Converting to the Standard Normal rv: Exponential PDF and CDF for x 0 Chapter 7: expected value of x

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1 Key Formula Sheet ASU ECN 22 ASWCC Chapter : no key formulas Chapter 2: Relative Frequency=freq of the class/n Approx Class Width: =(largest value-smallest value) /number of classes Chapter 3: sample and population means x = x i /n and µ = x i /N Weighted mean and geometric mean Chapter 4 continued: P(A B) = P(A) + P(B) P(A B) P(A B) = P(A B) P(B) P(A B) = P(B)P(A B) = P(A)P(B A) Multiplication Law for Independent Events x = w i x i /w i and xg = [(x)(x2) (xn)] /n P(A B) = P(B)P(A) Interquartile Range: IQR = Q3 Q Population and sample variance σ 2 = (x i µ) 2 N and s 2 = (x i x) 2 n Population and sample standard deviation σ = σ 2 and s = s 2 Coefficient of Variation ( Standard deviation Mean z-score: z i = x i x s Population and Sample Covariance σxy = (x i µx)(y i µy N 00 ) % and sxy = (x i x)(y i ȳ n Population and Sample Pearson Correlation ρxy = σxy/(σxσy) and rxy = sxy/(sxsy) Bayes Theorem P(A i B) = P(A i )P(B A i ) P(A)P(B A) + P(A2)P(B A2) + + P(An)P(B An) Chapter 5: Discrete Uniform Probability Mass Function: f (x) = /n Expected Value of a Discrete R V: E(x) = µ = x f (x) Variance of a Discrete R V: Var(x) = σ 2 = (x µ) 2 f (x) Number of Experimental Outcomes Providing Exactly x Successes in n Trials ( n x ) = n! x!(n x)! Binomial Probability Mass Function ( ) n P(X = x) = f (x) = p x ( p) (n x) x Chapter 4: Counting Rule for Combinations ( ) N C n N N! = = n n!(n n)! Counting Rule for Permutations ( ) P n N = n! N n = N! (N n)! Expected Value for Binomial Distribution: E(x) = µ = np Variance for Binomial Distr: Var(x) = σ 2 = np( p) Poisson Probability Mass Function: P(X = x µ) = f (x) = µx e µ Hypergeometric Probability Mass Function and Expected Value: x! Probability Rules: P(A) = P(A C ) f (x) = (r x )(N r n x ) N ( N n ) and E(x) = µ = nr Chapter 5 continued: Variance for the Hypergeometric Distribution: Var(x) = σ 2 = n ( r N )( r N )( N n N ) Chapter 6: Uniform PDF f (x) = { b a if a x b 0 otherwise Normal PDF The density function is f (x) = ( exp 2πσ 2 (x ) µ)2 2σ 2 Converting to the Standard Normal rv: z = x µ σ Exponential PDF and CDF for x 0 f (x) = µ e x/µ and P(x x0) = e x 0/µ Chapter 7: expected value of x E( x) = µ Standard Deviation of x (Standard Error) σx = σ n Expected Value and Std Dev (Standard Error) of p E( p) = p and σ p = p( p) Finite Pop Correction Factor: (N n)/(n ) Chapter 8: Interval Estimate of Population Mean, σ known and unknown n x ± z α/2 n and x ± t α/2 n σ s Necessary Sample Size for Interval Estimate of µ n = (z α/2) 2 σ 2 E 2

2 Chapter 8 continued: Interval Estimate of p p( p) p ± z α/2 n Necessary Sample Size for Interval Estimate of p n = (z α/2) 2 p ( p ) E 2 Chapter 9: Test Statistic for Hypothesis Tests About µ, σ known and unknown z = x µ 0 σ/ n and t = x µ 0 s/ n Test Stat for Hypothesis About p z = p p 0 p 0( p0) n Chapter 0: Point Estimate and Standard Error for Difference in Two Population Means x x2 and σ x x2 = σ 2 n + σ2 2 n2 Interval Estimate and Test Statistic for Difference in Two Means with Known Variances x x2 ± z α/2 σ x x2 = σ 2 n + σ2 2 n2 σ 2 and z = x x2 D0 n + σ2 2 n2 Interval Estimate and Test Statistic for Difference in Two Means with Unknown Variances x x2 ± t α/2 σ x x2 = s 2 n + s2 2 n2 s 2 and t = x x2 D0 n + s2 2 n2 Degrees of Freedom for t, Two Independent Random Samples d f = n ( s 2 n + s2 2 n2 ( s 2 ) 2 n + n2 ) 2 ( s 2 2 n2 ) 2 Chapter 0 continued: Test Statistic (Matched Samples) d t = µ d s d / n ANOVA Related: x j = n j i= x ij n j s 2 j = n j i= (x ij x j ) 2 n j x = k j= n j nt i= x ij MSTR = SSTR k SSTR = k n j ( x j x) 2 MSE = SSE j= nt k SSE = k (n j )s 2 j F = MSTR/MSE j= SST = k j= n j (x ij x) 2 SST=SSTR+SSE i= Chapter : not covered in this course Chapter 2: y = β0 + βx + ɛ E(y) = β0 + βx ŷ = b0 + bx b0 = ȳ b x b = (x i x)(y i ȳ) (x i x) 2 SSE = (y i = ŷ i ) 2 SST = (y i ȳ) 2 SSR = (ŷ i ȳ) 2 SST=SSR+SSE r 2 = SSR SST r xy = (sign of b) r 2 s 2 = MSE = SSE n 2 Standard Error of the Estimate, s = MSE σ b = σ s (xi x) 2 b = s t = b (xi x) 2 s b For simple regression, MSR = SSR because there is only one independent variable F = MSR MSE sŷ = s n + (x x) 2 (x x) 2 Confidence Interval for E(y ): ŷ α/a s ŷ s pred = s + n + (x x) 2 (x x) 2 Chapter 2 continued: Prediction Interval for y : ŷ ± t α/a s pred Residual for Observation i: y i ŷ i Chapter 3: y = β0 + βx + β2x2 + + βpxp + ɛ E(y) = β0 + βx + β2x2 + + βpxp ŷ = b0 + bx + b2x2 + + bpxp SST = SSR + SSE R 2 = SSR SST R 2 a = ( R 2 ) n n p MSR = SSR p MSE = SSE n p F = MSR MSE t = b i s bi Other Math Rule Reminders: e x = exp(x) ln = 0 lne = x! = (x)(x )(x 2) (2)() 0! = x 0 =

