Suggested Exercises 3
Math 150, Professor Mitchell
Directions: In this document you will find fifteen questions I have written for your practice along with a number of suggested exercises from your textbook. I would recommend you attempt as many of these as possible to make sure you’re familiar with not only the concepts presented in each section but also the types of questions that are associated with these topics. If you have any questions send me an email, ask during class, or see me during office hours.
1) Suppose you’re at a casino playing a slot machine. The slot machine consists of 4 wheels each with 6 images (a dollar sign, a spade, a club, a diamond, a heart, and a jackpot). The slot machine has the following payouts:
4 Jackpots: Win 500 dollars 4 Dollar Signs: Win 250 dollars
3 Jackpots, 1 Dollar Sign: Win 100 dollars 3 Jackpots, 1 Diamond: Win 25 dollars Otherwise you lose 1 dollar
(a) Calculate the expected value for this game. Interpret this expected value. Is this a good or bad game to play?
(b) Suppose you play this game 1000 times. What would you expect to happen? (c) How should the 4 Jackpot payout be adjusted to make this game fair?
2) Suppose you’re looking at playing a lottery. The lottery consists of picking a 5 digit num-ber. The chance of guessing a number correctly is .1 and each choice is independent. Suppose further that each ticket for the lottery costs 1 dollar. The payouts for the lottery are:
All 5 Numbers Correct: Win 20,000 dollars 4 Numbers Correct: Win 800 dollars Otherwise: Win nothing
(a) Calculate the probability that you win money on a single play of this lottery.
(b) Calculate the expected value of this game. Write a sentence interpreting this expected value. Is this game good or bad?
(c) Suppose you buy 100 tickets. What is probability that you lose money on every ticket?
3) Suppose that a coffee distribution company uses a machine to automatically fill their bags of ground coffee. The machine dispenses into each bag an average of 16.2oz with a standard deviation of .3oz. Assume that the amount of coffee the machine puts into a bag is normally distributed. Further suppose the company advertises each bag will have 16 oz of coffee.
(a) Find the chance a single bag has less than the advertised 16oz.
(b) Find the chance that in a sample of 4 bags all the bags have less than 16oz. (c) Find the chance that in a sample of 4 bags the average amount is less than 16oz.
(d) Suppose a sample of 100 bags turns up an average amount less than 16oz. What might you conclude about the machine?
4) I want to study how much American adults at age 30 have in savings. To do this I go on-line and find a study from 2012 that says that the average adult at age 30 has 13000 dollars in savings with a standard deviation of 9500 dollars.
(a) What is the most likely distribution for the amount of savings 30 year old adult Americans have? Why?
(b) Suppose my sample was of size 100. Describe the sampling distribution of the sample mean for samples of this size. How do you know it has the shape you claimed?
(c) What is the probability that if I repeated my sample of 100 people I would get an average below 12000 dollars?
(d) Would it be unsual for your sample of 100 people to come back with average above 16000 dollars? Why or why not?
5) Suppose that a company is studying their monthly profits. They know that their daily profit has an average of 550 dollars per day with a standard deviation of 340 dollars per day. The company needs to make a monthly profit of 13500 dollars to break even. Calculate the probability that make over this break even amount in a month (ie more than 13500).
6) Suppose you’re studying the bake time for your favorite type of brownies. On the box it adver-tises that it only takes 25 minutes to bake. You think that it actually takes longer. You take a random sample of 20 boxes of these brownies and find that the average bake time for your sample is 27 minutes with a standard deviation of 4 minutes.
(a) Write down appropriate hypotheses to test the claim that the bake time is actually longer than advertised.
(b) Explain what it would mean to fail to reject the null hypothesis in this situation. (c) Explain what it would mean to make a Type II error in this situation.
7) Suppose that you’re interested in studying the amount of time ten year old children in the US spend outdoors playing per day during the summer. You want to know if this number has changed since a 2010 study that reported the average child spends 45 minutes per day outside during the summer. To do this you take a random sample of 55 ten year old children and find that their average time was 33 minutes with a standard deviation of 17 minutes.
(a) Write down appropriate hypotheses to test the claim that amount of time 10 year old children spend outdoors playing has changed since 2010.
(b) Explain what it would mean to reject the null hypothesis in this situation. (c) Explain what it would mean to make a Type I error in this situation.
8) Suppose I am interested in studying the average number of cups of coffee El Camino faculty drink during the school week. I randomly select 40 faculty members and ask them to record how many cups of coffee they drink during the week. At the end of the week I calculate that they drank an average of 7.3 cups of coffee with a standard deviation of 4.8 cups of coffee.
(a) Construct a 98% confidence interval for the average number of cups of coffee that El Camino faculty drink during a school week. Write your interval as both a point estimate with a margin of error, and as an interval. Make sure to confirm all necessary conditions. Then write a sentence interpreting your confidence interval.
(b) Another faculty member has carried out a similar study and claims the average number of cups of coffee drank in a week is 5. Would you consider this likely, unlikely, or impossible? Why? (c) Would you feel comfortable using your interval to estimate the average number of cups of coffee that El Camino students drink during a school week? Why or why not?
