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Ch 6.1 to 6.2 Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition. Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution A1. Distinguish between Discrete and Continuous Random Variables Los Angeles Mission College Produced by DW Example 1: Determine whether the random variable is discrete or continuous. State the possible values of the random variable. (a) The number of fish caught during the fishing tournament. Discrete n 0, 1, 2, 3,... (b) The distance of a baseball travels in the air after being hit. Continuous Los Angeles Mission College d 0 Produced by DW A2. Discrete Probability Distributions Los Angeles Mission College Produced by DW Example 1: Determine whether the distribution is a discrete probability distribution. If not, state why. (a) x P(x) 0 0.34 1 0.21 2 0.13 3 0.04 4 0.01 P(x) 0.73 Not a discreet probability distribution because it does not meet P(x) 1 . Los Angeles Mission College Produced by DW (b) x P(x) 0 0. 40 1 0.31 2 0.23 3 0.04 4 0.02 P(x) 1 It is a discreet probability distribution because it meets P(x) 1 . Los Angeles Mission College Produced by DW Example 2 : (a) Determine the required value of the missing probability to make the distribution a discrete probability distribution. (b) Draw a probability histogram. Los Angeles Mission College x P(x) 0 0. 30 1 0.15 2 ? 3 0.20 4 0.15 5 0.05 Produced by DW (a) The required value of the missing probability P( x 0) P( x 1) P( x 2) P( x 3) P( x 4) P( x 5) 1 0.30 0.15 P( x 2) 0.20 0.15 0.05 1 P( x 2) 0.85 1 P( x 2) 1 0.85 0.15 (b) The probability histogram P(x) 0.50 0.40 0.30 0.20 0.10 x 0 Los Angeles Mission College 1 2 3 4 5 Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective B : Mean and Standard Deviation of a Discrete Random Variable Los Angeles Mission College Produced by DW Example 1: Find the mean, variance, and standard deviation of the discrete random variable. (a) Mean x [ x P( x)] (1) x P(x) x P(x) 0 0.073 1 0.117 0 (0.073) 0 1 (0.117) 0.117 2 0.258 2 (0.258) 0.516 3 0.322 3 (0.322) 0.966 4 0.230 4 (0.230) 0.920 x [ x P( x)] 2.519 Los Angeles Mission College Produced by DW (b) Variance ---> Use the definition formula x [(x x ) 2 P( x)] 2 (2a) x x x P(x) x P(x) 0 0.073 0 (0.073) 1 0.117 1 (0.117) 2 0.258 2 (0.258) 1 2.519 1.519 (1.519)2 (0.117) 0.269961237 2 2.519 0.519 (0.519)2 (0.258) 0.069495138 3 0.322 3 (0.322) 3 2.519 0.481 (0.481)2 (0.322) 0.074498242 4 0.230 4 (0.230) 4 2.519 1.481 (1.481)2 (0.230) 0.50447303 ( x x )2 P( x) 0 2.519 2.519 (2.519)2 (0.073) 0.463211353 x [ x P( x)] 2.519 Los Angeles Mission College x 2 [(x x ) 2 P( x)] 1.381639 x 1.381639 1.18 Produced by DW (c) Variance ---> Use the computation formula x [ x P( x)] x 2 2 2 (2b) x P(x) x P(x) 0 0.073 0 (0.073) 02 (0.073) 0 1 0.117 1 (0.117) 2 0.258 2 (0.258) 12 (0.117) 0.117 22 (0.258) 1.032 3 0.322 3 (0.322) 32 (0.322) 2.898 4 0.230 4 (0.230) 42 (0.230) 3.68 x [ x P( x)] 2.519 x 2 P( x) 2 [ x P(x)] 7.727 x 2 [ x 2 P( x)] x 2 x 2 7.727 2.5192 x 2 1.381639 x 1.381639 Los Angeles Mission College 1.18 Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective C : Mean Expected Value The mean of a random variable is the expected value, E( x) [ x P( x)] , of the probability experiment in the long run. In game theory x is positive for money gained and x is negative for money lost. Los Angeles Mission College Produced by DW Example 1: A life insurance company sells a $250,000 1-year term life insurance policy to a 20-year-old male for $350. According to the National Vital Statistics Report, 56(9), the probability that the male survives the year is 0.998734. Compute and interpret the expected