Test 1 Review. Review. Cathy Poliak, Ph.D. Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1)
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1 Test 1 Review Review Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Exam 1 Review Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 1 / 28
2 Outline 1 Test 1 Reveiw 2 Chapter 1 3 Chapter 2 4 Chapter 3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 2 / 28
3 Test 1 Scheduled February & 19 Covers chapters 1, 2, and 3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 3 / 28
4 What to Expect on the Exam The test has three parts 1. 24% of the grade is based on true/false questions. Three questions % of the grade is based on multiple choice questions. Four questions % of the grade is based on free response questions. Three free response questions with multiple parts. Each T/F and mutliple choice question is worth 8 points, and one free response is worth 16 points, the other two are worth 14 points each. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 4 / 28
5 Examples of True/False Questions 1. A sample is the set of all possible data values for a given subject under consideration. a. True b. False 2. X is the number of days it rained last month where you lived. X is an example of a discrete random variable. a. True b. False 3. X is the amount of rainfall in your state last month. X is an example of a continuous random variable. a. True b. False Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 5 / 28
6 Possible Free Response Questions From a list of observations find the mean, median, quartiles, variance and standard deviation. From a discrete probability distribution determine the mean, variance and standard deviation. Also, know the rules of means and variances. From a two-way table determine the probability of the events. Using the probability general rules to find probability. Given sets determine the unions and intersections for each case described. Finding probability of a binomial random variable. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 6 / 28
7 Example 1 Among 13 electrical components exactly 3 are known not to function properly. If 7 components are randomly selected, find the following probabilities: a) The probability that all selected components function properly. b) The probability that exactly 2 are defective. c) The probability that at least 1 component is defective. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 7 / 28
8 Example 2 Let A = {2, 7}, B = {7, 16, 22}, D = {34} and S = sample space = A B D. a) Identify (A c B c ) c. b) Identify A c B. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 8 / 28
9 Example 3 a) A researcher randomly selects 4 fish from among 8 fish in a tank and puts each of the 4 selected fish into different containers. How many ways can this be done? b) A person eating at a cafeteria must choose 4 of the 16 vegetables on offer. Calculate the number of elements in the sample space for this experiment. c) How many license plates can be made using 2 digits and 4 letters if repeated digits and letters are allowed? Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1) Review 9 / 28
10 Example 4 Suppose P(E) = 0.74, P(F) = 0.33, and P(E F) = Find each of the following: a) P(E F) d) P(E F) c b) P(E F c ) e) P(E F) c) P(E c F) f) Are events E and F independent? Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 10 / 28
11 Example 5 The following is a probability distribution: a) Find P(X = 4). X P(X) ? b) Find P(1 < X 3). Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 11 / 28
12 c) Find the mean of X. d) Find the variance of X. e) Find the standard deviation of X. f) Define a new random variable Y = 2X - 1. Determine the mean and standard deviation of Y. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 12 / 28
13 Example 6 According to government data, 20% of employed women have never been married. If 10 employed women are selected at random, what is the probability a) That exactly 2 have never been married? b) That at most 2 have never been married? c) That at least 8 have been married? d) What is the expected number of employed women that have never been married out of the 10? Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 13 / 28
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15 Example 7 The probability that a randomly selected person has high blood pressure (the event H) is P(H) = 0.4 and the probability that a randomly selected person is a runner (the event R) is P(R) = 0.4. The probability that a randomly selected person has high blood pressure and is a runner is 0.1. Find the probability that a randomly selected person either has high blood pressure or is a runner or both. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 14 / 28
16 What You Need an What is Provided Provided Formula sheet; see CASA calendar for the formula sheet provided. Online calculator; it will be a link you see in the exam. R studio; it will be a link you see in the exam. Can bring Calculator; if it is memory based CASA will remove the memory. Pencil; you will need something to write with for the free response questions. Your Cougar Card. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 15 / 28
17 Chapter 1 Section 1 Sample - Simple random sample Population versus sample Parameter versus statistic. Categorical variables Quantitative variables; continuous and discrete Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 16 / 28
18 Chapter 1 sections 2, 3 and 4 Describing distributions with numbers Center - mean, median, mode Spread - range, interquartile range (IQR), standard deviation Location - percentiles, quartiles (Q 1 and Q 3 ), z-scores, 1.5 IQR The five number summary Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 17 / 28
19 Chapter 1 section 5 Describing distributions with graphs Categorical variables - bar chart, pie chart Quantitative variables - histogram, stemplot, boxplot, dotplot Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 18 / 28
