Math-138 Unit 2 Packet Fall 2024 (Canvas) PDF

Name _________________________________ Instructor Name/Section _____________________ MATH-138 STATISTICS Unit 2 Classroom Activities and Examples Revised AY24-25 Howard Community College Allison Bell Lamont Vaughan...

Name _________________________________ Instructor Name/Section _____________________ MATH-138 STATISTICS Unit 2 Classroom Activities and Examples Revised AY24-25 Howard Community College Allison Bell Lamont Vaughan 1 Unit 2 Key Concepts Basic Probability Probability Rules & Definitions Contingency Tables & Venn Diagrams Probability Trees Random Probability Models Binomial Distributions The Normal Curve Central Limit Theorem for Means Central Limit Theorem for Proportions Chapter 5 Probability Rules and Definitions The ____________________ of an event is the proportion of times that the event occurs in the long run. The Law of Large Numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability. The collection of all the possible outcomes of a probability experiment is called a ____________________. If A denotes an event, the probability of event A is denoted by _____. The probability of an event is always between _____ and _____. In other words, for any event A, ____ ≤ 𝑃(𝐴) ≤ _____. If A cannot occur, then _______________. If A is certain to occur, then _______________. If a sample space has n equally likely outcomes and an event A has k outcomes, then ⬚ 𝑃(𝐴) = = ⬚ Rule of Thumb for Unusual Events: If 𝑃(𝐴) < 0.05 , then event A is considered unusual. General Addition Rule: 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵) Two events are said to be mutually exclusive if it is ____________________ for both events to occur. The ___________________ of an event says that the event does not occur and is denoted by 𝐴! The Rule of Complements: 𝑷(𝑨𝑪 ) = 𝟏 − 𝑷(𝑨) 2 The conditional probability of an event 𝐵 given an event 𝐴 is denoted _______________. 𝑃(𝐵|𝐴) is the probability that 𝐵 occurs, under the assumption that 𝐴 occurs. The conditional probability is computed as: 𝑃(𝐵|𝐴) = At Least One Probabilities: 𝑃( 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 1 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠) = 1 − 𝑃(𝑁𝑜 𝑒𝑣𝑒𝑛𝑡𝑠 𝑜𝑐𝑐𝑢𝑟) General Multiplication Rule: 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴|𝐵) ∙ 𝑃(𝐵) Two events are ______________________________ if the occurrence of one does not affect the probability that the other event occurs. If two events are not independent, we say they are dependent. When sampling with replacement, the draws are ______________________________. When sampling without replacement, the draws are ______________________________. Example 1 Consider the following table which presents the number of U.S. men and women (in millions) 25 years old and older who have attained various levels of education. Not a high High school Some Associate’s Bachelor’s Advanced school graduate college, no degree degree degree graduate degree Men 14.0 29.6 15.6 7.2 17.5 10.1 Women 13.7 31.9 17.5 9.6 19.2 9.1 A person is selected at random. a. What is the probability that the person is a man? b. What is the probability that the person is a man with a Bachelor’s degree? c. What is the probability that the person has a Bachelor’s degree, given that he is a man? d. What is the probability that the person is a woman who is a high school graduate? e. What is the probability that the person is a high school graduate? f. What is the probability that the person is a woman, given that the person is a high school graduate? 3 Example 2 In a certain city, 70% of high school students graduate. Of those who graduate, 40% attend college. Find the probability that a randomly selected high school student will attend college. Example 3 Items are inspected for flaws by three inspectors. If a flaw is present, each inspector will detect it with probability 0.8. The inspectors work independently. If an item has a flaw, what is the probability that at least one inspector detects it? Example 4 In its monthly report, the animal shelter states that it currently has 24 dogs and 18 cats available for adoption. 8 of the dogs are male and 6 of the cats are male. Suppose an animal from this shelter is randomly chosen. Find the following probabilities. a. P(Male) e. If you choose to randomly adopt two pets, find the following probabilities: i. P(Both are cats) b. P(Male | Cat) ii. P(Exactly one is a cat) c. P(Cat | Male) iii. P(At least one is a cat) d. Are the events of selecting a male and selecting a cat independent? Example 5 Suppose there is a box with seven red marbles, 3 green marbles, and 6 black marbles. We draw one marble, put it back, and then draw another (i.e., with replacement). a. Are drawing marbles with replacement independent events? Explain. 