value of this policy to the insurance company. Gain/Loss x P(x) Gain 350 0.998734 249650 1 0.998734 0.001266 Loss x P(x) 350 (0.998734) 349.5569 249650 (0.001266) 316.0569 E( x) [ x P( x)] 33.5 In the long run, the insurance company will profit $33.50 per 20-yearold male. Los Angeles Mission College Produced by DW Example 2: Shawn and Maddie purchase a foreclosed property for $50,000 and spend an additional $27,000 fixing up the property. They feel that they can resell the property for $120,000 with probability 0.15, $100,000 with probability 0.45, $80,000 with probability 0.25, and $60,000 with probability 0.15. Compute and interpret the expected profit for reselling the property. Gain/Loss x P(x) x P(x) Loss 50,000 27,000 77,000 0 Gain 120,000 77,000 43,000 0.15 77,000 (0) 0 43,000 (0.15) 6,450 Gain 100,000 77,000 23,000 0.45 23000 (0.45) 10,350 Gain 80,000 77,000 3,000 0.25 3,000 (0.25) 750 Loss 60,000 77,000 17,000 0.15 17,000 (0.15) 2,550 E( x) [ x P( x)] 15,000 In the long run, the expected gain is $15,000 per house. Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment The binomial probability distribution is a discrete probability distribution that obtained from a binomial experiment. Los Angeles Mission College Produced by DW Example 1: Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. (a) A random sample of 30 cars in a used car lot is obtained, and their mileages recorded. Not a binomial distribution because the mileage can have more than 2 outcomes. (b) A poll of 1,200 registered voters is conducted in which the respondents are asked whether they believe Congress should reform Social Security. A binomial distribution because – there are 2 outcomes. (should or should not reform Social Security) – fixed number of trials. (n = 1200) – the trials are independent. – we assume the probability of success is the same for each trial of experiment. Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.2 Binomial Probability Distribution Objective B : Binomial Formula Let the random variable x be the number of successes in n trials of a binomial experiment. Los Angeles Mission College Produced by DW Los Angeles Mission College Produced by DW Example 1: A binomial probability experiment is conducted with the given parameters. Compute the probability xof successes in the n independent trials of the experiment. n 15, p 0.85, x 12 P( x) n Cx p x (1 p)nx P( x 12) 15 C12 (0.85)12 (1 0.85)1512 455(0.85)12 (0.15)3 0.2184 Los Angeles Mission College Produced by DW Example 2: According to the 2005 American Community Survey, 43% of women aged 18 to 24 were enrolled in college in 2005. Twenty-five women aged 18 to 24 are randomly selected, and the number of enrolled in college is recorded. (a) Find the probability that exactly 15 of the women are enrolled in college. n 25, p 0.43, x 15 P( x) n Cx p x (1 p)nx P( x 15) 25 C15 (0.43)15 (1 0.43)2515 3268760(0.43)15 (0.57)10 0.0376 Los Angeles Mission College Produced by DW (b) Find the probability that between 11 and 13 of the women, inclusive, are enrolled in college. n 25, p 0.43, 11 x 13 P(11 x 13) P( x 11) P( x 12) P( x 13) 25 C11 (0.43)11 (1 0.43) 2511 25 C12 (0.43)12 (1 0.43) 2512 25 C13 (0.43)13 (1 0.43) 2513 4457400(0.43)11 (0.57)14 5200300(0.43)12 (0.57)13 5200300(0.43)13 (0.57)12 0.4027 Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.2 Binomial Probability Distribution Objective C : Binomial Table Another method for obtaining binomial probabilities is the binomial probability table. Table IV in Appendix A gives cumulative probabilities of a binomial random variable x such as P( x 6) . Table III in Appendix A gives the exact probability of a