Chapter 2 Organizing and Summarizing Data. Chapter 3 Numerically Summarizing Data. Chapter 4 Describing the Relation between Two Variables

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1 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. Chapter Organizing and Summarizing Data Relative frequency = frequency sum of all frequencies Class midpoint: The sum of consecutive lower class limits divided by. Chapter 3 Numerically Summarizing Data Population Mean: m = gx i N Sample Mean: x = gx i n Range = Largest Data Value - Smallest Data Value Population Standard Deviation: g(x s = i - m) gx (gx i ) i - N = B N R N Sample Standard Deviation s = B g(x i - x) n - 1 Population Variance: s Sample Variance: s (gx gx i ) i - n = R n - 1 Empirical Rule: If the shape of the distribution is bellshaped, then Approximately 68% of the data lie within 1 standard deviation of the mean Approximately 95% of the data lie within standard deviations of the mean Approximately 99.7% of the data lie within 3 standard deviations of the mean Population Mean from Grouped Data: m = gx i f i gf i Sample Mean from Grouped Data: x = gx i f i gf i Weighted Mean: x w = gw i x i gw i Population Standard Deviation from Grouped Data: s = B g(x i - m) f i gf i = R gx i f i - (gx i f i ) gf i gf i Sample Standard Deviation from Grouped Data: (gx g(x i - m) gx i f i ) i f f i - i gf s = = i B (gf i ) - 1 R gf i - 1 Population z-score: z = x - m s Sample z-score: z = x - x s Interquartile Range: IQR = Q 3 - Q 1 Lower and Upper Fences: Lower fence = Q 1-1.5(IQR) Upper fence = Q (IQR) Five-Number Summary Minimum, Q 1, M, Q 3, Maximum Chapter 4 Describing the Relation between Two Variables a a x i - x s x b a y i - y b s y Correlation Coefficient: r = n - 1 The equation of the least-squares regression line is yn = b 1 x + b 0, where yn is the predicted value, b 1 = r # s y s x is the slope, and b 0 = y - b 1 x is the intercept. Residual = observed y - predicted y = y - yn R = r for the least-squares regression model yn = b 1 x + b 0 The coefficient of determination, R, measures the proportion of total variation in the response variable that is explained by the least-squares regression line. Chapter 5 Probability Empirical Probability frequency of E P(E) number of trials of experiment Classical Probability number of ways that E can occur P(E) = number of possible outcomes = N(E) N(S) Addition Rule for Disjoint Events P(E or F ) = P(E) + P(F ) Addition Rule for n Disjoint Events P(E or F or G or g) = P(E) + P(F ) + P(G) + g General Addition Rule P(E or F ) = P(E) + P(F ) - P(E and F )

2 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. Complement Rule P(E c ) = 1 - P(E) Multiplication Rule for Independent Events P(E and F ) = P(E) # P(F ) Multiplication Rule for n Independent Events P(E and F and G g ) = P(E) # P(F) # P(G) # g Conditional Probability Rule P(F E) = P(E and F ) P(E) General Multiplication Rule = N(E and F ) N(E) Factorial n! = n # (n - 1) # (n - ) # g# 3 # # 1 Permutation of n objects taken r at a time: n P r = Combination of n objects taken r at a time: n! nc r = r!(n - r)! Permutations with Repetition: n! n 1! # n! # g # n k! n! (n - r)! P(E and F) = P(E) # P(F E) Chapter 6 Discrete Probability Distributions Mean (Expected Value) of a Discrete Random Variable m X = gx # P(x) Standard Deviation of a Discrete Random Variable s X = 3g(x - m) # P(x) = 3gx P(x) - m X Binomial Probability Distribution Function P(x) = n C x p x (1 - p) n-x Mean and Standard Deviation of a Binomial Random Variable m X = np s X = np(1 - p) Chapter 7 The Normal Distribution Standardizing a Normal Random Variable z = x - m s Finding the Score: x = m + zs Chapter 8 Sampling Distributions Mean and Standard Deviation of the Sampling Distribution of x Sample Proportion: pn = x n m x = m and s x = s n Mean and Standard Deviation of the Sampling Distribution of pn m np = p and s np = B p(1 - p) n Chapter 9 Estimating the Value of a Parameter Confidence Intervals A (1 - a) # 100% confidence interval about p is pn(1 - pn) pn { z a/ #. B n A (1 - a) # 100% confidence interval about m is x { ta/ # s 1n. Note: t a/ is computed using n - 1 degrees of freedom. Sample Size To estimate the population proportion with a margin of error E at a (1 - a) # 100% level of confidence: n = pn(1 - pn)a z a/ E b rounded up to the next integer, where pn is a prior estimate of the population proportion, or n = 0.5 a z a/ E b rounded up to the next integer when no prior estimate of p is available. To estimate the population mean with a margin of error E at a (1 - a) # 100% level of confidence: n = a z a/ # s E b rounded up to the next integer.

