Probability and Statistics

1. (5 pts) Meredith Watts, police chief of Shorewood, knows that the probability of any rape that occurs in the city being reported to the police is .5. If a rape is reported, the probability that someone will be arrested is .6. If someone is arrested, the probability of a conviction is .9. If a person is convicted of rape the probability of actually serving jail time is .7. Given that a rape occurs, what is the probability of a person serving jail time for the crime?
 
2. (10 pts) The governor of a large, northeastern state is trying to decide whether to veto a bill that she opposes that has been passed by the state legislature. As a politically minded actor, she has adopted the decision rule that if the bill has a probability greater than .5 of becoming a law even if she vetoes it, then she will sign the bill despite her opposition (and no doubt share credit for its passage). There are two ways in which the bill may become law over her veto. First, the state legislature may overturn her veto by voting resoundingly for passage. The governor estimates the probability of this outcome to be .3. The second way is through a referendum involving the voters of the state. If she vetoes the bill, and the legislature does not pass it over her veto, she further estimates that the probability of a referendum movement occurring is .9 (a powerful interest group supports the bill). She further estimates that the probability that the referendum movement will obtain the required number of signatures to qualify for a vote of the citizens of the state is .75.  Finally, she estimates that given a referendum, the voters have a 50/50 chance (P = .5) of approving the bill. Based on the governor’s decision rule, should she veto the bill? 
 
3. (5 pts) The average time a visitor spends at the River Park Art Exhibit is 62 minutes. The standard deviation is 12 minutes. If a visitor is selected random, find the probability that he or she will spend at least 82 minutes at the exhibit.
 
4. (5 pts) A Channel 4 News story claims that the city environmental preservation unit pays too much for routine automobile maintenance of the vehicle fleet. The average (mean) amount paid by the unit is $104.15 with a standard deviation of $9.71 (normal distribution).  As an initial step, the unit head decides to investigate the most expensive 25% of the routine maintenance bills.  According to this criterion, above what dollar amount should bills be investigated?
 
5. (10 pts) A recent study of 25 commuters showed that they spent an average of $18.23 on public transportation per week. The standard deviation of the sample was $3.00. Find the 95% confidence interval of the true population mean. Assume the variable is normally distributed. (10 points)
 
Hypothesis Test (Use the FIVE Steps)
 
6. (10 pts) A maker of frozen meals claims that the average caloric content of its meals is 800, and the standard deviation its meals (population) is 25. A researcher tested 12 meals and found that the average number of calories was 873. Is there enough evidence to reject the claim at a = 0.05? How about a = 0.02?
 
7. (10 pts)  A college professor claims that the average cost of a paperback textbook is greater than $27.50. A sample of 50 books has an average cost of $29.30. The standard deviation of the sample is $5.00. Find the P-value for the test. On the basis of the P-value, should the null hypothesis be rejected at a = 0.05?
 
8. (10 pts) The police department of Youbet, Michigan, is considering the purchase of new radar guns.  On the average, the guns presently used by the department can detect a speeding vehicle at a distance of 2217 feet.  In order to demonstrate the new radar guns, the manufacturer has allowed the Youbet police department to use the guns for a short period of time.  A random sample of 20 days in action shows that the new guns can detect a speeding vehicle at 2230 feet (mean) with a standard deviation of 72 feet. In your opinion, should the department purchase the new radar guns?
 
I need a hundred percent!