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MDM4U The Normal Distribution Test 2 TOTAL MARK /30 1. What is, usually, the random variable in an exponential probability distribution? [K2] Answer: Waiting time between succesive events in Poisson process (a process in which events occur continuously and independently at a constant average rate). 2. What is the standard normal distribution? [K2] Answer: A normal distribution with 𝜇 = 0 and 𝜎 = 1is called the standard normal distribution. 3. What does a z-score indicate? [C2] Answer: The z-score of a datum x is 𝑧 = 𝑥−𝜇 𝜎 . It indicates the number of standard deviations an observation or datum is above or below the mean. Thus, a positive z-score indicates a datum above the mean, while a negative z-score indicates a datum below the mean. 4. Calculate 𝑃(𝑋 < 9) for a normal distribution with 𝜇 = 10 and 𝜎 = 1. [K2] Soluton: 𝑃(𝑋 < 9) = 𝑃 (𝑍 < 9−10 1 ) = 𝑃(𝑍 < −1) = 0.1587. 5. What is a continuity correction? [C2] Answer: A continuity correction is an adjustment that is made when a discrete distribution is approximated by a continuous distribution. 6. Use the normal approximation to find 𝜇 and 𝜎 in a binomial distribution with n = 1000 and p = 0.5. [A2] Soluton: 𝜇 = 𝑛𝑝 = 500, 𝜎 = √𝑛𝑝𝑞 = √250 = 5√10 7. A really tough trivia quiz has 50 multiple-choice questions, each with four possible answers. Use a normal approximation to calculate the probability of randomly guessing the correct answers for 10 to 15 (inclusive) of the questions. [A3] Soluton: 𝜇 = 𝑛𝑝 = 50(0.25) = 12.5 , 𝜎 = √𝑛𝑝𝑞 = √50(0.25)(0.75) ≈ 3.062, 𝑃(9.5 < 𝑋 < 15.5) ≈ 𝑃 ( = 𝑃( 9.5 − 12.5 3.062 9.5 − 12.5 3.062 <𝑍< <𝑍< 15.5 − 12.5 3.062 15.5 − 12.5 3.062 ) ) ≈ 𝑃(−0.98 < 𝑍 < 0.98) = 𝑃(𝑍 < 0.98) − 𝑃(𝑍 < −0.98) = 0.8365 − 0.1635 = 0.8365 − 0.1635 = 0.673. 8. Suppose the commuting time from North Bay to Barrie varies uniformly from 135 min to 180 min. What is the probability that the drive will take lesss than 160 min? [A3] Answer: 5 9 9. The life spans of a particular species of turtle are normally distributed with a mean of 180 years and a standard deviation of 40 years. What is the probability that one of these turtles, selected at random, will live more than a century? Answer: 97.5% 10. The masses of Statistics students are belived to normally distributed. The masses (in kg) of random sample of 36 sytudents are: [T6] 62 67 66 70 70 74 63 67 67 70 71 75 63 68 67 70 71 75 64 68 67 71 71 77 64 68 68 71 71 77 65 68 68 71 72 78 a) Determine the mean and standard deviation of these data. Answer: 𝜇 = 69.31, 𝜎 = 4.02 b) What is the probability that the mass of a randomly selected Statistics student will be at least 70 kg? Answer: 43.08% c) Of 120 Statistics students, how many would you expect to have a mass greater than 65 kg? Answer: 103. 11. Given the pdf 2𝑥 if 𝑥 ∈ [0; 3] f (x) = { 9 0 otherwise, a) Sketch the graph of f (x). b) Determine: P (2 < x < 3). 5 Answer: . 9 c) Determine: P (x = 3) d) Determine: P (2 ≤ x ≤ 3).