4.13 If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. (a) The table below gives the probability of each color for a randomly chosen plain M&M: Color Brown Red Yellow Green Orange Blue Probability 0.3 0.2 0.2 0.1 0.1 ? What must be the probability of drawing a blue candy? (b) The probabilities for peanut M&M’s are a bit diﬀerent. Here they are: Color Brown Red Yellow Green Orange Blue Probability 0.2 0.1 0.2 0.1 0.1 ? What is the probability that a peanut M&M chosen at random is blue? (c) What is the probability that a plain M&M is any of red, yellow, or orange? What is the probability that a peanut M&M has one of these colors? 4.22 An automobile manufacturer buys computer chips from a supplier. The supplier sends a shipment containing 5% defective chips. Each chip chosen from this shipment has probability 0.05 of being defective, and each automobile uses 12 chips selected independently. What is the probability that all 12 chips in a car will work properly? Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: Number of cars X 0 1 2 3 4 5 Probability 0.09 0.36 0.35 0.13 0.05 0.02 Find the mean and standard deviation of number of cars per household. 4.40 The academic motivation and study habits of female students as a group are better than those of males. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures these factors. The distribution of SSHA scores among the women at a college has mean 120 and standard deviation 28, and the distribution of scores among men students has mean 105 and standard deviation 35. You select a single male student and a single female student at random and give them the SSHA test. (a) Explain why it is reasonable to assume that the scores of the two students are independent. (b) What are the mean and standard deviation of the diﬀerence (female minus male) of their scores? (c) From the information given, can you ﬁnd the probability that the woman chosen scores higher than the man? If so, ﬁnd this probability. If not, explain why you cannot. (d) Regardless of your answer in (c) find the probability that the woman chosen scores higher than the man. 4.38 Laboratory data show that the time required to complete two chemical reactions in a production process varies. The ﬁrst reaction has a mean time of 40 minutes and a standard deviation of 2 minutes; the second has a mean time of 25 minutes and a standard deviation of 1 minute. The two reactions are run in sequence during production. There is a ﬁxed period of 5 minutes between them as the product of the ﬁrst reaction is pumped into the vessel where the second reaction will take place. What is the mean time required for the entire process? 4.39 The times for the two reactions in the chemical production process described in the previous exercise are independent. Find the standard deviation of the time required to complete the process. 5.11 “How would you describe your own personal weight situation right now—very overweight, somewhat overweight, about right, somewhat underweight, or very underweight?” When the Gallup Poll asked an SRS of 1005 adults this question, 51% answered “about right.” (Lydia Saad, “Hold the gravy! Six in ten Americans want to lose weight,” www.gallup.com/poll/releases/, November 21, 2001.) Suppose that in fact 51% of the entire adult population think their weight is about right. (a) Many opinion polls announce a “margin of error” of about ±3%. What is the probability that an SRS of size 1005 has a sample proportion ˆp that is within ±3% (±0.03) of the population proportion p = 0.51? (b) Answer the same question if the population proportion is p = 0.06. (This is the proportion who told Gallup that they were “very overweight.”) How does the probability change as p moves from near 0.5 to near zero? 1.61 In 2003, scores on the math part of the SAT approximately followed a normal distribution with mean 519 and standard deviation 115. (a) What proportion of students scored above 500? (b) What proportion scored between 400 and 600? 4.23 A string of holiday lights contains 20 lights. The lights are wired in series, so that if any light fails the whole string will go dark. Each light has probability 0.02 of failing during a 3-year period. The lights fail independently of each other. What is the probability that the string of lights will remain bright for 3 years? 4.25 Government data show that 27% of employed people have at least 4 years of college and that 14% of employed people work as laborers or operators of machines or vehicles. Can you conclude that because (0.27)(0.14) = 0.038 about 3.8% of employed people are college-educated laborers or operators? Explain your answer. 8.9 A question in a Christmas tree market survey was “Did you have a Christmas tree last year?” Of the 500 respondents, 421 answered “Yes.” Give a 90% conﬁdence interval for the proportion of Indiana households who had a Christmas tree this year.