stat400-hw07-Sp24-soln

Homework 07 Solutions STAT 400, Spring 2024, D. Unger Each Exercise or lettered part of an Exercise is worth 5 points. Homework assignments are worth 50 points. Exercise 1 Based on a fictitious study, the amount of pure alcohol (in grams) a randomly selected Illinois student on Green Street consumes on a Friday night has a mean of 57 grams and a standard deviation of 18 grams. (a) If we assume that the amount of pure alcohol an Illinois student drinks on a Friday night follows a normal distribution, then calculate the proportion of students on Green Street who consume between 30 and 80 grams of pure alcohol. Solution Let X = alcohol consumption in grams where X ~ Normal(57, 18 2 ). P ሺ 30 ൑ 𝑋 ൑ 80 ሻ ൌ P ൬ 30 െ 57 18 ൑ 𝑋 െ 57 18 ൑ 80 െ 57 18 ൰ ൌ P ሺെ 1.50 ൑ 𝑍 ൑ 1.28 ሻ ൌ P ሺ𝑍 ൑ 1.28 ሻ െ P ሺ𝑍 ൑ െ 1.50 ሻ ൌ 0.8997 െ 0.0668 ൌ 0.8329 (b) According to the National Institute on Alcohol Abuse and Alcoholism, a “standard drink” contains roughly 14 grams of pure alcohol. What is the probability that a randomly selected Illinois students consumes no more than two standard drinks on a Friday night? Solution Let X = alcohol consumption in grams where X ~ Normal(57, 18 2 ). P ሺ𝑋 ൑ 28 ሻ ൌ P ൬ 𝑋 െ 56 18 ൑ 28 െ 57 18 ൰ ൌ P ሺ Z ൑ െ 1.61 ሻ ൌ 0.0537 (c) How much would an Illinois student have consumed if they consumed more than 75% of all other Illinois students on Green Street on a Friday night?

Solution Let X = alcohol consumption in grams where X ~ Normal(57, 18 2 ). Then x 0.75 is the value such that P( X ≤ x 0.75 ) = 0.75. P ሺ𝑋 ൑ 𝑥 ଴ . ଻ହ ሻ ൌ 0.75 ⇒ P ሺ𝑍 ൑ 𝑧 ଴ . ଻ହ ሻ ൌ 0.75 ⇒ P ሺ𝑍 ൑ 0.68 ሻ ൌ 0.75 ⇒ 𝑧 ଴ . ଻ହ ൌ 0.68 𝑥 ଴ . ଻ହ ൌ μ ൅ 𝑧 ∙ σ ൌ 57 ൅ 0.68 ∙ 18 ൌ 69.24 grams Exercise 2 A chocolatier produces caramel-filled chocolates that have a labeled weight of 20.4 grams. Assume that the distribution of the weights of these caramel-filled chocolates is N (21.37, 0.16). (a) Let X denote the weight of a single chocolate selected at random from the production line. Find P ( X > 22.07). Solution Let X = weight of a single chocolate where X ~ Normal(21.37, 0.16). P ሺ X ൐ 22.07 ሻ ൌ P ൬ 𝑋 െ 21.37 √ 0.16 ൐ 22.07 െ 21.37 √ 0.16 ൰ ൌ P ሺ𝑍 ൐ 1.75 ሻ ൌ P ሺ𝑍 ൏ െ 1.75 ሻ ൌ 0.0401 (b) Suppose that 15 caramel-filled chocolates are selected independently and weighed. Let Y equal the number of these chocolates that weigh less than 20.857 grams. Find P ( Y ≤ 2). Solution Let X = weight of a single chocolate where X ~ Normal(21.37, 0.16). Let Y = number of chocolates weighing less than 20.857 out of 15 total, where Y ~ Binomial(15, p = P( X < 20.857)) P ሺ X ൏ 22.07 ሻ ൌ P ൬ 𝑋 െ 21.37 √ 0.16 ൏ 20.857 െ 21.37 √ 0.16 ൰ ൌ P ሺ𝑍 ൏ െ 1.28 ሻ ൌ 0.1003, 𝑡ℎ𝑜𝑢𝑔ℎ 𝑖𝑡 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 𝑜𝑘𝑎𝑦 𝑡𝑜 𝑟𝑜𝑢𝑛𝑑 𝑡𝑜 0.10. Thus, Y ~ Binomial(15, 0.10).

