MATH Statistics & Probability Performance Objective Task Analysis Benchmarks/Assessment Students:

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1 1. Demonstrate understanding and give If one card drawn from an ordinary deck of 52 cards, what is the union of sets probability that it will be either a club intersection of sets or a face card (king, queen, or jack)? independent events dependent events mutually exclusive Venn diagrams complements sample space 1. Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. An artist who has entered a large oil painting and a small painting in a show feels that the probabilities are, respectively, 0.15, 0.18, and 0.11 that she will sell the large oil painting, the small one, or both. What is the probability that she will sell a) either or both of the two paintings b) neither of the two paintings Given P(K) = 0.45, P(L) = 0.27, and P(KnL) = 0.13, draw a Venn diagram for each, shade in the areas associated with the various regions, and determine the probabilities of: a) P(KnL')= b) P(K'UL)= c) P(K'nL)= d) P(K'nL')= e) P(KUL)= f) P(K'UL')= 1

2 A warning system installation consists of two independent alarms having probabilities of operating in an emergency of 0.95 and 0.90 respectively. Find the probability that at least one alarm operates in an emergency. (TIMSS, adapted) Arlene and her friend want to buy tickets to an upcoming concert, but the tickets are difficult to obtain. Each outlet will have its own lottery, so that everyone who is in line at a particular outlet to buy tickets when they go on sale has an equal chance of purchasing tickets. Arlene goes to a ticket outlet where she estimates that her chance of being able to buy tickets is 1/2. Her friend goes to another outlet, where Arlene thinks that her chance of being able to buy tickets is 1/3. What is the probability that both Arlene and her friend are able to buy tickets? What is the probability that at least one of the two friends is able to buy tickets? (CERT HS Standards) 2

3 1. Demonstrate understanding and give conditional probability sample space A given B 2. Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces. For burglaries in a certain city, police records show that the probability is 0.35 that an arrest will be made. The probability that an arrest and conviction will occur is What is the probability that a person arrested for burglary will be convicted? The probability that there will be a shortage of cement is 0.28, and the probability that there will not be a shortage of cement and a construction job will be finished on time is What is the probability that the construction job will be finished on time given that there will not be a shortage of cement? A whole number between 1 and 30 is chosen at random. If the digits of the number that is chosen add up to 8, what is the probability that the number is greater than 12? 3

4 1. Demonstrate understanding and give Radios are packed in cartons of 12. A carton of radios is inspected and the random variable number of defective radios found is discrete random variable recorded. Identify the random variable continuous random variable used and list its possible values. probability distribution 3. Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses. Determine whether the following meets the condition of a probability distribution of a random variable. Explain why or why not? f(1) =.025, f(2) = 0.35, f(3) = 0.35, f(4) = 0.10 A company manufactures insulin needles and packages them in boxes of 100. Based on historical data from sampling, it is known that 90% of the boxes contain no defective needles, 7% contain exactly one defective needle, and 3% contain exactly two defective needles. Based on this information, what is the probability distribution for x, where x represents the number of defective needles per box. A random variable X has the following distribution: x P(X=x) Find: P(X>1) P(X 2 <2) 4

5 1. Demonstrate understanding and give Four cards are selected, one at a time, from a standard deck of 52 cards. Let binomial random variable x represent the number of aces drawn binomial expansion in the set of 4 cards. * factorials a) If this experiment is completed * combinations without replacement, explain why Pascal's Triangle x is not a binomial random binomial distribution variable. * at most b) If this experiment is completed * at least with replacement, explain why x is * not more than a binomial random variable. * less than * more than other kinds of distributions * normal * exponential 4. Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families. If the probability is 0.40 that a divorcee will remarry within three years, find the probabilities that of ten divorcees: a) at most three will remarry within 3 years b) at least seven will remarry within 3 years c) from 2 to 5 will remarry within 3 years A basketball player has a history of making 80% of the foul shots taken during games. What is the probability that he will miss three of the next five shots he takes? 5