binomial random variable such as ( = 6). Example 1: Use the Binomial Table to find P( x 6) with n 12 and p 0.4 . From Cumulative Binomial Probability Distribution (Table IV), P( x 6) 0.8418 . From Binomial Probability Distribution (Table III), P( x 6) P( x 6) P( x 5) P( x 4) P( x 3) P( x 2) P( x 1) P( x 0) 0.1766 0.2270 0.2128 0.1419 0.0639 Los Angeles Mission College 0.0174 0.0022 0.8418 Produced by DW Example 2: According to the American Lung Association, 90% of adult smokers started smoking before turning 21 years old. Ten smokers 21 years old or older are randomly selected, and the number of smokers who started smoking before 21 is recorded. (a) Explain why this is a binomial experiment. – There are 2 outcomes (smoke or not) – The probability of success trial is the same for each trial of experiment – The trials are independent – Fixed numbers of trials Los Angeles Mission College Produced by DW (b) Use the binomial formula to find the probability that exactly 8 of them started smoking before 21 years of age. n 10, p 0.9, x 8 From Binomial Probability Distribution (Table III), P( x 8) 0.1937 (c) Use the binomial table to find the probability that at least 8 of them started smoking before 21 years of age. n 10, p 0.9, x8 From Binomial Probability Distribution (Table III), P( x 8) P( x 8) P( x 9) P( x 10) 0.1937 0.3874 0.3487 0.9298 Los Angeles Mission College Produced by DW (d) Use the binomial table to find the probability that between 7 and 9 of them, inclusive, started smoking before 21 years of age. n 10, p 0.9, 7 x9 From Binomial Probability Distribution (Table III), P(7 x 9) P( x 7) P( x 8) P( x 9) 0.0574 0.1937 0.3874 0.6385 Los Angeles Mission College Produced by DW Chapter 6.1 Discreet Random Variables Objective A : Discrete Probability Distribution Objective B : Mean and Standard Deviation of a Discrete Random Variable Objective C : Mean Expected Value Chapter 6.2 Binomial Probability Distribution Objective A : Criteria for a Binomial Probability Experiment Objective B : Binomial Formula Objective C : Binomial Table Objective D : Mean and Standard Deviation of a Binomial Random Variable Objective E : Applications Los Angeles Mission College Produced by DW Chapter 6.2 Binomial Probability Distribution Objective D : Mean and Standard Deviation of a Binomial Random Variable Example 1: A binomial probability experiment is conducted with the given parameters. Compute the mean and standard deviation of the random variable x . n 9, p 0.8 x n p 9 0.8 7.2 x np(1 p) 9(0.8)(1 0.8) 9(0.8)(0.2) 1.2 Los Angeles Mission College Produced by DW Example 2: According to the 2005 American Community Survey, 43% of women aged 18 to 24 were enrolled in college in 2005. (a) For 500 randomly selected women ages 18 to 24 in 2005, compute the mean and standard deviation of the random variable x , the number of women who were enrolled in college. n 500, p 0.43 x n p 500 0.43 215 x np(1 p) 500(0.43)(1 0.43) 500(0.43)(0.57) 11.070 (b) Interpret the mean. An average of 215 out of 500 randomly selected women aged 18 to 24 were enrolled in college. Los Angeles Mission College Produced by DW (c) Of the 500 randomly selected women, find the interval that would be considered "usual“ for the number of women who were enrolled in college. Data fall within 2 standard deviations of the mean are considered to be usual. Interval x 2 x 215 2 (11.07) 215 22.14 (192.86, 237.14) (d) Would it be unusual if 200 out of the 500 women were enrolled in college? Why? No, because 200 is within the interval obtained in part (c). It is not unusual to find 200 out of 500 women were enrolled in college in 2005. Los Angeles Mission College Produced by DW