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Continuous Probability Distributions A function f is called the probability density function of a continuous random variable X if f satisfies the following properties: i) f (x) ≥ 0 for all real number x. ii) Total area under the graph of f is 1. Definition. The probability that X lies between a and b is defined as P (a < X < b) = area between a and b (under the graph of f above the X-axis). Note: This course covers only Normal distribution. Standard Normal Distribution 1. Find each of the following probabilities for the standard normal random variable z (a) P (−1 ≤ z ≤ 1) (b) P (−1.96 ≤ z ≤ 1.96) (c) P (−2.57 ≤ z ≤ 2.57) (d) P (−1.23 ≤ z ≤ −0.24) (e) P (1.34 ≤ z ≤ 2.81) (f) P (z ≤ −1.24) (g) P (z ≥ 1.85) (h) P (z ≤ 1.96) 2. Find a value of the standard normal random variable z, call it z0 , such that the following equations hold (you may earn partial credit if you correctly draw the graph and shade the area): (a) P (z ≤ z0 ) = 0.0401 (b) P (−z0 ≤ z ≤ z0 ) = 0.95 (c) P (−z0 ≤ z ≤ 0) = 0.2967 (d) P (z ≤ z0 ) = 0.0057 (e) P (−2 ≤ z ≤ z0 ) = 0.9010 (f) P (z ≥ z0 ) = 0.10 (g) P (z ≥ z0 ) = 0.86 (h) P (z ≤ z0 ) = 0.0405 3. Suppose the random variable x has a normal distribution with µ = 440 and σ = 50. Find x0 such that 90% of the values of x are less than x0 1 4. Suppose that the scores, x, on a college entrance examination are normally distributed with a mean score of 540 and a standard deviation of of 100. (a.) What percentage of scores are less than 560? (b.) What percentage of scores are between 520 and 565? (c) A certain prestigious university will consider for admission only those applicants whose scores exceed the 90th percentile of the distribution. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university. 5.) Which of the following is not a property of the normal curve? A) P(µ − 3σ < x < µ + 3σ) = 0.997 B) P(µ − σ < x < µ + σ) = 0.95 C) mound-shaped (or bell shaped) D) symmetric about µ 6. If a data set is from an approximately normal distribution, approximately what percentage of measurements would you expect to fall within each of the intervals x̄ ± s, x̄ ± 2s, x̄ ± 3s? A) 68%, 98%, 100% respectively. B) 68%, 95%, 100% respectively. C) 60%, 95%, 98% respectively. D) 65%, 98%, 100% respectively. 7. The board of examiners that administers the real estate broker’s examination in a certain state found that the mean score on the test was 591 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score? Assume that the scores are normally distributed. 8. The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 6200 miles. If the manufacturer guarantees the tread life of the tires for the first 52,560 miles, what proportion of the tires will need to be replaced under warranty? 9. The average life of a certain type of motor is 10 years, with a standard deviation of 2 years. If the manufacturer is willing to replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the lives of the motors follow a normal distribution. 10. A toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3, 000 and σ = 50). What is the probability that the car will go more than 3,100 yards without recharging? 11. A company pays its employees an average wage of $3.25 an hour with a standard deviation of 50 cents. Wages are normally distributed. Determine the proportion of workers getting wages between $2.75 and $3.69 an hour. Determine the minimum wage of the highest 5%. 12. TRUE/FALSE. a. The number of children in a family can be modelled using a continuous random variable. b. For any continuous probability distribution, P(x = c) = 0 for all values of c. Descriptive Methods for Assessing Normality Read items 1, 2, 3 on page 249, 250. Approximate Binomial Probability with a Normal Distribution when n is large and p is approximately equal to 0.5. If n is large and p is approximately equal to 0.5, then binomial probability can be calculated using normal table as follows: The binomial probability p(X = c) is equal to the normal probability p(c − 0.5 < X < c + 0.5) for any c. Use µ = np and σ 2 = np(1 − p) from binomial distribution. 2 13. According to the American Cancer Society, melanoma, a form of skin cancer, kills 60% of Americans who suffer from the disease each year. Consider a sample of 1000 melanoma patients. Find the probability that X (the number of melanoma deaths in the sample) will exceed 620 patients per year. 14. Historical data suggest that only 70% of all students earn C or better grade in large sections of STA-2023. Find the probability that more than 225 but fewer than 270 students in your class of 350 students will have C or better grade. 15. The Statistical Abstract of the United States reports that 30% of the country’s households are composed of one person. If 200 randomly selected homes are to participate in a Nielson survey to determine television ratings, find the probability that fewer than 75 of these homes are one-person households. 16. The probability that a person responds to a mailed questionnaire is 0.4. What is the probability that of 250 questionnaires, more than 140 will not be returned? Answers: 1. 0.6826; 0.95; 0.9898; 0.2959; 0.0876; 0.1075; 0.0322; 0.975 2. -1.75; 1.96; 0.83; -2.53; 1.43; 1.28; -1.08; -1.74 3. 504; 4. 0.5793; 0.178; 668; 5. B; 6. B; 7. 530.52; 8. 0.1151; 9. 6.24; 10. 0.0228; 11. 0.6519; 4.07; 12. (a) False; (b) True; 13. 0.0934; 14. 0.9866; 15. 0.9871; 16. 0.8907 (or 0.8888 is acceptable) 3 *** This Section is Intended for Honors Students Only *** Extra Practice Problems 1. A physical-fitness association is including the mile run in its secondary-school fitness test for boys. The time for this event for boys in secondary school is normally distributed with a mean of 450 seconds and a standard deviation of 40 seconds. What is the probability that at least 9 of 10 randomly selected boys finish the mile in 400 seconds? (Ans. (10)(0.1056)9 (0.8944) + (0.1056)10 ) 2. A physical-fitness association is including the mile run in its secondary-school fitness test for boys. The time for this event for boys in secondary school is normally distributed with a mean of 450 seconds and a standard deviation of 40 seconds. What is the probability that at least 60 of 100 randomly selected boys finish the mile in 400 seconds? (Ans. Approximately zero) 3. Assume that the GPA of all UCF students is normally distributed with mean 2.75 and standard deviation 0.25. Twenty students are randomly selected from all UCF students. What is the probability that eight of these twenty students have GPA exceeding 3.00? (Ans.(125970)(0.1587)8 (0.8413)12 ) 4. Assume that the weight of adults is normally distributed with mean 150 pounds and standard deviation 20 pounds. If twenty adults are randomly selected, what is the probability that eight of these twenty adults are over weight? An adult is considered overweight if weight exceeds 200 pounds. (Ans.(125970)(0.0062)8 (0.9938)12 ) Two more distributions Probability distribution for an exponential random variable x. x f (x) = θ1 e− θ , x > 0. mean (µ) = θ, Variance (σ 2 ) = θ2 . Probability distribution for a uniform random variable x. 1 f (x) = d−c , c < x < d. mean (µ) = c+d Variance (σ 2 ) = 2 , d−c √ . 12 4