Exam 3 Review Guide
Math 150, FALL 2018
Directions: The exam will consist of 6 free response questions. All questions will be drawn from the questions written below. All answers on the exam must either have work shown or attached explanations. You are allowed one sheet of notes written on a 8.5x11 sheet of paper. You are also allowed the use of a calculator. It is HIGHLY recommended that you have a calculator for this exam. Question 1) Suppose that the commute times for El Camino students is normally distributed with a mean of 37 minutes and a standard deviation of 10 minutes.
(a) Find the probability that a randomly selected El Camino student has a commute time above 45 minutes.
(b) Find the probability that in a random sample of 4 El Camino students ALL of them have a commute time above 45 minutes.
(c) Find the probability that in a random sample of 9 El Camino students the AVERAGE commute time is above 45 minutes.
(d) Find the probability that in a random sample of 16 El Camino students the total commute time for the group was above 640 minutes. (Hint: Convert this into a statement about an average) Question 2) I want to study how often El Camino students attend math classes. According to data from the school, students from the fall semester who are taking math classes attend class an average of 27.8 days with a standard deviation of 5.2 days.
(Note: The maximum possible number of days to attend was 30).
(a) What is the most likely shape of the distribution for how many days an El Camino student attends their math class during a semester? Why?
(b) Now suppose I take a random sample of 100 El Camino students from the fall semester who are taking math classes. Describe the sampling distribution of the sample mean for samples of this size. How do you know it has the shape you claimed?
Question 3) Suppose there is a slot machine with 3 wheels. On each wheel there is one of six images, a Jackpot, a Crown, a Strawberry, a Joker, a Dollar Sign, and a Lightning Bolt. The pay out structure is as follows:
3 Jackpots: Win 150 dollars
2 Jackpots, 1 Dollar Sign: Win 10 dollars 1 Jackpot, 2 Dollar Signs: Win 5 dollars Otherwise you lose 1 dollar.
(a) Construct a probability model for this machine.
(b) What is the probability that you win money on any one play of the slot machine?
(c) Calculate the expected value of this slot machine. Interpret this expected value. Is this a good or bad game to play in the long run?
(d) Interpret the expected value if you played this game 1000 times.
(e) What payout should be given for getting 3 Jackpots if you want the game to be fair? (f) What is the probability that you lose this game 10 times in a row?
Question 4) Suppose there is a milk jug toss game at a carnival. You must pay 5 dollars to play the game. If you play you get 5 throws. You know that you can make 40% of your throws. (Assume each throw is independent) The payouts for the game are as follows:
Make 5 Throws, Win a Big Prize valued at 35 dollars Make 4 Throws, Win a Medium Prize valued at 20 dollars Make 3 Throws, Win a Small Prize valued at 10 dollars (a) Construct a probability model for this game.
(b) What is the chance that you win when you play this game?
(c) What is the chance that you win a prize at this game three times in a row? (Assume each game is independent)
(d) What is the expected value of this game? Interpet this value. (e) Interpret this expected value if you played the game 50 times.
Question 5) In the year 2000 it was reported that the average amount of sleep an adult American gets per night was 7.2 hours. You want to know if that value has decreased. To study this you randomly sample 13 American adults and record the number of hours of sleep they get during a single night. You get the following data in hours:
4.7,5.3,5.8,6.0,6.2,6.4,6.8,7.0,7.1,7.2,8.5,8.6,9.1
(a) Carry out a hypothesis test at α = .05 to see if there is evidence that the average amount of time adult Americans sleep per night has decreased since 2000. Make sure to check all necessary conditions and state your conclusion in the context of this problem.
(b) Describe what it would mean to make a Type II error in this situation.
(c) Build a 95% confidence interval for the average amount of sleep adult Americans get per night. Make sure to interpret your interval.
(d) Would it be correct to say that 95% of American adults get an amount of sleep that lands in your interval from part (c)? Why or why not?
Question 6) Suppose I am studying the average amount of money adult Americans have saved in their retirement account at age 55. After conducting a study using a random sample, I build a 95% confidence interval. My interval is ($64,000,$100,000). Decide if each of the following are true or false interpretations of this interval.
(a) The average for my sample was $82,000.
(b) There is a 95% chance that the true average amount of savings for adult Americans at age 55 is between $64,000 and $100,000.
(c) 95% of American adults, age 55, have between $64,000 and $100,000 saved in their retirement account.
(d) It is impossible that the true average savings for Americans, age 55, is $60,000.
(e) It is likely that the true average amount of savings for an American adult is between $64,000 and $100,000.