Solution E ሾ𝑋 ଶ ሿ ൌ ෍ 𝑥 ଶ ∙ 𝑓 ௑ ሺ𝑥ሻ ௔௟௟ ௫ ൌ ෍ 𝑥 ଶ ∙ 2 𝑥 ൅ 3 12 ௔௟௟ ௫ ൌ ෍ 2 𝑥 ଷ ൅ 3 𝑥 ଶ 12 ௔௟௟ ௫ ൌ ሺ 2 ൅ 3 ሻ ൅ ሺ 16 ൅ 12 ሻ 12 ൌ 33 12 Var ሾ𝑋ሿ ൌ E ሾ𝑋 ଶ ሿ െ ሺ E ሾ𝑋ሿሻ ଶ ൌ 33 12 െ ൬ 19 12 ൰ ଶ ൌ 35 144 ൌ 0.2431 or… Var ሾ𝑋ሿ ൌ ෍ ෍ ሺ𝑥 െ 𝜇 ௑ ሻ ଶ ∙ 𝑓ሺ𝑥 , 𝑦ሻ ௔௟௟ ௬ ௔௟௟ ௫ ൌ ෍ ෍ ൬𝑥 െ 19 12 ൰ ଶ ∙ 𝑥 ൅ 𝑦 24 ௔௟௟ ௬ ௔௟௟ ௫ ൌ ⋯ ൌ 35 144 ൌ 0.2431 (d) What is the probability that you will choose to eat lunch in the Union more than on Green Street in one week? Solution P ሺ Y ൐ X ሻ ൌ P ሺ Y െ X ൐ 0 ሻ ൌ P ሺሾ X ൌ 1 ∩ Y ൌ 3 ሿ ∪ ሾ X ൌ 2 ∩ Y ൌ 3 ሿ ∪ ሾ X ൌ 1 ∩ Y ൌ 2 ሿሻ ൌ P ሺ X ൌ 1 ∩ Y ൌ 3 ሻ ൅ P ሺ X ൌ 2 ∩ Y ൌ 3 ሻ ൅ P ሺ X ൌ 1 ∩ Y ൌ 2 ሻ ൌ 𝑓ሺ 1,3 ሻ ൅ 𝑓ሺ 2,3 ሻ ൅ 𝑓ሺ 1,2 ሻ ൌ 1 ൅ 3 24 ൅ 2 ൅ 3 24 ൅ 1 ൅ 2 24 ൌ 1 2 (e) What is the probability that the total number of times you choose to eat on Green Street or in the Union in one week is at least three times? Solution P ሺ X ൅ Y ൒ 3 ሻ ൌ 1 െ P ሺ X ൅ Y ൏ 3 ሻ ൌ 1 െ ሾ P ሺ X ൌ 1, Y ൌ 0 ሻ ൅ P ሺ X ൌ 1, Y ൌ 1 ሻ ൅ P ሺ X ൌ 2, Y ൌ 0 ሻሿ ൌ 1 െ ሾ𝑓ሺ 1,0 ሻ ൅ 𝑓ሺ 1,1 ሻ ൅ 𝑓ሺ 2,0 ሻሿ ൌ 1 െ ൤ 1 ൅ 0 24 ൅ 1 ൅ 1 24 ൅ 2 ൅ 0 24 ൨ ൌ 1 െ ൤ 1 24 ൅ 2 24 ൅ 2 24 ൨ ൌ 19 24