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Chapter 7 Review AP Statistics Vocabulary & Formulas: discrete & continuous random variables, formulas for discrete/continuous random variables probability distributions, formulas for mean and standard deviation of discrete/continuous random variables, formulas for mean, variance, and standard deviation of discrete random variables 1. Inspecting meat is a very important part of the process of ensuring healthy food. So that meat inspectors do not fall into a pattern of inspection, they use a 4-sided die to decide whether or not to sample the meat from a particular "side" of beef. The meat inspector rolls the die and if it comes up a 4, a sample is taken of that meat. Forty sides of beef are available for possible inspection. Define random variable x = number of sides selected for inspection out of the 40 available. a) What is the mean of the random variable x? Notice that this is an example of a binomial distribution: mean: x np 40(.25) 10 b) What is the standard deviation of the random variable x? standard deviation: x np(1 p) 2.7386 2. From your own experience, give an example of a continuous random variable and a discrete random variable. In a few sentences, explain why each is categorized as continuous or discrete. Explanation for continuous random variable must include the idea of an interval of values on a number line. The variable you selected should be a quantity that is measured. This should contain a few sentences. Explanation for discrete random variable must include the idea of isolated values on a number line. The variable you selected should be a quantity that is counted. This should contain a few sentences. AP Statistics Page 1 of 5 TEKS: STAT.1A-G, STAT.2E & F, STAT 5A-D 3. One common method used to forecast the results of elections is to take an "exit poll," asking people who they supported as they come out of the voting place. Suppose Smith and Jones are running for City Dogcatcher. In point of fact, 60% of the voting public in the neighborhood of the polling place support Smith. Assuming that the candidate of choice for a voter leaving the polling place is independent of the candidate of choice for the next person to leave, what is the probability that the first 5 voters to leave the polling place would have this sequence for candidate of choice? Smith, Smith, Smith, Jones, Smith Assuming independence, P(Smith,Smith,Smith,Jones,Smith) = (.6)(.6)(.6)(.4)(.6)=0.05184 4. For a variable z that has a standard normal distribution, a) What is the probability that z < - 2.45 ? 0.0071 b) What is the probability that z < + 1.67 ? 0.9525 c) What is the probability that z is between - 1.43 and + 1.43 0.8472 d) What value of z separates the smaller 10% of the standard normal distribution from the larger 90%? z = -1.28 e) What values of - z and + z separate the middle 95% of the standard normal distribution from the extreme 5%? z = -1.960 to z = 1.960 AP Statistics Page 2 of 5 TEKS: STAT.1A-G, STAT.2E & F, STAT 5A-D 5. While playing Monopoly, Laura estimated the probabilities of the non-zero rents according to the following probability distribution: x P( x) $10 0.20 $20 0.20 $50 0.40 $130 0.20 Consider the random variable x = dollar amount in rent in a Monopoly roll. a) If the table above specifies the probability distribution f ( x) , what is the mean of the random variable x? x x P(x) $52 b) If the probabilities are associated with the outcomes as in the table above, what is the standard deviation of the random variable x? x (x x ) 2 P( x) $42.14 6. In a major study by the statistics classes at Jefferson High School, parking spaces were examined for compliance with the requirement to put money in the meters. Overall, the students found that 80% of metered parking places had meters that had not expired, and 20% had expired. If the traffic officer in charge of ticketing cars with expired meters, and starts at a random location in the City, what is the probability he or she will find an expired meter before the 3rd one? Again, think back to our probability rules from Chapter 6: P(expired before 3rd) = 1 - P(1st not expired and 2nd not expired) = 1 – (0.80)(0.80) = 0.36. AP Statistics Page 3 of 5 TEKS: STAT.1A-G, STAT.2E & F, STAT 5A-D 7. In a study of the reaction times of left- and right-handers' reaction times to tones delivered to the right ear, the right-handers' scores were approximately normally distributed with a mean of 240 milliseconds and standard deviation of 32 milliseconds. The mean score for right-handers was 210 ms. a) Sketch a normal distribution that describes left-handers' reaction times, and locate the mean reaction time for right-handers in this distribution. Sketch of left-handers’ reaction times with mean of right-handers marked at 210 ms. Mean of the distribution should be at 240. mean of right handers 100 200 240 300 b) About what proportion of left-handers’ reaction times would be "better" (i.e. smaller) than the mean reaction time for right-handers? xm ) s 210 240 P( z ) 32 Pz .0.94 P L R P ( z 0.1743 AP Statistics Page 4 of 5 TEKS: STAT.1A-G, STAT.2E & F, STAT 5A-D 8. In a certain very large city, the Department of Transportation (D.O.T.) has organized a complex system of bus transportation. In an advertising campaign, citizens are encouraged to use the new "GO-D.O.T!" system and head for the nearest bus stop to be transported to and from the central city. Suppose that at one of the bus stops the amount of time (in minutes) that a commuter must wait for a bus is a uniformly distributed random variable, T. The values of T run from 0 minutes to 30 minutes. a) Sketch the probability distribution of T. Density curve is a rectangle, length 20, height 1/30 (at left below). 0.05 0.05 12 20 20 b) What is the probability that a random commuter will spend more than 12 minutes waiting for GO-D.O.T? 1 PT 12 18 0.60 . Illustrated at right above. 30 AP Statistics Page 5 of 5 TEKS: STAT.1A-G, STAT.2E & F, STAT 5A-D