4  b. Find the probability of drawing two black marbles in a row. c. Find the probability of drawing one green marble and then one red marble. d. Find the probability of drawing one red marble and then one marble that is not red. Example 6 Suppose we have the same box of marbles as in the problem above, but this time we do not replace each marble after we draw the first (i.e., without replacement). a. Find the probability of drawing one red marble and then one marble that is not red. b. Find the probability of drawing two black marbles in a row. Example 7 A junk box in your room contains a dozen batteries, five of which are totally dead. You start picking batteries one at a time and test them. a. Will the trials be independent? b. The first two you choose are both good. c. The first 4 you pick are all good. d. At least one of the first 3 works. e. You have to pick 5 batteries before you get a good one (the 5th one is good). Example 8 A regular deck of card has 52 cards. Assuming that you do not replace the card you had drawn before the next draw, what is the probability of drawing three aces in a row? 5  Contingency Tables A ______________________________ is a table that displays two qualitative variables and their relationships. Example 1 A survey of automobiles parked in student and staff lots at a large university classified the brands by country of origin, as seen in the table. Driver Origin Student Staff Total American 107 105 European 33 12 Asian 55 47 Total a. What percent of cars surveyed are foreign? Write as a fraction first and then find the percent. b. What percent of American cars are owned by students? Write as a fraction first and then find the percent. c. What percent of the students own American cars? Write as a fraction first and then find the percent. d. What percent of cars were of Asian origin and driven by students? Write as a fraction first and then find the percent. e. What percent of student driven cars are American? Write as a fraction first and then find the percent. 6 Example 2 A study was undertaken at a certain college to determine what relationship, if any, exists between mathematics ability and interest in mathematics. The ability and interest for 150 students were determined, with the results in the following table. INTEREST ABILITY Low Average High Below grade level 40 8 11 Grade level 15 17 19 Above grade level 5 10 25 If one of the participants in the study is randomly chosen, find the following probabilities: a. P(the person has low interest in mathematics) b. P(the person is at grade level ability) c. P(the person has low interest and has grade level ability) d. P(the person has high interest or is below grade level ability) e. P(the person has a high interest given the person has grade level ability) f. Are interest and ability independent? 7 Venn Diagrams A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Often, they serve to graphically organize things, highlighting how the items are similar and different. They are very useful in probability. Example 1 Records show that 80% of the students preparing for the high school equivalency exam need help in math, 70% need help in English, and 55% need work in both. If a student is randomly chosen, find the probability that the person needs help in math or English. a. Use the General Addition Rule to find the probability that a randomly chosen student needs help in math or English. Venn Diagram b. Create a Venn Diagram of the problem. c. Find P(Math but not English). d. Use the diagram to find P(Math or English). e. Find P(neither Math nor English). Example 2 In a large doll store, 47% of the dolls blink, 56% have moveable legs and arms, and 26% blink and have movable arms. a. What is the probability that a doll selected at random only has movable legs and arms? b. What is the probability that a doll selected at random blinks given it has movable legs and arms? 8 Example 3 1) Out of 62 people surveyed, 22 of them own a laptop, 39 own a desktop and 8 own both a laptop and a desktop. Complete the diagram and answer the questions using your picture. a. What is the probability that a person selected at random from those surveyed does not own a computer? b. What is the probability that a person selected at random from those surveyed owns only a desktop computer? c. What is the probability that a person selected at random from those surveyed owns a laptop or a desktop? Tree Diagrams Tree diagrams are often used to solve probability problems when provided with conditional probabilities. Example 1 In a small biology classroom, the probability a student is interested in dissecting frogs is 0.75. Given a student is interested in dissection, the probability they attend the extra credit event is 0.85. Given a student is not interested in dissection, the probability they attend the extra credit event is 0.6. Create a tree diagram of the situation, then answer the follow-up questions. a. What is the probability a student selected at random is interested in dissection and did not attend the extra credit event? 