation- s=√∑ ( x− x̄ ) 2 n−1 = sum of squared deviations Empirical Rule- 68% = , 95% = , All or nearly all = Z-score = z= x− x̄ s Correlation r : r= 1 n−1∑ z x z y= 1 n−1∑ ( x− x̄ s x )( y− ȳ s y ) n is the number of points, x̄ and ȳ are means, and s x s y are standard deviations for x and y. the sum is taken over all n observations Regression Line- equation Let ŷ denote the predicted value of y ŷ=a+bx (a denotes y-intercept ( a= ȳ−b ( x̄ ) ) and b denotes slope ( b=r ( s y s x ) Residual sum of squares = ∑ (residual)2 = ∑ ( y− ŷ ) 2 This squares each vertical distance between a point and the line and gets the sum Margin of Error when using a simple random sample = 1 √n ´ 100% If the events are disjoint then, P(A and B) = 0), so P(A or B) = P(A) + P(B) For the intersection of two independent events: P(A and B) = P(A) X P(B) For the union of two events, P(A or B) = P(A) + P(B) – P(A and B) Conditional Probability of Event A, given ( | ) that event B has occurred = P(A|B )= P(AandB ) P(B) P(AandB )=P( A|B)´ P(B ) P(AandB )=P(B|A )´ P(A ) Independent- P(A|B )=P( A ) or P(B|A )=P(B ) , P(AandB )=P( A )´ P(B) Probability Distribution: mean = m=∑ xP (x ) called weighted average Normal Distribution: Cumulative Probability- falling below the point m+ zs = x Complement Probability- falling above the point m+ zs =x Z-score for a value x of a random variable- the number of standard deviations that x falls from the mean m . Equation: z= x−m s Standard Normal Distribution- the normal distribution with mean m =0 and S.D. s =1. It is the distribution of normal z-scores When a random variable’s values are converted to z-scores by subtracting the mean and dividing by the S.D., the z-scores have the standard normal dist (=0, =1) Conditions for Binomial Distribution: Formula for binomial probabilities- P( x )= n ! x ! (n−x )! pn−x , x=0,1,2, .. . , n n! is called n factorial = 1 x 2 x 3 x … x n Mean m and Standard Deviation s : m=np , s=√np(1−p ) Sampling Distribution: For binary data: Mean = p and Standard Deviation = √ p (1−p) n If n is sufficiently large that the expected numbers of outcomes of the two types: [np and n(1−p ) ] ³ 15 , then this sampling distribution is approx normal Standard error of x̄ - the standard deviation of the sampling distribution of the sample mean x̄ For a quantitative variable, the sampling distribution of x̄ has: center- mean = m and spread- standard error = s /√n Sampling distribution of x̄ more bell shaped as n increases 90% C.I.: p̂±1 .645 (se ) 95% C.I.: p̂±1 .. 96 (se ) 99% C.I.: p̂±2 .58 (se ) Confidence Interval estimating a Population Proportion: Point Estimate: p̂ Standard Error: se=√ p̂(1− p̂ )/n Then multiply se by the z-score to get m Confidence Interval: p̂±z ( se ) Sample size for estimating a population proportion: n = (z)2 p̂(1− p̂ ) /m2 Sample size needed for large-sample confidence interval: n p̂ ³ 15 and n(1− p̂ ) ³ 15 Confidence Interval estimating a Population Mean- Point Estimate: x̄ Standard Error: . Then multiply se by the t-score to get m Degrees of freedom- df = n – 1 Confidence Interval: x̄±t (se ) Sample Size for Margin of Error m: n=4 s 2 /m2 Parameter Proportion Mean 1. Assumptions Categorical Variable Randomization Expected #s of successes and failures ³ 15 Quantitative Variable Randomization Approximately normal population 2. Hypothesis H0: p = p0 Ha: p ¹ p0 (two sided) Ha: p> p0 (one-sided) Ha: p< p0 (one-sided) H0: m=m0 Ha: m ¹m0 (two sided) Ha: m>m0 (one-sided) Ha: m<m0 (one-sided) 3. Test Statistic z= p̂−p0 √ p0 (1−p0 )/n t= x̄−m0 s /√n 4. P-value Two-tail (Ha: p ¹ p0 ) Or right tail (Ha: p> p0 ) Or left tail (Ha: p< p0 ) Probability from standard normal distribution Two-tail (Ha: m ¹m0 ) Or right tail (Ha: m>m0 ) Or left tail (Ha: m<m0 ) Probability from t distribution (df = n – 1) 5. Conclusion Interpret P-value in context Reject H0 if P-value ¿α (significance lvl) Interpret P-value in context Reject H0 if P-value ¿α (significance lvl) Mean of differences = difference of means: for dependent samples, the difference ( x̄1− x̄2 ) between the means of the two samples equals the mean x̄d of the difference scores for the matched pairs 95% Confidence interval for the population mean difference: x̄d±t . 025(sd /√n) To test the hypothesis H0: m1=m2 of equal means, we can conduct the single-sample test of H0: md=0 with the difference scores. The test statistic is t=( x̄−0d )/ (sd /√n) Comparing means of dependent samples: construct confidence intervals and significance tests using sample of difference scores, d = observation in sample 1 – observation in sample 2 The 95% confidence interval x̄d±t . 025(se ) and the test statistic t=( x̄d−0)/ se are the same as for a single sample Type I and Type II Errors When H0 is true, a Type I error occurs when H0 is rejected P(type I error) = significance level