3 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. Chapter 10 Hypothesis Tests Regarding a Parameter Test Statistics z 0 = pn - p 0 p 0 (1 - p 0 ) C n t 0 = x - m 0 s 1n Chapter 11 Inferences on Two Samples Test Statistic Comparing Two Population Proportions (Independent Samples) z 0 = pn 1 - pn - (p 1 - p ) pn(1 - pn) B 1 n n where pn = x 1 + x n 1 + n. Confidence Interval for the Difference of Two Proportions (Independent Samples) pn 1 (1 - pn 1 ) (pn 1 - pn ) { z a/ + pn (1 - pn ) C n 1 n Test Statistic Comparing Two Proportions (Dependent Samples) z 0 = 0 f 1 - f f 1 + f 1 Test Statistic for Matched-Pairs Data Confidence Interval for Matched-Pairs Data s d { t a/ # d 1n Note: t a/ is found using n - 1 degrees of freedom. Test Statistic Comparing Two Means (Independent Sampling) t 0 = (x 1 - x ) - (m 1 - m ) s 1 + s Cn 1 Confidence Interval for the Difference of Two Means (Independent Samples) s 1 (x 1 - x ) { t a/ + s Cn 1 n Note: t a/ is found using the smaller of n 1-1 or n - 1 degrees of freedom. n t 0 = d - m d s d 1n where d is the mean and s d is the standard deviation of the differenced data. Chapter 1 Additional Inferential Procedures Chi-Square Procedures Expected Counts (when testing for goodness of fit) E i = m i = np i for i = 1,, p, k Expected Frequencies (when testing for independence or homogeneity of proportions) (row total)(column total) Expected frequency = table total Chi-Square Test Statistic x 0 = a (observed - expected) expected i = 1,, p, k = a (O i - E i ) E i All E i Ú 1 and no more than 0% less than 5.

4 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. Inference on the Least-Squares Regression Model Standard Error of the Estimate s e = C g(y i - yn i ) n - Standard error of b 1 s b1 = = C g residuals n - s e g(x i - x) Test statistic for the Slope of the Least-Squares Regression Line b 1 - b 1 t 0 = = b 1 - b 1 s en g(xi - x) s b1 Confidence Interval about the Mean Response of y, yn yn { t a/ # 1 (x* - x) se + Cn g(x i - x) where x* is the given value of the explanatory variable and t a/ is the critical value with n - degrees of freedom. Prediction Interval about an Individual Response, yn yn { t a/ # se C n + (x* - x) g(x i - x) Confidence Interval for the Slope of the Regression Line s b 1 { t a/ # e g(x i - x) where t a/ is computed with n - degrees of freedom. where x* is the given value of the explanatory variable and t a/ is the critical value with n - degrees of freedom.

5 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. TABLE I Random Numbers Column Number Row Number Table ii Critical Values for Correlation Coefficient n n n n

6 Table III Binomial Probability Distribution This table computes the probability of obtaining x successes in n trials of a binomial experiment with probability of success p. p n x Note: means the probability is rounded to four decimal places. However, the probability is not zero. Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

7 Table III (continued ) p n x Note: means the probability is rounded to four decimal places. However, the probability is not zero. Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

8 Table III (continued) p n x Note: means the probability is rounded to four decimal places. However, the probability is not zero. Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

9 Table III (continued ) p n x Note: means the probability is rounded to four decimal places. However, the probability is not zero. Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

10 Table IV Cumulative Binomial Probability Distribution This table computes the cumulative probability of obtaining x successes in n trials of a binomial experiment with probability of success p. p n x Note: means the probability is rounded to four decimal places. However, the probability is not zero means the probability is rounded to four decimal places. However, the probability is not one. Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

11 Table Iv (continued ) n x Note: means the probability is rounded to four decimal places. However, the probability is not zero means the probability is rounded to four decimal places. However, the probability is not one. p Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

12 Table Iv (continued ) n x Note: means the probability is rounded to four decimal places. However, the probability is not zero means the probability is rounded to four decimal places. However, the probability is not one. p Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

13 Table IV (continued ) n x Note: means the probability is rounded to four decimal places. However, the probability is not zero means the probability is rounded to four decimal places. However, the probability is not one. p Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc.

14 Tables and Formulas for Sullivan, Fundamentals of Statistics, 4e 014 Pearson Education, Inc. Area z Table v TABLE V Standard Normal Distribution z Confidence Interval Critical Values, z A/ Level of Confidence Critical Value, z A/ 0.90 or 90% or 95% or 98% or 99%.575 Hypothesis Testing Critical Values Level of Significance, A Left-Tailed Right-Tailed Two-Tailed { { {.575

MTH 225: Introductory Statistics

Marshall University College of Science Mathematics Department MTH 225: Introductory Statistics Course catalog description Basic probability, descriptive statistics, fundamental statistical inference procedures