9  b. What is the probability a student selected at random did not attend the extra credit event? c. What is the probability a student selected at random is interested in dissection, given they did not attend the extra credit event? Example 2 An airline offers discounted “advance-purchase” fares to customers who buy tickets more than 30 days before travel and charges “regular” fares for tickets purchased during those last 30 days. The company has noticed that 60% of its customers take advantage of the advance-purchase fares. The “no-show” rate among people who paid regular fares is 30%, but only 5% of customers with advance-purchase tickets are no shows. a. Represent the situation above using a tree diagram. b. What is the probability that a customer has an advance-purchase ticket and did not show for the flight? c. What percent of all ticket holders are no-shows? d. What’s the probability that a customer with an advance-purchase ticket did not show? e. What’s the probability that a customer who didn’t show had an advance-purchase ticket? (Reverse the conditioning) 10 Example 3 In a certain population of caribou, the probability of a caribou being sickly is 0.4. If a caribou is sickly, the probability wolves eat it is 0.3. If a caribou is not sickly, the probability wolves eat it is 0.2. If a caribou is chosen at random from the herd, answer the following questions. a. Create a tree diagram before answering the questions below. b. What is the probability of the caribou being sickly and not being eaten by wolves? c. What is the probability of the caribou not being eaten by wolves? d. If a caribou is sickly, what is the probability that it will be eaten by wolves? e. If wolves ate a caribou, what is the probability that it was sickly? 11  Chapter 6 Discrete random variables are random variables whose possible values can be listed. Continuous random variables are random variables that can take on any value in an interval. A ______________________________ for a discrete random variable specifies the probability for each possible value of the random variable. Example 1 Following is the probability distribution of a random variable that represents the number of extracurricular activities a college freshman participates in. 𝒙 0 1 2 3 4 𝑷(𝒙) 0.06 0.14 0.45 0.21 0.14 a. Find the probability that a student participates in exactly two activities. b. Find the probability that a student participates in more than two activities. c. Find the probability that a student participates in at least one activity. Example 2 There are 5000 undergraduates registered at a certain college. Of them, 478 are taking one course, 645 are taking two courses, 568 are taking three courses, 1864 are taking four courses, 1357 are taking five courses, and 88 are taking six courses. Let X be the number of courses taken by a student randomly sampled from this population. Find the probability distribution of X. The__________________________________________________________ provides a measure of center for the probability distribution of a random variable. Another name for the mean of a random variable is the ______________________________. Expected Value Formula : 𝑬(𝒙) = ∑ 𝒙 ∙ 𝑷(𝒙) 12 Example 1 $ If you bet $5 on the number 17 in a game of roulette, your probability of winning is P(17)= %&. If the player wins, they get $30. Find the expected value of this game. Outcome X P(X) Example 2 An insurance company sells a one-year term life insurance policy to a 70-year-old man. The man pays a premium of $400. If he dies within one year, the company will pay $10,000 to his beneficiary. The probability that a 70-year-old man is still alive one year later is 0.9715. Let X be the profit made by the insurance company. Find the expected value of the profit. 13  MEAN/STANDARD DEVIATION OF A RANDOM VARIABLE ON THE TI-84 PLUS Step 1: Enter the values of the random variable into L1 and the associated probabilities in L2. Step 2: Press STAT and highlight the CALC menu and select 1-Var Stats with L1 and L2 as the arguments. Example 3 Suppose that out of 120 students, 10 complete 10th grade (10 years of school), 50 complete high school (12 years of school), 45 complete college (16 years of school), and the rest complete a Ph.D. (20 years of school). a) Complete the following probability model for the number of years of schooling students complete. b) Give at least 2 reasons why you know this is a valid probability distribution. c) Find the expected number of years of schooling for this group of students. d) Find the standard deviation for the expected number of years of schooling for this group of students. 14 The Binomial Distribution A random variable that represents the number of successes in a series of trials has a probability distribution called the binomial distribution. The conditions for the Binomial Distribution: A fixed number of trials are conducted. There are two possible outcomes for each trial. One is labeled “success” and the other is labeled “failure.” The probability of success is the same on each trial. The trials are independent. This means that the outcome of one trial does not affect the outcomes of the other trials. The random variable X represents the number of successes that occur Example 1 Decide in each case whether X is a binomial random variable. a. A coin is tossed 10 times. Let X be the number of times the coin lands heads. b. Five basketball players each attempt a free