20 Chapter 2 section 1: Counting Techniques If an experiment can be described as a sequence of k steps with n 1 possible outcomes on the first step, n 2 possible outcomes on the second step, and so on, then the total number of experimental outcomes is given by (n 1 )(n 2 )... (n k ). Permutations allows one to compute the number of outcomes when r objects are to be selected from a set of n objects where the order of selection is important. The number of permutations is given by Pr n n! = (n r)! Combinations counts the number of experimental outcomes when the experiment involves selecting r objects from a (usually larger) set of n objects. The number of combinations of n objects taken r unordered at a time is ( ) n Cr n = = r n! r!(n r)! Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 19 / 28
21 Chapter 2 section 2: Sets and Venn Diagrams Notation Description a A The object a is an element of the set A. A B Set A is a subset of set B. That is every element in A is also in B. A B Set A is a proper subset of set B. That is every element that is is in A is also in set B and there is at least one element in set B that is no in set A. A B A set of all elements that are in A or B. A B A set of all elements that are in A and B. U Called the universal set, all elements we are interested in. A C The set of all elements that are in the universal set but are not in set A. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 20 / 28
22 Chapter 2 section 3: Basic Probability Models For any event A, the probability of A is P(A) = number of times A occurs total number of outcomes. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 21 / 28
23 General Rules of Probability 1. The probability P(A) of any event A satisfies 0 P(A) If S is the sample space in a probability model, then P(S) = Complement rule: For any event A, P(A C ) = 1 P(A) 4. General rule for addition: For any two events A and B P(A B) = P(A) + P(B) P(A B) 5. General rule for multiplication: For any two events A and B P(A B) = P(A) P(B, given A) or P(A B) = P(B) P(A, given B) Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 22 / 28
24 Chapter 2 section 4: General Probability Models Two events are disjoint if the occurrence of one prevents the other from happening. If two events A and B are disjoint then P(A and B) = P(A B) = 0. Two events are independent if the occurrence of one does not change the probability of the other. If two events A and B are independent then P(A and B) = P(A B) = P(A) P(B). Conditional Probability: For any two events A and B, the probability of A given B is P(A given B) = P(A B) = P(A B) P(B) Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 23 / 28
25 Chapter 3 section 1: Random Variables Discrete random variables has either a finite number of values or a countable number of values, where countable refers to the fact that there might be infinitely many values, but they result from a counting process. Continuous random variables are random variables that can assume values corresponding to any of the points contained in one or more intervals. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 24 / 28
26 Discrete Random Variable Probability Distribution Suppose that X is a discrete random variable whose distribution is Values of X x 1 x 2 x 3 x k Probability p 1 p 2 p 3 p k To find the mean of the random variable X, multiply each possible value by its probability, then add all the products: µ X = x 1 p 1 + x 2 p 2 + x 3 p x k p k k = x i p i. i=1 The variance of a discrete random variable X is σx 2 = (x 1 µ X ) 2 p 1 + (x 2 µ X ) 2 p (x k µ X ) 2 p k k = (x i µ X ) 2 p i i=1 The standard deviation of X is the square root of the variance σ = σ 2. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 25 / 28
27 Rules for Means and Variances If X is a random variable and a and b be are fixed numbers, such that we add a to X and multiply b to X, a + bx, then The mean of a + bx changes by what is added and multiplied: µ (a+bx) = a + bµ X The variance of a + bx changes by the square of the multiplied value: σ 2 (a+bx) = b2 σ 2 X The standard deviation of a + bx is the square root of the variance: σ (a+bx) = b 2 σx 2 = bσ X Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 26 / 28
28 Chapter 3 section 2: Binomial Distribution The distribution of the count X of successes in the Binomial setting has a Binomial probability distribution. Where the parameters for a binomial probability distribution is: n the number of observations p is the probability of a success on any one observation The possible values of X are the whole numbers from 0 to n. The probability of selecting k successes out of n observations uses the formula: P(X = k) = n C k (p) k (1 p) n k Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c (Department Reveiw of Mathematics University of Houston Exam 1 ) Review 27 / 28
29 Popper Set Up Fill in all of the proper bubbles. Make sure your ID number is correct. Make sure the filled in circles are very dark. This is popper number 04. Cathy Poliak, Ph.D. Office in Fleming 11c Section (Department 4.1 of Mathematics University of Houston Lecture ) / 29
30 Popper 04 Questions Newsweek in 1989 reported that 60% of young children have blood lead levels that could impair their neurological development. Assuming a random sample from the population of all school children at risk. Determine the type of distribution we have. 1. The probability that at least 5 children out of 10 in a sample taken from a school may have a blood lead level that may impair development. a) Geometric b) Binomial c) None of these 2. The probability you will need to test 10 children before finding a child with a blood lead level that may impair development. a) Geometric b) Binomial c) None of these 3. The probability you will need to test no more than 10 children before finding a child with a blood lead level that may impair development. a) Geometric b) Binomial c) None of these Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 4.1 of Mathematics University of Houston Lecture ) / 29
31 Popper 04 Questions For each random variable, determine if it is: a. Discrete b. Continuous 4. The number of cars passing a busy intersection between 4:30 PM and 6:30 PM. 5. The weight of a fire fighter. 6. The amount of soda in a can of Pepsi. Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Section (Department 4.1 of Mathematics University of Houston Lecture ) / 29
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