Review of Math 526: Applied Mathematical Statistics for Midterm I c Fall 2013 by Professor Yaozhong Hu

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1 Review of Math 526: Applied Mathematical Statistics for Midterm I c Fall 2013 by Professor Yaozhong Hu Some Concepts: statistics, population, sample, parameter, data, statistic, sample size, maximum, minimum, frequency distribution, frequency table, frequency histogram, relative frequency histogram, bar chart, pie chart, ogive, box plot, stem-and-leaf plot, Class intervals, lower limits and upper limits, class size, class mark, class boundary. Steps to follow in forming a frequency distribution: 1) Find the Range. Range = largest value - smallest value 2) Find the number of classes: 3) Find the class width: The number of classes = log (Sample size) The approximate class width = 4) Determine the class (determine the starting value) 5) Tally to find the Frequency and then form the frequency table Steps to draw a histogram (or a bar chart): Range Number of classes. 1) Find the frequency table (Find the frequency or relative frequency table) 2) Find the class boundaries 3) Draw the rectangular system and the histogram Steps to draw an Ogive (Cumulative plot): 1) Find the frequency table (Find the frequency or relative frequency table) 2) Find the class boundaries 3) Find the cumulative frequency. Find the frequency which is less than the class boundaries 4) Draw the rectangular system and the histogram Steps to draw pie chart: 1) Find the relative frequency and percentage or percentage frequency. 2) Find the angle which is equal to 360 relative frequency 3) Draw the pie chart 1

2 Mean, Median, Mode (Arithmetic) Mean of a variable = P xi n. Median n Median = the middle value if the number of observation is odd the mean of the two middle values if the number of observation is even Mode is the value(s) which occurs most often (with the greatest frequency) and the Sample range is max min. First quartile Q 1 and the third quartile Q 3 The way in TI-83 to find Q 1 (or Q 3 ) is as follows: 1) Arrange the data in increasing order. 2) Find the median and throw away the median and the second (first) half of the data. 3) The median of the remaining data is Q 1 (or Q 3 ). Sample variance is defined as P s 2 (xi x) 2 = n 1 " = 1 X x 2 i n 1 ( P # x i ) 2. n sample standard deviation s. Some Concepts: Sample space, event, union (or), intersection (and), complement (not), Venn Diagram. A probability (measure) is a function from certain subsets of sample space to R such that (1) P (A) 0 (2) P ( ) = 1 (3) P (A 1 [ A 2 [A n [ )=P(A 1 )+P(A 2 )+ P(A n )+ for mutually disjoint A i, i =1, 2,. The conditional probability of A given B is defined as P (A B) =P (A \ B)/P (B). Dependence If P (A \ B) =P (A)P (B), then we say that A and B are independent. In this case P (A B) = P (A). Some other useful formulas P (A [ B) =P (A)+P (B) P (A c )=1 P (A) P (A [ B) =P (A)+P (B) P (A \ B) if A and B are mutually exclusive. P (A \ B) =P (A B)P (B) =P (B A)P (A) P (B) =P (A 1 )P (B A 1 )+P (A 2 )P (B A 2 )+ + P (A n )P (B A n ), where A i,i=1, 2,,n are mutually exclusive and their union is the sample space. 2

3 Bayes formula: P (B i A) = P (B i)p (A B i ) P (A) If each outcome occurs equally likely, then = P (B i )P (A B i ) P (B 1 )P (A B 1 )+P (B 2 )P (A B 2 )+ + P (B n )P (A B n )). P (A) =r/n = number of outcomes in A number of outcomes in S. Counting techniques 1. The multiplication law: If a certain experiment can be performed in r ways and another can be performed in k ways, then the combined experiment can be performed in rk ways. 2. Addition law: P (A [ B) =P (A)+P (B) ifa and B are mutually exclusive. 3. The number of permutations (ordered arrangements) of n distinct objects taken r is np r = n!/(n r)!. 4. The number of combinations of n distinct objects taken r is given n n! = r r!(n r)!. Random Variable is a function from sample space S to R. It associates with a probability density function (pdf) and a cumulative distribution function (cdf). A (finite) discrete pdf is given by x x 1 x 2 x n f(x) f 1 f 2 f n nx where f i > 0, i =1, 2, and f i = 1. i=1 A continuous pdf is given by a function f(x), x 2 R, wheref(x) 0 and f(x)dx = 1. If X has pdf f(x), then ( P a<xappleb f(x) if X discrete P (a <Xapple b) = f(x)dx if X continuous R b a The cdf is defined as P F (a) =P (X apple a) = R xapplea a 1 f(x) if X discrete f(x)dx if X continuous Examples Thus P (a <Xapple b) =F (b) F (a) and P (X = a) =F (a) F (a ). Example 1 A sample consists of 5 numbers. 10, 11, 5, 12. What is the other number? We know the mean is 9 and also the four numbers are Example 2 (i) Find the median of 1, 1, 3, 4, 5, 6, 3. (ii) Find the median of 1, 1, 3, 3, 4, 5, 6, 7. (iii) Find the mode of 1.0, 1.1, 1.2, 1.2, 1.2, 1.4, 1.4,