throw. Let X be the number of free throws made. c. Ten cards are in a box. Five are red and five are green. Three of the cards are drawn at random. Let X be the number of red cards drawn. CALCULATOR PROCEDURES BINOMIAL PROBABILITIES ON THE TI-84 PLUS In the TI-84 PLUS Calculator, there are two primary commands for computing binomial probabilities. These are binompdf and binomcdf. These commands are on the DISTR (distributions) menu accessed by pressing 2nd, VARS. binompdf(n,p,x): – Computes the probability of getting exactly x successes in n trials, if the probability of success is p. binomcdf(n,p,x): – Computes the probability of getting x or fewer successes in n trials, if the probability of success is p. 15 Let X be a binomial random variable with n trials and success probability p. The mean, variance, and standard deviation of X are: Mean: Variance: Standard Deviation: Example 1 The Pew Research Center recently reported that approximately 30% of Internet users in the United States use the image sharing website Pinterest. Suppose a simple random sample of 15 Internet users is taken. Use your calculator to find the following probabilities. a. Find the probability that exactly four of the sampled people use Pinterest. b. Find the probability that fewer than three of the people use Pinterest. c. Find the probability that more than one person uses Pinterest. d. Find the probability that the number of people who use Pinterest is between 1 and 4, inclusive. Example 2 Suppose that 20% of all college students smoke cigarettes. If 15 are selected at random: a) What is the probability that 10 smoke? b) What is the probability that at most 12 of the students smoke? c) Find the mean and standard deviation of the number of college students who smoke. 16 Example 3 A student takes an exam consisting of 12 true or false questions. At least 8 correct answers are required to pass. If the student guesses, what is the probability that he will pass? Example 4 A quality control inspector has drawn a sample of 13 light bulbs from a recent production lot. If the number of defective bulbs is 2 or less, the lot passes inspection. Suppose 10% of the bulbs in the lot are defective. What is the probability that the lot will pass inspection? Example 5 A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find the probability that: a. Exactly 3 fail b. Fewer than 2 fail c. More than 3 fail d. Between 2 and 5 fail (inclusive) e. Find the mean and standard deviation of the number of failing alarm components. 17  Chapter 7 The Standard Normal Curve A probability density curve tells what proportion of the data falls within a given interval. For any probability density curve, the area under the entire curve is 1. Normal Curves (or distributions) have one mode, and the distributions are symmetric around the mode. The normal distribution follows The Empirical Rule. CALCULATOR PROCEDURES Finding Areas with the TI-84 plus On the TI-84 PLUS calculator, the normalcdf command is used to find areas under a normal curve. Four numbers must be used as the input. The first entry is the lower bound of the area. The second entry is the upper bound of the area. The last two entries are the mean and standard deviation. This command is accessed by pressing 2nd, Vars. Finding 𝒛-Scores or x-values Given an Area with the TI-84 plus The invNorm command on the TI-84 PLUS calculator returns the z -score with a given area to its left. This command takes three values as its input. The first value is the area to the left, the second and third values are the mean and standard deviation. This command is accessed by pressing 2nd, Vars. 18 Example 1 What percent of a standard Normal model is found in each of the following regions? a. -1 < z < 2 b. z > 2.1 c. z < -1.55 Example 2 Some IQ Tests are standardized to a Normal model with a mean of 100 and a standard deviation of 16. a. Draw the model. b. In what interval would you expect the central 95% of the scores? c. About what percent should have scores between 68 and 84? d. About what percent of the people should have scores above 116? e. About what percent of the people should have scores above 132? 19 Example 3 Suppose a Normal model describes the fuel efficiency of cars currently registered in Maryland. The mean is 24 mpg with a standard deviation of 6 mpg. a. What percent of all cars get less than 15 mpg? b. What percent of all cars get between 20 and 30 mpg? c. What percent of all cars get more than 40 mpg? d. Describe the fuel efficiency of the worst 20% of all the cars. e. What gas mileage represents the 3rd quartile? f. Describe the fuel efficiency of the most efficient 5% of the cars. Example 4 A study reported that the length of pregnancy from conception to birth is approximately normally distributed with mean 𝜇 = 272 days and standard deviation 𝜎 = 9 days. What proportion of pregnancies last longer than 280 days? Example 5 Ithaca, New York gets an average of 35.4” of rain each year with a standard deviation of 4.2”. Assume the Normal model applies to their yearly rainfall. a. What percentage of years did Ithaca get more than 40” of rainfall? b. Describe the driest 20% of all years in Ithaca. 