# Probability Distributions

Problem 1
A. The random variable x has the discrete probability distribution shown here:
a.
b.
c.
d.
Find P (x ≥ 1)
Find P (x < -1) Find P (-1 < x ≤ 2) Find P (x < 1) B. Consider the probability distribution for the random variable x shown here: a. Find: μ, σ2, and σ (Hint: use Excel) b. Graph P (x) c. Locate μ and the interval μ ± σ on the graph (Hint: add vertical lines at each of these three points). d. What is the probability that x will fall within the interval μ ± σ (approximately, using the probabilities given in the table above)? C. According to the National Center for Education Statistics, 43.7% of all Washington students were eligible for free-or-reduced lunches in 2011. If x equals the number of students in a random sample of seven students who were eligible for free-or-reduced lunches in 2011, then the probability distribution of x is: a. Find the probability that more than 4 students are eligible for free-or-reduced lunch in a random sample of 7 students. b. Find the expected value of x and interpret the result in 1-2 sentences. Problem 2 A. Suppose 20% of the individuals in a population of interest have hypertension. Coding a case of hypertension by 1, and not hypertension by 0, show that we can interpret the probability of a randomly drawn person as having hypertension as the expected value (or the expected number of cases per person). B. What is the expected number of cases of hypertension in a group of 50 people selected at random from that population? C. Now imagine that we have an automated blood pressure machine. ● The machine correctly gives a positive result (says the patient is hypertensive) for 80% of those who really do have hypertension. ● The machine incorrectly gives a positive result (says the patient is hypertensive) for 5% of those who don't have hypertension. ● As stated above, the rate of hypertension in the population of interest is 20%. a. What is the probability that an individual in the population is hypertensive given that he/she tests positive? b. What is the probability that an individual in the population is not hypertensive given that she/he tests negative? Problem 3 Suppose you work for the Seattle Public Schools and are tasked with planning a series of “Family Outdoor Learning” events for the first day of summer. The events would take place at local parks throughout the city. Based on the school district’s records for similar events in the past, you estimate that attendance will depend on the weather according to the following table. A. What is the expected (mean) attendance? B. Registration costs \$5 for each attendee, no matter the age. The costs will be \$0.75 per attendee for cleaning and parking attendants, plus \$8,000 for administration and facilities, plus \$3,000 for rental outdoor education equipment. Would you advise your supervisor to go ahead with the events? Why or why not? [Hint: Consider fixed costs, variable costs, expected total costs, and expected total revenues.] C. Suppose your supervisor decides to go ahead with the events, but a week before the events, you receive a gloomy weather forecast that changes the probabilities for the above weather conditions to 0.30, 0.50, 0.10, and 0.10, respectively. If the events are cancelled, then the school district will still owe one-half of the administrative and facility fees of \$8,000, plus a 15% cancellation fee for the rental outdoor education equipment. Would you advise your supervisor to cancel? Why or why not? Problem 4 Complete the following problems. A contractor has concluded from his experience that the cost of building a luxury home is a normally distributed random variable with a mean of \$500,000 and a standard deviation of \$40,000. a. What is the probability that the cost of building a home will be between \$380,000 and \$550,000? b. The probability is 0.7 that the cost of the building will be less than what amount? c. The contractor is bidding on a contract to build a luxury home. He wants to present a range of costs such that he is 80% sure the true cost of buildingthe home will be within this range. What is the shortest range of costs that he might bid?

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