Free Response 1. lFX= the number oftelevisions in a randomly selected LLS. household. the probability distribution of'Xis given here. Number of'TVs nun-n Pmbabili m a] Explain 1.vhy this is a valid probability distribution. b} Is X a discrete or a continuous random variable? Explain. c] What is the probability that a randomly selected household has at least 2 televisions? 2. Refer to exercise 1. a} Make a histogram to display the probability distribution of'X. b]- llifalculate and interpret the expected value oth'. c] Calculate and interpret the standard deviation ot'X. 3. To test whether someone has extrasensory perception (ESPJ. choose ] of4 cards at random a star. wave. cross. or circle. fish the person to identiy the card without seeing it Do this a total ofEl] times and see how many cards the person identies correctly. Let X= number of correct identications. assuming that the person does not have ESP and isjust guessing on each card. a] Explain why Xis a binomial random variable. % El b} Find and interpret the mean ofX. c] Find and interpret the stande deviation of'X. .| El 4. Refer to Exercise 3. a] Use the binomial probability Formula to calculate P{X= 5}. Interpret this result in context. b} Alec makes 3 correct identications out ofl cards. Does this provide convincing evidence that Alec has ESP":J Find the probability that a person without ESP identies E or more cards correctly and use the result to answer the question. 5. Kristen likes to throw tennis balls for her dog Bucket. Let T = the time it takes for Bucket to chase down a ball, return to Kristen and drop the ball at her feet on a randomly selected throw. T is equally likely to take any value in the interval form 8 seconds to 15 seconds. a] Sketch a graph ofthe probability distribution of T. Be sure to include a vertical scale. b} Find the probability that, on a randomly selected throw, Bucket returns the ball within I3 seconds. c] Find the mean of T. d. Bigger animals tend to carry their young longer before birth. The length of'horse pregnancies from conception to birth varies according to a roughly normal distribution with mean 336 days and standard deviation E- days. a] Sketch a graph ofthe distribution oflengths of'horse pregnancies. Label the mean and the points that are I, 2. and 3 stande deviations From the mean. b} Use the ll-9539.? rule to estimate the percentage of'horse pregnancies that are longer than 342 days