9) Suppose I want to study the GPA’s of El Camino students based on their intended field of study. I randomly select 20 students who have indicated that they want to eventually major in psychology. Their GPA’s are listed in increasing order below.
2.76,2.84,2.88,2.93,2.94,2.97,2.98,3.02,3.05,3.05
3.08,3.09,3.10,3.13,3.16,3.24,3.28,3.35,3.37,3.46
(a) Carry out a hypothesis test at α = .01 to test the claim that students in psychology have a higher than 3.0 GPA. Make sure to confirm all necessary conditions and interpret your conclusion in the context of the problem.
(b) Describe what it would mean to make a Type I error in this situation. (c) Describe what it would mena to make a Type II error in this situation.
10) Suppose you work for a local private school with about 700 students. You want to run an advertisement that says, ”Our students score better on the ACT!” To research this you look up the national average for the ACT and find it is 21 points. You then take a random sample of 32 of your own students and find out their average ACT score was 23.2 with a standard deviation of 4.7 points.
(a) Carry out a hypothesis test at α=.05 to see if you can run your advertisment. Make sure to check all necessary conditions and interpret your conclusion in context of this situation.
(b) If you were worried about running a false advertisement would you want to retest at α= .01 or.1?
11) Suppose you are conducting consumer research for a bar. You want to study if male customers tip more if their bartender is female as opposed to male. You take collect a random sample of 12 men and ask them to come to a bar two nights in a row. On the first night they have a male bartender. On the second night they have a female bartender. You record the tips as percentages at the end of the night. You get the following data:
Subject Night 1 Night 2
1 18 21
2 15 22
3 25 35
4 16 30
5 15 19
6 20 25
7 10 30
8 20 36
9 10 22
10 20 33
11 20 26
12 12 30
(a) Carry out a hypothesis test atα=.01 Make sure to confirm all the necessary conditions. State your conclusion in context of the situation.
(b) Construct a 95% confidence interval for the difference between tip percentages for male and female bartenders with male customers. Write a sentence explaining your interval.
12) Suppose a professor wants to know if his morning Calculus class (a large lecture with around 250 people) performs differently than his afternoon Calculus class (also a large lecture, but with around 350 people). On the next exam he randomly selects 12 students from his morning class and 12 students from his afternoon class and records their scores. He gets the following data:
Morning Class: 56,63,64,71,74,76,78,80,83,83,87,92 Afternoon Class: 55,68,75,76,78,84,86,87,88,88,90,95
(a) Carry out an appropriate hypothesis test at α = .05 to test the professor’s claim. Make sure to check all necessary conditions. State your conclusion in context of the problem.
(b) Using the same data, construct a 95% confidence interval for the average difference between the morning class and the afternoon class. Make sure to interpret your interval.
13) Suppose a drug company is testing a new weight loss pill. They would like to claim that the pill can help you lose more than 5 pounds in 2 weeks. To test this they gather a random sample of 10 adults. They record the initial weight of each adult, give them a 2-week supply of the weight loss pills, and then record their weights again after 2 weeks. They get the following data:
Subject Weight Before Weight After
1 219 200
2 188 189
3 230 227
4 198 184
5 250 237
6 265 256
7 174 165
8 300 293
9 254 247
10 176 172
(a) Carry out a hypothesis test at α = .05 to see if there is enough evidence for the company to run their advertisement. Make sure to check all necessary conditions. State your conclusion in the context of the problem.
(b) Describe what it would mean to make a Type II error in this situation.
(c) Construct a 95% confidence interval for the average weight loss from this pill. Using your in-terval could you justify that the pill helps you lose weight? Why or why not?
14) Suppose that a researcher at a large university wants to examine the amount of time stu-dents spend studying per night. Specifically, the researcher wants to know if STEM stustu-dents at this university (there are around 10,000 STEM students) study more per night than non-STEM students (there are around 20,000 non-STEM students). To do this the researcher randomly sam-ples 150 STEM students and 200 non-STEM students. The reseracher finds that the average for the STEM group is 125 minutes per night with a standard deviation of 50 minutes while for the non-STEM group the average was 95 minutes with a standard deviation of 55 minutes.
(a) Carry out an appropriate hypothesis test at α = .05 to test the professor’s claim. Make sure to check all necessary conditions. State your conclusion in the context of the problem. (b) Describe what a Type I error would be in this situation.
15) Suppose a researcher is studying the average age at which women have their first child. Ac-cording to previous research done in 2000 the average age of a woman when she had her first child was 24.9 years old. The researcher wants to to know if this value has changed. To study this the researcher randomly samples 250 women who have children and records the age they were when they had their first child. The researcher gets an average in their sample of 26.1 years old with a standard deviation of 6.8 years.
(a) Carry out an appropriate hypothesis test at α = .05 to test the claim that the average age that women have their first child has changed since 2000. Make sure to check all necessary condi-tions. State your conclusion in the context of the problem.
(b) Construct a 95% confidence interval for the average age that women have their first child. Make sure to interpret your interval.