20  Central Limit Theorem for Means If several samples are drawn from a population, a sample mean x can be calculated along with its probability. The probability distribution of those sample means x is called the _______________________________ ___________________________ of x. M MEAN AND STANDARD DEVIATION OF A SAMPLING DISTRIBUTION FOR 𝒙 The mean of the sampling distribution 𝜇!̅ =𝜇 # The standard deviation of the sampling distribution 𝜎!̅ = √% The Central Limit Theorem for Means Let 𝑥̅ be the mean of a large (𝑛 > 30) simple random sample from a population with mean 𝜇 and standard deviation 𝜎. Then 𝑥̅ has an approximately normal distribution, with mean 𝜇'̅ = 𝜇 and standard deviation 𝜎'̅ = ). √+ Conditions to Check: 𝑛 > 30 or an approximately normal population Example 1 Decide whether it is appropriate to use the normal distribution to find probabilities for x. a. A simple random sample of size 45 will be drawn from a population with mean 𝜇 = 15 and standard deviation 𝜇 = 3.5. b. A simple random sample of size 8 will be drawn from a normal population with mean 𝜇 = −60 and standard deviation 𝜇 = 5. c. A simple random sample of size 24 will be drawn from a normal population with mean 𝜇 = 35 and standard deviation 𝜇 = 1.2. 21 Example 2 College’s data about the incoming freshmen is normally distributed and indicates that the mean of their high school GPAs is 3.4 with a standard deviation of 0.35. The students are randomly assigned to freshmen writing seminars in groups of 25. a. Describe the sampling distribution for all samples of size 25. b. Find the probability a student has a GPA greater than 3.1. c. Find the probability that one of the groups has an average GPA greater than 3.1. Example 3 The weight of potato chips in a medium size bag is stated to be 10 ounces on the package. The amount the packaging machine puts in these bags is believed to follow a normal model with a mean of 10.2 ounces and a standard deviation of 0.12 ounces. a. What percentage of the bags are underweight (under 10 oz.)? b. Some of the chips are sold in “bargain packs” of 3 bags. Describe the sampling distribution of these bargain packs. c. What is the probability that the mean of the bargain packs (3 bags) is “overweight”? d. What is the probability that the mean weight of the bargain packs (3 bags) is below the stated amount? e. What’s the probability that the mean weight of a 24-bag case of potato chips is below 10 ounces? (Hint: find the sampling distribution for 24 bag cases first) 22  The Central Limit Theorem for Proportions x The________________________________________ is pˆ =. If several samples are drawn from a n population, they are likely to have different values for p̂. The probability distribution of p̂ is called the ______________________________________________________________. T MEAN AND STANDARD DEVIATION OF A SAMPLING DISTRIBUTION FOR 𝒑 The mean of the sampling distribution 𝜇!" = 𝑝 !($%!) The standard deviation of the sampling distribution 𝜎!" = % ' Central Limit Theorem for Proportions Let 𝑝̂ be the sample proportion for a sample of size 𝑛 from a population with population proportion 𝑝. If 𝑛𝑝 ≥ !($%!) 10 and 𝑛(1 − 𝑝) ≥ 10, then the distribution of 𝑝̂ is approximately normal with 𝜇!" = 𝑝 and 𝜎!" = %. ' Conditions to Check: 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10 Example 1 Decide whether it is appropriate to use the normal distribution to find probabilities for 𝑝̂. a. A simple random sample of size 20 is drawn from a population with population proportion 𝑝 = 0.7. b. A simple random sample of size 55 is drawn from a population with population proportion 𝑝 = 0.8 c. A simple random sample of size 1000 is drawn from a population with population proportion 𝑝 = 0.03. 23 Example 2 According to a Harris poll, chocolate is the favorite ice cream flavor for 27% of Americans. If a sample of 100 Americans is taken, what is the probability that the sample proportion of those who prefer chocolate is greater than 0.30? Example 3 According to a national survey, 71% of millennials are either not engaged or are actively disengaged in the workplace. If we were to randomly survey 100 millennials with jobs, what is the probability less than 65% of them would not be engaged at work? a. First, find the mean and standard deviation for the sampling distribution for the proportion of 100 millennials who are not engaged in the workplace. b. Using your sampling distribution from part a, what is the probability less than 65% of them would not be engaged at work? Example 4 According to a national survey, 62% of adults have sought care for neck or back pain. If we randomly survey 250 adults, what is the probability more than 70% of them have sought care for neck or back pain? a. First, find the mean and standard deviation for the sampling distribution for the proportion of 250 adults who sought care for neck and back pain. b. Using your sampling distribution from part a, what is the probability more than 70% of them have sought care for neck or back pain? 24