Assignment 3
Due: November 9th (Monday) In class
Total points: 100
Instructions
Can hand in a paper copy of your assignment in class
Assignment
Problem set – Hand in
Assignment
1. Suppose that we have a sample space S = [E1, E2, E3, E4, E5, E6, E7], where E1, E2, …, E7 denote the sample points. The following probability assignments apply: P(E1) = .05, P(E2) = .20, P(E3) = .20, P(E4) = .25, P(E5) = .15, P(E6) = .10, and P(E7) = .05. Let
A = {E1, E4, E6}
B = {E2, E4, E7}
C = {E2, E3, E5, E7}
a. Find P(A), P(B), and P(C).
b. Find A ? B and P(A ? B).
c. Find A n B and P(A n B).
d. Are events A and C mutually exclusive?
e. Find BC and P(BC).
2. Forty-three percent of Americans use social media and other websites to voice their opinions about television programs (The Huffington Post, November 23, 2011). Below are the results of a survey of 1400 individuals who were asked if they use social media and other websites to voice their opinions about television programs.
Uses Social Media and Other Websites to Voice Opinions About Television Programs
Doesn’t Use Social Media and Other Websites to Voice Opinions About Television Programs
Female
395
291
Male
323
355
a. Show a joint probability table.
b. What is the probability a respondent is female?
c. What is the conditional probability a respondent uses social media and other websites to voice opinions about television programs given the respondent is female?
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d. Let F denote the event that the respondent is female and A denote the event that the respondent uses social media and other websites to voice opinions about television programs. Are F and A independent events?
3. Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a .90 probability of detecting a missile attack. Use a binomial probability distribution to answer the following questions.
a. What is the probability that a single detection system will detect an attack?
b. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
c. If three systems are installed, what is the probability that at least one of the systems will detect the attack?
d. Would you recommend that multiple detection systems be used? Explain.
4. Given that z is a standard normal random variable, compute the following probabilities.
Note: using excel might be easier – show your command
a. P(-1.98 = z = .49)
b. P(.52 = z = 1.22)
c. P(-1.75 = z = -1.04)
5. Given that z is a standard normal random variable, find z for each situation.
Note: using excel might be easier – show your command
a. The area to the left of z is .9750.
b. The area between 0 and z is .4750.
c. The area to the left of z is .7291.
d. The area to the right of z is .1314.
e. The area to the left of z is .6700.
f. The area to the right of z is .3300.
6. The American Automobile Association (AAA) reported that families planning to travel over the Labor Day weekend would spend an average of $749 (The Associated Press, August 12, 2012). Assume that the amount spent is normally distributed with a standard deviation of $225.
a. What is the probability that family expenses for the weekend will be less than $400?
b. What is the probability that family expenses for the weekend will be $800 or more?
c. What is the probability that family expenses for the weekend will be between $500 and $1000?
d. What are the Labor Day weekend expenses for 5% of the families with the most expensive travel plans?
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7. There has been a trend toward less driving in the last few years, especially by young people. From 2001 to 2009 the annual vehicle miles traveled by people from 16 to 34 years of age decreased from 10,300 to 7900 miles per person (U.S. PIRG and Education Fund website, April 6, 2012). Assume the standard deviation was 2000 miles in 2009. Suppose you would like to conduct a survey to develop a 95% confidence interval estimate of the annual vehicle-miles per person for people 16 to 34 years of age at the current time. A margin of error of 100 miles is desired. How large a sample should be used for the current survey?
8. How large a sample should be selected to provide a 95% confidence interval with a margin of error of 10? Assume that the population standard deviation is 40.