6 You are playing a game in which the probability that you'll win is 1/3, the probability that you'll lose or play to a tie is 2/3. If you play this game 8 times, what is the probability that you'll win exactly 3 times? 5. Students determine the mean and the standard deviation of a normally distributed random variable. 1. Demonstrate understanding and give mean variance standard deviation formulas Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1 that 0, 1, 2, or 3 hurricanes will hit a certain coast area in any given year. Find the mean, variance, and standard deviation. 75% of the foreign-made autos sold in the United States in 1994 are now falling apart. Determine the probability distribution of x, the number of autos that are falling apart in a random sample of 5 cars. Draw a histogram of the distribution. Calculate the mean and the standard deviation of this distribution. Suppose that x is a normally distributed random variable with mean µ. Find P(X<µ). 6

7 1. Demonstrate understanding and give Recruits for a police academy were required to undergo a test that mean (average) measures their exercise capacity in median (middle) minutes. Find the mean, median, * depth mode, and midrange. midrange 25, 27, 30, 33, 30, 32, 30, 34, 30, 27 percentile 26, 25, 29, 31, 31, 32, 34, 32, 33, 30 quartile inter quartile range mode (most often) range frequency distribution 6. Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations. The following are the numbers of restaurant meals that 13 persons ate during a given week. 3, 10, 5, 1, 8, 5, 6, 12, 15, 1, 0, 6, 5 Find the mean, median, mode, and midrange. Find the mode and median for the following seven numbers: Students compute the variance and the standard deviation of a distribution of data. 1. Demonstrate understanding and give measures of dispersion (spread) range deviation from the mean * absolute value variance summation sum of squares Comment of the following statement: "The mean loss for customers at First State Bank (which was not insured) was $150. The standard deviation of the losses was $125." 7

8 Consider the following two sets of data: Set 1: 46, 55, 50, 47, 52 Set 2: 30, 55, 65, 47, 53 Both sets have the same mean, which is 50. Compare these measures for both sets: Sum of the variances Sum of the squares Range Comment on the meaning of these comparisons. Fifteen randomly selected high school seniors were asked to state the number of hours they slept last night. The resulting data are: 5, 6, 6, 8, 7, 7, 9, 5, 4, 8, 11, 6, 7, 8, 7. Find the variance and the standard deviation. 8 Find the mean and standard deviation of the following seven numbers: Make up another list if seven numbers with the same mean and a smaller standard deviation. Make up another list if seven numbers with the same mean and a larger standard deviation. (ICAS)

9 1. Demonstrate understanding and give frequency tables histograms line and bar graphs * intervals * class: limits, marks, intervals stem and leaf box and whisker plots cumulative distributions (ogive) 8. Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and whisker plots. The following are the grades that 50 students obtained on a statistics test: 73, 65, 82, 70, 45, 50, 70, 54, 32, 75, 75, 67, 65, 60, 75, 87, 83, 40, 72, 64, 58, 75, 89, 70, 73, 55, 61, 78, 89, 93, 43, 51, 59, 38, 65, 71, 75, 85, 65, 85, 49, 97, 55, 60, 76, 75, 69, 35, 45, 63. Prepare a stem-and-leaf display of these values. Group the grades into classes and convert into a cumulative "less than" distribution. Make a histogram of the distribution. Make a box and whisker plot. Scientists have observed that crickets move their wings faster in warm temperatures than in cold temperatures. By noting the pitch of cricket chirps, it is possible to estimate the air temperature. Below is a graph showing 13 observations of cricket chirps per second and the associated air temperature. (graph goes here) On the graph, draw in an estimated line of best fit for these data. Using your line, estimate the air temperature when cricket chirps of 22 per second are heard. (TIMSS) 9

10 A fundraising group sells 1,000 raffle tickets at $5 each. The first prize is an $1,800 computer. Second prize is a $500 camera and the third prize is $300 cash. What is the expected value of a raffle ticket? (ICAS) Carla has made an investment of $100. She understands that there is a 50% chance that after a year her investment will have grown to exactly $150. And there is a 20% chance that she'll double her money in that year, but there is also a 30% chance that she'll lose the entire investment. What is the expected value of her investment after a year? (CERT HS Standards0 10

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