3 ECN22 Exam A Spring 205, ASU-COX Choose the best answer Do not write letters in the margin or communicate with other students in any way If you have a question note it on your exam and ask for clarification when your exam is returned In the meantime choose the best answer Neither the proctors nor Dr Cox will answer questions during the exam Dr Cox will post a key the day after the exam or in the case of the final exam the day after all finals are given Grades will be posted on Bb after scores are returned from the testing center Please check each question and possible answers thoroughly as questions at the bottom of a page sometimes run onto the next page Relax You studied You know the material You can nail it The sample size (a) can be larger than the population size (b) can be larger or smaller than the population size (c) is always smaller than the population size (d) is always equal to the size of the population 2 Data collected over several time periods are (a) time controlled data (b) time series data (c) crossectional data (d) time crossectional data 3 Statistical inference (a) is the same as Data and Statistics (b) refers to the process of drawing inferences about the sample based on the characteristics of the population (c) is the same as a census (d) is the process of drawing inferences about the population based on the information taken from the sample 4 Product brand is an example of (a) categorical data (b) quantitative data (c) either categorical or quantitative data 3

4 (d) time series data 5 Consider the data set below: customer information customer $ spent days since billing tenure Which variables are quantitative? (a) $ spent, days since billing, and tenure (b) customer and $ spent (c) customer, $ spent and tenure (d) they are all quantitative 6 What is true about the data set below: customer information customer $ spent days since billing tenure (a) the data are time series (b) the data are time series and cross-sectional (c) the data are cross-sectional (d) some variables are time-series and some are cross sectional 7 Which of the following is a quantitative variable (a) gender (b) education level (c) employment status (d) interest rate 4

5 8 What is the mode in the following example? (a) 3 (b) 7 (c) 37 (d) 49 9 The display below is an example of a (a) histogram (b) bar chart (c) scatter plot (d) pie chart 0 The display below has 5

6 (a) too many classes (b) the right number of classes (c) too few classes (d) a class range that is too narrow The relative frequency of platinum cards in the portfolio is: charge/credit card type card frequency relative frequency blue 43 green 78 gold 76 platinum 25 (a) (b) 59% (c) 80 (d) The relative frequency of blue cards in the portfolio is: charge/credit card type card frequency relative frequency blue 43 green 78 gold 76 platinum 25 6

7 (a) 66 (b) 42 (c) 34% (d) The cumulative frequency of credit card accounts with a balance less than $30,000? credit card balance card cum freq cum relative freq <= 9, ,000-9, ,000-29, , (a) 25 (b) 94% (c) 397 (d) The cumulative relative frequency of cards with a balance under $20,000 is: credit card balance card cum freq cum relative freq <= 9, ,000-9, ,000-29, , (a) 76 (b) 42 (c) 339 (d) 94 5 A set of credit card accounts have the following estimated probabilities of default: 067, 743, 042, 022, 03, 94, 00 Find the 30th percentile (a) 2 (b) 036 (c) 03 7

8 (d) The variance of a set of cereal prices is 38 The standard deviation is (a) 7 (b) 90 (c) 38 (d) 69 7 Find the z-score for an observation with a value of 722 when the mean is 86 and the variance is 227 (a) -6 (b) 6 (c) -292 (d) The average of 22, 25, 26, 29 is (a) 255 (b) 25 (c) 26 (d) The average is always greater than the standard deviation (a) true (b) false (c) true for ratio data but false for interval data (d) false for ratio data but true for interval data 20 Suppose the covariance between two variables is 57 while their individual standard deviations are 8 and 452 The correlation coefficient is: (a) 403 (b) 57 (c) 635 (d) 00 8