Question 7) I want to know if gas prices are generally higher in Chicago, IL than in the rest of the country. (Note: There are approximately 500 gas stations around the greater Chicago area) Currently the nationwide average gas price is 3.12 dollars per gallon. To study this I drive around Chicago and sample the prices of every 4th gas station that I see. I get the following data:
3.00,3.02,3.06,3.09,3.11,3.11,3.13,3.14,3.15,3.17,3.18,3.18,3.19,3.20,3.22,3.24,3.24,3.25,3.27,3.28 (a) What type of sampling was used here?
(b) Carry out a hypothesis test atα=.05 to see if the average gas price in Chicago is higher than the national average. Make sure to confirm all necessary conditions.
(c) Build a 95% confidence interval for the mean price of gas in Chicago. Make sure to interpret your interval.
(d) Explain what it would mean here to make a Type I error in this situation.
Question 8) Suppose you work for a fast food company. You are interested in studying if peo-ple spend different amounts when going through the drive-thru versus actually ordering inside the restaurant. You take a random sample of 81 bills from the drive thru and find that the average amount spent was 9.38 with a standard deviation of 3.74. You also take a random sample of 61 bills from dine-in customers and find that the average amount spent was 10.79 with a standard deviation of 4.31.
(a) Carry out a hypothesis test at α =.05 to test this claim. Make sure to confirm all necessary conditions.
(b) Regardless of your answer to (a), construct a 95% confidence interval for the average difference in spending. Make sure to interpret your interval.
(c) If your company was worried about missing a difference between drive thru sales and order-in sales, should you retest atα=.01 or α=.10?
Question 9) Suppose Cheapo Airlines claims that their flights are on average more than $50 cheaper than their competitors. You want to test this. To do so you randomly select 12 different flights paths (ie starting city and destination city) and find the price that Cheapo Airlines offers and the price their competitors offer. You get the following data.
Flightpath Cheapo Airlines Competitors
1 232 268
2 201 272
3 305 355
4 117 175
5 97 142
6 310 371
7 233 287
8 113 161
9 75 143
10 100 152
11 135 200
12 80 142
(a) Carry out a hypothesis test to test the company’s claim atα =.01. Make sure to confirm all necessary conditions.
(b) Construct a 99% confidence interval for the true difference in average price between Cheapo Airlines and their competitors. Make sure to interpret your interval.
(c) Explain what it would mean to make a Type II error in this situation.
Question 10) Suppose a college student is interested in estimating the average starting salary for students who graduate with a Business degree and for students who graduate with an Economics degree. The student randomly samples 45 recently graduated students with a buisness degree and 32 recently graduated students with an economics degree. The student finds that the business stu-dents have an average starting salary of 47 thousand with a standard deviation of 8 thousand while the economics students have an average starting salary of 55 thousand with a standard deviation of 6 thousand.
Question 11) Suppose that Dish TV wants to study how many hours a week the average American spends watching television. They take a random sample of 150 of their subscribers and ask them how many hours they spend each week watching television. They get an average of 15.3 hours with a standard deviation of 4.6 hours.
(a) Would it be appropriate here to use this information to estimate the average number of hours Americans spend watching television? Why or why not? If not, what population could we make inferences about?
(b) Construct a 99% confidence interval for the true average of an appropriate population. Make sure to check all conditions and interpret your interval.
(c) If the data showed a right skew would this make us worried about the validity of our interval? Why or why not?
(d) If someone else reported that they also did a study and got a sample average of 16.3 hours would you consider this sample average likely, unlikely, impossible, or you can’t decide from this information?
(e) If a person told you that they personally watch around 18 hours of television each week would you consider this likely, unlikely, impossible or you can’t decide from this infromation?
Question 12) Suppose I want to study if professors who hand out candy to their classes get higher student evaluations than professors who don’t. I go to a large university. I randomly select 30 students from the professors who hand out candy and give them a short survey, asking them to rate their teacher on a scale of 1 to 10. I get an average of 8.3 with a standard deviation of 2.3. I then randomly select 50 students from the professors who do not hand out any sort of candy. I give them the same survey and get an average of 7.4 with a standard deviation of 2.8. (Note: For this problem you may assume that there were lots of students who had professors who gave out candy and lots of students who had professors that did not give out candy).
(a) Carry out a hypothesis test at α = .10 to test this claim. Make sure to check all necessary conditions.
(b) Regardless of your answer to (a), if you did decide to reject the null hypothesis, why would you not be able to necessarily conclude that handing out candy causes the professor to get higher student evaluations?
