1. (12 pts) An acquaintance of mine likes to bet on football games. A few years ago he was bragging to me about how well he was doing. Specifically, he told me that he had bet on about 100 games and had won at least 65 of his bets. (This is completely true!) Assuming that he was telling me the truth, does this result suggest that this person has a knack for picking games or could it be attributed to luck? Answer this question by computing the probability that someone betting on 100 games could have won 65 or more just by luck. Assume that someone randomly placing bets will have a 50-50 chance of winning any one bet placed. 2. The distribution of incomes of employees of a large corporation (rounded to the nearest $10,000), together with years of service at this corporation, are shown in the table below. Years 0 5 10 $30,000 .18 05 .02 Income $40,000 .09 .08 .03 $50,000 .13 .12 .05 $60,000 .05 .10 10 a. (8 pts.) Find the mean and standard deviation of incomes for all employees of this company. (Hint: you will likely find the arithmetic slightly easier if you work in units of $1000's.) b. (6 pts.) Find the mean salary of all employees that have been with this company for 5 years. c. (5 pts.) Are salary level and time of employment independent? Justify your conclusion. 3. (1 1 pts.) A door-to-door salesperson has a 30% chance of making a sale at the first house that they visit. After the first house, this person loses some of their enthusiasm so that all at all subsequent houses their chance of making a sale drop to 25% What is the probability that this person has to visit more than 5 houses to make a sale? 4. (17 pts.) A gas station sells diesel fuel, unleaded gasoline, and super unleaded. The cost per gallon of diesel is $1.00, the cost per gallon of unleaded is $1.10, and the cost per gallon of super unleaded is $1.30. Daily sales of diesel fuel have been normally distributed with a mean of 500 gallons and a standard deviation of 100 gallons. Daily sales of unleaded have been normally distributed with a mean of 2000 gallons and a standard deviation of 400 gallons. Finally daily sales of super unleaded have been normal with a mean of 1000 gallons and standard deviation of 300 gallons.Sales of diesel are independent of sales of the other two fuels. The sales of unleaded and super unleaded, though, are highly correlated, with the correlation coefficient being 0.7. On any one day, what is the probability that the total sales for this gas station exceed $4500? 5. A magazine has, in a letter to potential advertisers, claimed that the average income level of people reading the magazine is $79,000. This claim is obviously being made to convince the potential advertisers that it will be to their advantage to advertise in this magazine. a. (15 pts.) A random sample of 90 readers of this magazine is taken, and it is found that their average income is $76,400. Does this result cast doubt upon the accuracy of the magazine's claim? Answer this question by computing the probability of a random sample of 90 readers having an average income of $76,400 or less if the true mean of incomes is really $79,000. Make the assumption that the standard deviation of incomes of the readers of this magazine is $10,000. b. (4 pts.) In practice, the distribution of incomes is almost always skewed to the right, due to the influence of a small number of very high incomes. Will this effect adversely affect the accuracy of your calculations in part (a)? Briefly explain. 6. (12 pts.) A company that manufactures auto parts has a large number of machines producing these auto parts. Most of these machines were designed and built in 2010. These 2010 models are slowly being phased out in favor of a newer model of auto parts-producing machine that on 15 came out in 2018. As of now, 75% of the company's machines are 2010 models and 25% are the newer 2018 models. Of the parts produced by the 2010 models, 20% are defective. The 2018 models only produce at a 10% defective rate. Bluetoo A container holds a number of auto parts that have been produced by one machine, though it is not known which model it was. One of these parts is tested and is not defective. What is the probability that the machine that produced it was a 2010 model? 7. (10 pts.) A consumer finance company makes 5-year auto loans. They have found that of loans made to their most creditworthy customers, 50% go the entire five years without problems, 12% go the five years with at least one missed payment, 28% are paid off early, and 10% are defaulted upon. Twenty-give loans have recently been originated. If past experience on loans is still valid, what is the probability that of these 25 loans 12 will go the five years without problems, 5 will miss at least one payment, 5 will pay off early, and 3 will default