9 2 Suppose WP Carey has 4 internships lined up with local companies Suppose that there are 24 applicant for the internships that meet the GPA requirement; WP Carey will not select an applicant that does not meet the GPA requirement How many possible combinations of students filling the internships are there? (a) (b) (c) (d) A really big number not given here 22 A study by the Institute for Higher Education Policy found the values in the joint probability table below The underlying data are for former college students that had taken out student loans The table shows whether the student received a college degree versus whether they are successfully making their student loan payments What is the probability that a former student completed their degree? satisfactory delinquent total (a) 58 (b) 42 (c) 26 (d) 6 23 A study by the Institute for Higher Education Policy found the values in the joint probability table below The underlying data are for former college students that had taken out student loans The table shows whether the student received a college degree versus whether they are successfully making their student loan payments What is the probability that a former student completed their degree and is currently satisfactory on their loan payments? satisfactory delinquent total (a) 26 9

10 (b) 42 (c) 92 (d) A study by the Institute for Higher Education Policy found the values in the joint probability table below The underlying data are for former college students that had taken out student loans The table shows whether the student received a college degree versus whether they are successfully making their student loan payments What is the probability that a former student did not complete their degree and is currently delinquent on their loan payments? satisfactory delinquent total (a) 26 (b) 74 (c) 50 (d) Suppose you observe the following prices for cereals which are given in dollars, 3, 3, 5, 5, 7 where the $7 cereal is a high end organic granola Find the z-score of the organic granola (a) z=76, and it is an outlier (b) z=76, which means it is not an outlier (c) z=298, which means it is not an outlier (d) z=288, which means it is close to being an outlier 26 A study by the Institute for Higher Education Policy found the values in the joint probability table below The underlying data are for former college students that had taken out student loans The table shows whether the student received a college degree versus whether they are successfully making their student loan payments What is the probability that a former student did not complete their degree given that they are currently delinquent on their loan payments? satisfactory delinquent total

11 (a) 34 (b) 68 (c) 50 (d) Consider the Let s Make a Deal game we played in class Instead of three doors suppose that there are four doors, a, b, c and d What is the probability that you will win if you switch from your original guess after one door is opened? You can apply Bayes Rule or some other technique to solve this (a) /4 (b) /2 (c) 2/3 (d) 3/8 (e) 6/8 28 A study by the Institute for Higher Education Policy found the values in the joint probability table below The underlying data are for former college students that had taken out student loans The table shows whether the student received a college degree versus whether they are successfully making their student loan payments What is the probability that a former student with a loan is delinquent on their loan payments? satisfactory delinquent total (a) 6 (b) 34 (c) 50 (d) I have checked that my ID is bubbled in correctly If it is bubbled in incorrectly I will get this question wrong (a) True (b) False

12 Key c 2 b 3 d 4 a 5 a 6 c 7 d 8 c 9 d 0 c d To find the answer we need to complete the table Then we can read the answer out of the table charge/credit card type card frequency relative frequency blue green gold platinum d To find the answer we need to complete the table Then we can read the answer out of the table charge/credit card type card frequency relative frequency blue green gold platinum c The value is found from noting that there are a total of 422 observations and the values are found in the table: 2

13 credit card balance card cum freq cum relative freq <= 9, ,000-9, ,000-29, , a The value is found from noting that there are a total of 422 observations and the values are found in the table: credit card balance card cum freq cum relative freq <= 9, ,000-9, ,000-29, , c c n = 7 and p = 30 so the index point is i = (7)(30)/00 = 2 which we round up to 3 The third value is 03 6 a a 38 = 7 7 c c Find the z-score by using the formula and plugging in, z = = = a 9 b 20 c Compute: r = 57 (8)(452) = b b Use the formula ( ) 24 = 24! 4 0!4! = b b Take the value from the table satisfactory delinquent total

14 23 a a Take the value from the table satisfactory delinquent total d d Take the value from the table satisfactory delinquent total b b x = 66 and s = 589 so that z = = 76 and we would not consider it to be an outlier because 3 < z < 3 26 b b Take the values from the table to make the calculations P (A B) = P (A B) P (B) = 34 5 = 68 satisfactory delinquent total d d Suppose that you pick a (or any door) and switch to d (or any unrevealed door) after b or c is revealed (or whichever other two doors there are) P (d picked a (b or c revealed)) P (d)p (picked a (b or c revealed) d) = K where K = P (a)p (picked a (b or c rev) a) + P (b)p (picked a (b or c rev) b) + P (c)p (picked a (b or c rev) c) + P (d)p (picked a (b or c rev) d), = (/4)(/4) (/4)(/6) + (/4)(/8) + (/4)(/8) + (/4)(/4) = 3 8 4

15 A simpler way to get the same answer would be to fix the idea that an initial guess would give the prize door with probability /4 That means the other doors are right with probability 3/4 Then, after one is revealed there are only two left so picking one of them gives you (/2)(3/4)=3/8 as the probability you will get the right door 28 c c Take the value from the table satisfactory delinquent total a 5