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Statistics 5.2 Quiz 6 A quick quiz consists of a true/false question followed by a multiple-choice question with four possible answers (a,b,c,d). An Unprepared student makes random guesses for both answers. a. What is the probability of that both answers are correct b. Is guessing a good strategy? Random Variables Everyone look at figure 5-1 on page 204 of our book. • A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Random Variables Everyone look at figure 5-1 on page 204 of our book. • A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure • A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula. Random Variables Everyone look at figure 5-1 on page 204 of our book. • A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure • A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula. Random Variables • A discrete random variable has either a finite number of values or a countable number of values. Random Variables • A discrete random variable has either a finite number of values or a countable number of values. • A continuous random variable has infinitely many values, and those value can be associated with measurements on a continuous scale without gaps or interruptions. Random Variables • A discrete random variable has either a finite number of values or a countable number of values. • A continuous random variable has infinitely many values, and those value can be associated with measurements on a continuous scale without gaps or interruptions. Random Variables Determine whether the given random variable is discrete or continuous. a. The total amount in (ounces) of soft drinks that you consumed in the past year. b. The number of cans of soft drinks that you consumed in the past year. c. The number of movies currently playing in U.S. theaters. d. The running time of a randomly selected movie. e. The cost of making a randomly selected movie. Random Variables We use probability histograms to graph a probability distribution x (number of peas with green pods) P(x) 0 0.001 1 0.015 2 0.088 3 0.264 4 0.396 5 0.237 Random Variables We use probability histograms to graph a probability distribution P(x) 0 0.001 1 0.015 2 0.088 3 0.264 4 0.396 5 0.237 Series 1 Axis Title x (number of peas with green pods) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 Axis Title 4 5 Random Variables Requirements for a Probability Distribution 1. () = 1 where x assumes all possible values. 2. 0 ≤ ≤ 1 for every individual value of x Random Variables Requirements for a Probability Distribution 1. () = 1 where x assumes all possible values. 2. 0 ≤ ≤ 1 for every individual value of x. Random Variables Based on a survey conducted by Frank N. Magid Associates, Table 5-2 lists the probabilities for the number of cell phones in use per household. Does the table below describe a probability Distribution? x P(x) 0 0.19 1 0.26 2 0.33 3 0.13 Random Variables Does = (where can be 0, 1, 2, 3, or 4) determine a 10 probability distribution? Random Variables Does = (where can be 0, 1, 2, 3, or 4) determine a 10 probability distribution? X P(x) 0 0/10 1 1/10 2 2/10 3 3/10 4 4/10 Random Variables Does = (where can be 0, 1, 2, 3, or 4) determine a 10 probability distribution? X P(x) 0 0/10 1 1/10 2 2/10 3 3/10 4 4/10 Total 10/10=1 Random Variables Mean, Variance, and Standard Deviation • = [ ∙ ] Mean for a probability distribution Random Variables Mean, Variance, and Standard Deviation • = [ ∙ ] • 2 = − Mean for a probability distribution 2 ∙ Variance for a probability distribution Random Variables Mean, Variance, and Standard Deviation • = [ ∙ ] Mean for a probability distribution • 2 = − 2 • 2 = 2 ∙ ∙ − 2 Variance for a probability distribution Variance for a probability distribution Random Variables Mean, Variance, and Standard Deviation • = [ ∙ ] Mean for a probability distribution • 2 = − 2 • 2 = 2 ∙ − 2 Variance for a probability distribution • = 2 ∙ − 2 Standard Deviation for a probability distribution ∙ Variance for a probability distribution Random Variables Mean, Variance, and Standard Deviation • = [ ∙ ] Mean for a probability distribution • 2 = − 2 • 2 = 2 ∙ − 2 Variance for a probability distribution • = 2 ∙ − 2 Standard Deviation for a probability distribution ∙ Variance for a probability distribution Lets do example 5 in excel!, and then do problem 3 on the worksheet Random Variables Determine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small) x P(x) 0 0.528 1 0.360 2 0.098 3 0.013 4 0.001 5 0+ Random Variables Determine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small) () ⋅ () 0 0.528 0 1 0.360 0.360 2 0.098 0.196 3 0.013 0.039 4 0.001 0.004 5 0+ 0 Random Variables Determine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small) () ⋅ () 0 0.528 0 1 0.360 0.360 2 0.098 0.196 3 0.013 0.039 4 0.001 0.004 5 0+ 0 Total 1 .599 Random Variables Determine whether the following is a probability distribution and if so find its mean and standard deviation . Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (0+ denotes a positive probability value that is very small) () ⋅ () 0 0.528 0 1 0.360 0.360 2 0.098 0.196 2 − .599 2 ⋅ 0.098 3 0.013 0.039 3 − .599 2 ⋅ 0.013 4 0.001 0.004 4 − .599 2 ⋅ 0.001 5 0+ 0 Total 1 .599=0.6 − ⋅ 0 − .599 2 ⋅0 1 − .599 2 0.360 5 − .599 2 ⋅0 0.5269 = 0.725=0.7 Random Variables Round off rule for , Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round , 2 to one decimal place. Random Variables Round off rule for , Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round , 2 to one decimal place. Recall the range rule of thumb Random Variables Round off rule for , Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round , 2 to one decimal place. Recall the range rule of thumb = + Random Variables Round off rule for , Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round , 2 to one decimal place. Recall the range rule of thumb = + = − Random Variables Round off rule for , Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round , 2 to one decimal place. Recall the range rule of thumb = + = − Random Variables Use the range rule of thumb to identify a range of values containing the usual number of peas with green pods. Based on this is it unusual to get only one pea with a green pod? Explain. x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 Random Variables Rare Event Rule for Inferential Statistics If, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct. Random Variables Rare Event Rule for Inferential Statistics If, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct. • Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. ( ) ≤ 0.05 Random Variables Rare Event Rule for Inferential Statistics If, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct. • Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. ( ) ≤ 0.05 • Unusually low number of successes: x successes among n trials is an unusually low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less. ( ) ≤ 0.05 Random Variables Rare Event Rule for Inferential Statistics If, under a given assumption(such that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is not correct. • Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. ( ) ≤ 0.05 • Unusually low number of successes: x successes among n trials is an unusually low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less. ( ) ≤ 0.05 Random Variables x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 a) Find the probability of getting exactly 3 peas with green pods . b) Find the probability of getting 3 or fewer peas with green pods. c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not? Random Variables x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 a) Find the probability of getting exactly 3 peas with green pods . 0.023 b) Find the probability of getting 3 or fewer peas with green pods. c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not? Random Variables x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 a) Find the probability of getting exactly 3 peas with green pods . 0.023 b) Find the probability of getting 3 or fewer peas with green pods. 0.027 c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). d) Is 3 and unusually low number of peas with green pods? Why or why not? Random Variables x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 a) Find the probability of getting exactly 3 peas with green pods . 0.023 b) Find the probability of getting 3 or fewer peas with green pods. 0.027 c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). Part (b) d) Is 3 and unusually low number of peas with green pods? Why or why not? Random Variables x (# of peas with green pods) P(x) 0 0+ 1 0+ 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.276 8 0.100 a) Find the probability of getting exactly 3 peas with green pods . 0.023 b) Find the probability of getting 3 or fewer peas with green pods. 0.027 c) Which Probability is relevant to determine whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b). Part (b) d) Is 3 and unusually low number of peas with green pods? Why or why not? Yes since ≤ 3 = 0.027 ≤ 0.05 Random Variables Expected Value The expected value of a discrete random variable is denoted by E, and it represents the mean value of its outcomes. It is obtained by finding the value of [ ∙ ] = [ ∙ ] Random Variables Expected Value The expected value of a discrete random variable is denoted by E, and it represents the mean value of its outcomes. It is obtained by finding the value of [ ∙ ] = [ ∙ ] Random Variables You are considering placing a bet on the number 7 in roulette or red for roulette. Random Variables • If you bet $5 on the number 7 in roulette, the probability of losing $5 is 37/38 and the probability making a net gain of $175 is 1/38. Let’s find the expected value if you bet on 7. Random Variables • If you bet $5 on the number 7 in roulette, the probability of losing $5 is 37/38 and the probability making a net gain of $175 is 1/38. Let’s find the expected value if you bet on 7. Event () ⋅ () Lose −5 37/38 −$4.87 $175 1/38 $4.61 Win(net) Total −$0.26 Random Variables • If you bet $5 on red, the probability of losing $5 is 20/38 and the probability making a net gain of $5 is 18/38. Let’s find the expected value if you bet on red. Random Variables • If you bet $5 on red, the probability of losing $5 is 20/38 and the probability making a net gain of $5 is 18/38. Let’s find the expected value if you bet on red. Event () ⋅ () Lose −$5 20/38 −$4.87 $5 18/38 $4.61 Win(net) Total −$0.26 Homework!!! • 5-2: 1-17 odd ,21, 25, 27