4 Example 3 Compute the first quartile and the third quartile of the following sample: 390, 406, 446, 420, 370, 328, 410, 320, 368, 392, 280, 325, 382, 290, Example 4 The mean and median salary of ten people is $50,000 and $48,000 respectively. Assume that the highest paid person is the only one who will get a raise of $5,000 next year. What is the new mean and median salary for the next year (for this ten people)? Example 5 The following data give the amounts (in dollars) spent on fast-food meals by a young couple during forty weeks The Frequency Table is Class Frequency Relative Frequency Percentage Frequency Sum Example 6 Find the stem-and-leaf plot of the following data Example 7 Let ={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Let A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8, 10} and C = {1, 3, 5, 7}. Find(A \ B) [ (A \ C c ). Example 8 A pair of dice are rolled once. Find the sample space. Example 9 How many points are there in a sample space when two dice are thrown once. Example 10 An experiment consists of flipping a coin and then flip it a second time if a hear occurs. If a tail occurs, then a die is tossed. Find the sample space. Example 11 S = {3, 4} and S 1 = x x 2 7x + 12 = 0 are the same set. Example 12 A = {1, 2, 4, 7}, B{1, 2, 3, 6}, C = {1, 3, 4, 5} Then AB = {1, 2, }, BC = {1, 3}, ABC = {1}. A \ C = {1, 2, 3, 4, 5, 7}, B 0 A = {4, 7}. (A \ B)C 0 = {2, 6, 7}. Example 13 If S = {x 0 <x<12}, M = {x 1 <x<10} and N = {x, 1/2 <x<6}, find M \ N; (b) M [ N; (c) M 0 \ N 0. (a) Example 14 How many even four-digit numbers can be formed from the digits 0, 1, 2, 5, 6 and 9 if each digit can be used only once? 4

5 Example 15 A president and a treasurer are to be selected from a student club consisting of 50 people. How many di erent choices of o cers are possible if (a) (b) (c) (d) there are no restriction; A all serve only if he is president; B and C will serve together or not at all; D and E will not serve together Example 16 In a college football training session, the defensive coordinator needs to have 10 players standing in a row. Among these 10 players, there are 1 freshman, 2 sophomores, 4 juniors, and 3 seniors. How many di erent ways can they be arranged in a row if only their class level will be distinguished? Example 17 How many di erent letter arrangements can be made from the letters in the word STATIS- TICS? Example 18 Two balls are randomly selected from a bowl containing 6 white balls and 5 black balls. What is the probability that one of the drawn balls is white and the other black? Example 19 In a family of three, it is know that there is at least one girl. What is the probability that this family has at least one boy? Example 20 A basketball team consists of 6 black and 6 white players. The players are to be paired in groups of two for the purpose of determining roommates. if the pairings are done at random, what is the probability that none of the black players will have a white roommate? Example 21 A box contains 5 pairs of shoes.if 4 shoes are randomly selected, what is the probability that there is exactly two complete pair? Example 22 A box contains 5 pairs of shoes.if 4 shoes are randomly selected, what is the probability that there is exactly one complete pair? Example 23 Acommitteeofsize5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women? Example 24 In the game of bridge, each of the four players gets 13 cards. If North and South have 8 spades between them, what is the probability of East has 3 spades and West has 2? Example 25 Given P (A) =1/3 and P (B) =1/4, P (A \ B) =1/6. Find P (A ), P(A [ B), P(A [ B ), P(A [ B ), P(A \ B ) Example 26 Let = {x : set function. 1 <x<1}. Findaconstant such that P (C) = R C e x dx is a probability Example 27 Mr. Perez figures that there is a 30% chance her company will set up branch o ce in Phoenix. If it does, she is 60% certain that she will be made manager of this new operation. What is the probability that Perez will be a Phoenix branch o ce manager? Example 28 One bag contains 4 white balls and 3 black balls, and a second bag contain 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the prob that a ball now drawn from the second bag is back? 5

6 Example 29 A system consists of two components C 1 and C 2. Let the probability that C 1 works without failure be 0.9 and the probability that C 2 works without failure be 0.8. Find the probability that the system works without failure when (a) C 1 and C 2 are connected in series (b) C 1 and C 2 are connected in a parallel way. Example 30 Let F (x) = (a) Plot this distribution function. (b) Compute P ( 3 <Xapple 1/2) and P (X = 0). ( 0 x<0 x apple x<1 1 1 apple x Example 31 Suppose that X is a continuous random variable whose probability density function is given by f(x) = C(4x 2x 2 ) if 0 <x<2 0 otherwise (a) What is C? (b) Find P (X >1). Example 32 Let the pdf of a random variable X have the following form (a) Find c. (b) Compute P (1 apple X apple 5). (c) Compute P (X 2 > 3). f(x) =ce 2 x. 6