b) What is a good approximation of S and why can you use it? d) What is the approximate probability exactly 38 plants sprout?
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1 Ex. 3: As a green thumb, you know that when you plant seeds, they are not guaranteed to sprout into plants. If each seed sprouts independently of one another, and each has a.68 probability of sprouting, let S be the number of plants that sprout in the 44 seeds you plant. a) What is the distribution and parameters of S? b) What is a good approximation of S and why can you use it? c) As you water the soil you wonder what the probability is that 38 of your plants sprout. What is the exact probability 38 seeds sprout? d) What is the approximate probability exactly 38 plants sprout? e) It has been several days and the seeds have started to sprout. So far 34 have sprouted. What is the approximate probability that no more than 40 sprout all together?
2 Ex. 4: 52% of the residents in New York City are in favor of outlawing cigarette smoking in publicly owned areas. Approximate the probability that greater than 50 percent of a random sample of n people from New York are in favor of this prohibition when: a) n = 11 b) n = 101 c) n = 1,001 More Practice: Ex. 5: The life of a certain type of automobile tire is normally distributed with mean 34,000 miles and standard deviation 4,000 miles. a) What is the probability that such a tire lasts over 40,000 miles? b) What is the probability that it lasts between 30,000 and 35,000 miles? c) Given that it has survived 30,000 miles, what is the conditional probability that it will survive another 10,000 miles?
3 Ex. 6: The annual rainfall in Cleveland, Ohio is approximately normally distributed with mean 40.2 inches and standard deviation 8.4 inches. Find the following: a) Next year's rainfall will exceed 44 inches. b) Assuming rainfall from year to year is independent what is the probability that in exactly three of the next seven years the rainfall will exceed 44 inches? Ex. 7: Suppose that the travel time from your home to your office is normally distributed with mean 40 minutes and standard deviation 7 minutes. If you want to be 95 percent certain that you will not be late for an appointment at 1 p.m., what is the latest time that you should leave for home?
4 Ex. 8: Chips Ahoy is famous for its slogan 1,000 chips delicious. In fact, they guarantee that if your box of Chips Ahoy cookies do not have at least 1,000 chips, they will give you a full refund and a free case of cookies. A box of Chips Ahoy cookies has 80 cookies. The number of chocolate chips in each cookie follows a Poisson distribution with an average of 13 chips. Let C be the number of chocolate chips in a box of Chips Ahoy cookies. a) What is the approximate distribution of C? b) You decide to buy a box of Chips Ahoy cookies. What is the probability you get a free case? If all boxes of cookies are independent and you buy 20 boxes, how many do you expect to have less than 1,000 chips? Ex. 9: The distribution of adult human weights has a mean of 150 pounds with a standard deviation of 50 pounds. The elevator at a local hotel holds a maximum weight of 5,000 pounds. Let W be the total weight of 30 people (all independent) that step into this elevator. a) What is the approximate distribution of W? b) What is the probability the elevator breaks down? c) This elevator is really busy and carries 30 people in each load for the next 65 loads. Each elevator run takes 2 minutes and it takes 15 additional minutes to fix the elevator that breaks down. How much time do you expect it to have taken these 65 loads to complete?
5 Ex. 10: Over summer break you become very bored and decide to play following game. Roll 40 fair sixsided dice and take the sum of the up faces as the outcome of the game. a) By using the CLT approximate the probability the outcome of any given game is exactly 140. Exact Probability: b) If someone offered you 30:1 odds with this game would you play it? Ex. 11: Business Week conducted a survey of graduates from 30 top MBA programs (Business Week, 9/22/2003). On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $168,000 and $117,000, respectively. Assume the standard deviation for the male graduates is $40,000, and for female graduates it is $25,000. a) What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean? b) What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $10,000 of the population mean?
6 c) In which of the preceding two cases, a) or b), do we have a higher probability of obtaining a sample estimate within $10,000 of the population mean? Why? d) What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $4,000 below the population mean? Ex. 12: Let S = X_1 + X_2 + + X_40 be a sample of size 40 from a distribution with the following PDF. f(x) = 1.5x^2; -1 < x < 1 a) Find the mean and variance of the distribution. b) By using the CLT approximate P(S > 3.5)
7 Ex. 13: Let S = X_1 + X_2 + + X_30 be the sum of a random sample of size n = 30 from a Poisson distribution with λ = 2/3. a) Using the CLT approximate P(15 < S 22) b) Using the CLT approximate P(21 S 27) Ex. 14: Let X_ be the mean of a random sample of size n = 36 from an Exp(λ = 3) distribution. Find a good approximation for X_. Then find P(2.5 X 4)
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