STA 2014 Statistics Week 8, Individual Work Solution Correct Answers 6.1 Skill Building In problem 7 - 10, determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. 8. (a) The time it takes for a light bulb to burn out. (b) The weight of a T - bone steak. (c) The number of free - throw attempts before the first shot is made. (d) In a random sample of 20 people, the number with type A blood. 20. Parental involvement, In the following probability distribution, the random variable X represents the number of activities a parent of a 6 th - 8 th grade student is involved in. (a) Verify that this is a discrete probability distribution. (c) Compute and interpret the mean of the random variable X. (d) Compute the variance of the random variable X. (e) Compute the standard deviation of the random variable X. 6.2 Skill Building 10. A poll of 1,200 registered voters is conducted in which the respondents are asked whether they believe Congress should reform Social Security. In problem 29 - 34 (a) construct a binominal probability distribution with the given parameters, (c) compute the mean and standard deviation, using the methods of this section 30. n=8,p=0.5 Applying the Concepts 36. Smokers, 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. (b) Find the probability that exactly 8 of them started smoking before 21 years of age. (c) Find the probability that at least 8 of them started smoking before 21 years of age (d) Find the probability that fewer than 8 of them, inclusive, started smoking before 21 years of age. 26. - 1 1 3 5 7 9 11 x 30. σ 11.5 14 16.5 x c) Interpretation 1: 11.51 % of refrigerators distributed last more than 17 years Interpretation 2: The probability is 0.1151 that refrigerators last more than 17 years 34. You explain it! Miles per Gallon, Elena conducts an experiment in which she fills up the gas tank on her Toyota Camry 40 times and records the miles per gallon for each fill - up. A histogram of the miles per gallon indicates that a normal distribution with mean of 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon could be used to model the gas mileage for her car. (a) The following figure represents the normal curve with μ=24.6 miles per gallon and σ =3.2 miles per gallon. The area under the curve to the right of the x = 26 is 0.3309. Provide two interpretations of this area. (b) The following figure represents the normal curve with μ = 24.6 miles per gallon and σ – 3.2 miles per gallon. The area under the curve between x =18 and x= 21 is 0.1107. Provide two interpretations of this area. 7.2 Skill Building 6. Determine the area under the standard normal curve that lies to the left of: (a) z = -- 3.49 (b) z = - 1.99 (c) z = 0.92 (d) z = - 2.90 8, Determine the area under the standard normal curve that lies to the right of: (a) z = - 3.49 (b) z = - 0.55 (c) z = 2.23 (d) z = 3.45 10. Determine the area under the standard normal curve that lies between: (a) z = - 2.55 and z = 2.55 (b) z = - 1.67 and z = 0 (c) z = - 3.03 and z = 1.98 16. Find the z - score such that the standard normal curve to its left is 0.2. 20. Find the z - score such that the area under the standard normal curve to its right is 0.89. 24. Find the z - scores that separate the middle 70% of the distribution from the area in the tails of the standard normal distribution. In Problems 27 - 32, find the value of 𝑍𝛼 28. Z 0.35 In Problem 33 - 34, find the indicated probability of the standard normal random variable z. 34. P(Z ˂ - 0.61) 7.5 Skill Building 22. Smokers, According to information Please Almanac, 80% of adult smokers started before they were 18 years old. Suppose 100 smokers 18 years old or older, are randomly selected. Use the normal approximation to the binomial to: From the parameter n=100 and p = 0.8, we get n ᵪ = np = 100 x 0.8=80 and σ ᵪ = ඥ 𝑛𝑝 ( 1 − 𝑝 ) = ඥ 100 𝑥 0 . 8 ( 1 − 0 . 8 ) = √ 16 = 4 Note that np (1 - p) = 16 ≥ 10, the normal approximation to the binominal distribution can be used. (a) Approximate the probability that exactly 80 of them started smoking before they were 18 years old. (b) Approximate the probability that at least 80 of them started smoking before they were 18 years old. (c) Approximate the probability that fewer than 70 of them started smoking before they were 18 years old. (d) Approximate the probability that between 70 and 90 of them, inclusive, started smoking before they were 18 years old. 28. Liars, According to a USA Today “Snapshot,” 3% of Americans surveyed lie frequently. You conduct a survey of 500 college students and find that 20 of them lie frequently. From the parameters n - 500 and p= 0.03 we get μ ᵪ = np = 500 x 0.03 = 15 and σ ᵪ = ට 𝑛𝑝 ( 1 − 𝑝 ) = ඥ 500 𝑥 0 . 03 ( 1 − 0 . 03 ) = √ 14 . 55 = 3 . 814 Note that np (1 - p) = 14.55 ≥ 10 so the norma;l approximation to the biominal distribution can be used. (a) Compute the probability that, in a random sample of 500 college students, at least 20 lie frequently, assuming the true percentage is 3%. (b) Does this result contradict the USA Today “Snapshot”? Explain. STA 2014 Statistics Week 8 , Individual Work solution correct answers 6.1 Skill Building In problem 7 - 10, determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. 8. (a) The time it takes for a light bulb to burn out. Answer: The time it takes for a light bulb to burn out is continuous because time is measured. The random variable T might describe the time it takes to blow out T70. (b) The weight of a T - bone steak. Answer: The weight of the T - bone steak is x, is a discrete random variable because the value of the possible values is x=01 1,2,3,4,5,6,7,8,9,10,… etc. (c) The number of free - throw attempts before the first shot is made. Answer: The number of free – throw attempts before the first shot is made is discrete random variable because the value of the possible values x= 0,1,2,3,4,5,6,7,8,9,10,…etc. (d) In a random sample of 20 people, the number with type A blood. Answer: In a random sample of 20 people the number with type A blood would be a discrete random variable because the value of possible value 5x= 0,1,2,3,4,,5,6,7,8,9,10,…etc. 20. Parental involvement, In the following probability distribution, the random variable X represents the number of activities a parent of a 6 th - 8 th grade student is involved in. X P(x) 0 0.073 1 0.117 2 0.258 3 0.322 4 0.230 (a) Verify that this is a discrete probability distribution. Answer: This is a discrete probability distribution because all the probabilities are between 0 and 1 (Inclusive) and the sum of the probability is 1 0.073+0.258+0.322+0.230=1