Q1 The number of passengers who use an elite chauffeured-limousine service from the airport to the city CBD in a 24-hour period is distributed as follows: (a) Graph the probability distribution. (b) What is the probability that no more than 5 passengers use the service? (c) What is the probability that more than 6 passengers use the service? (d) What is the probability that less than 7 passengers use the service? (e) What is the probability that between 5 and 7 passengers (inclusive) use the service? (f) Find the mean and standard deviation for the number of passengers who use the service 24-hour period. Q2 The probability of a property being sold by a real-estate agent in the first month of listing is 0.4. If 12 properties are selected at random, use the following binomial probability table to answer parts (a) to (d): Binomial Probabilities Table (a) What is the probability that exactly five properties are sold in the first month of listing? (b) What is the probability that at least five properties are sold in the first month of listing? (c) What is the probability that more than eight properties are sold in the first month of listing? (d) What is the probability that between 3 and 7 (inclusive) properties are sold in the first month of listing? (e) Show that P(X = 0) = 0.002177. (f) Find the mean and standard deviation of the number of properties that are sold in the first month of listing. Q3 Let the random variable X follow a normal distribution with µ = 50 and σ 2 = 36. (a) Find the probability that X is greater than 48. (b) Find the probability that X is greater than 45 and less than 55. (c) Find the probability that X is less than 40. (d) If P(X > a) = 0.1, determine the value of a. Q4 Applicants for a graduate position in a bank need to sit an aptitude test. The marks for the test are normally distributed with a mean of 100 and a standard deviation of 25. (a) What is the probability that a person will obtain a mark below 90? (b) What is the probability that a person will obtain a mark above 120? (c) What is the probability that a person will obtain a mark between 95 and 110? (d) If 200 people sit the test, how many will be offered a follow up interview if the selection criterion is to interview the top 10% of applicants? What is the lowest score a selected interviewee can have? Q5 The mean time taken in a supermarket checkout queue is normally distributed with a mean of 6 minutes and a standard deviation of 2 minutes. (a) What is the probability that an individual customer waits more than 6.5 minutes? (b) If a sample of 16 customers is taken, what is the probability that the sample mean waiting time is more than 6.5 minutes? (c) If a sample of 36 customers is taken, what is the probability that the sample mean waiting time is more than 6.5 minutes? (d) Compare and comment on the probabilities in parts (a), (b) and (c). Include an explanation about the difference between the standard deviation and the standard error and the shape of the sampling distribution. Q6 A consumer research firm takes a sample of 64 people in order to estimate the mean number of hours per month that adults watch television. If the population mean is 90 hours with a standard deviation of 20 hours, (a) Explain how the Central Limit Theorem is used to determine the distribution of the sample mean in this question. (b) What is the probability that the sample mean exceeds 95 hours? (c) What is the probability that the sample mean is between 88 and 92 hours? Q7 The tax office reports that 55% of tax returns lead to a refund. If a random sample of 100 tax returns is taken, (a) What is the probability that the sample proportion exceeds 0.6? (b) What is the probability that the sample proportion is between 0.4 and 0.6?
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