Final Exam Review Guide
Math 150, FALL 2018
Directions: The exam will consist of 8 free response questions and will take the entire class (2 hours and 5 minutes). All questions will be related to 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. You will need all your statistical tables for the final Exam (Table A, C, D and E).
Keep in mind that half of the exam will focus on hypothesis testing and confidence intervals while the other half of the exam will focus on all the earlier material such as probability, correla-tion/regression, normal and sampling distributions, and descriptive statistics.
Question 1) For each of the following studies or surveys identify as least one source of potential bias. (a) You want to know people’s opinions about changing the electoral college system. You ran-domly generate 1000 phone numbers and call them between the hours of 12pm and 2pm, and ask if they like the idea of changing the electoral college system.
(b) You are interested in studying college students’ views on online textbooks for college courses. You go out to a college campus and decide to ask every fifth person that walks out of the math building there the following question Would you support having online only textbooks for your courses considering that textbooks are massively overpriced and online textbooks could reduce this costs by a lot, ultimately saving you a ton of money?”
(c) A city council wants to know how many businesses don’t use proper fire alarms in their build-ings. To study this they randomly choose 25 local businesses and send fire officials with city safety inspectors to ask business owners if they use proper fire alarms in their buildings.
(d) A person wants to know what genre of movie is most popular in America. They take a poll by asking all their friends to tell them their favorite genre of movie.
Question 2) Suppose that you are doing psychological research and are studying how long it takes adults to solve a series of puzzles. You find that the completion times are normally distributed with an average of 29 minutes and a standard deviation of 6 minutes.
(a) Find the probability that a randomly selected adult takes between 25 and 35 minutes to finish these puzzles.
(b) If a person is at the 70th percentile (ie they take longer than 70% of other adults) how long do they take to solve these puzzles?
(c) Find the probability that a randomly selected adult takes more than 30 minutes to finish these puzzles.
(d) Find the probability that in a group of 4 randomly selected people all of them take more than 30 minutes to finish these puzzles.
(e) Find the probability that in a group of 8 randomly selected adults their average finishing times is above 30 minutes.
(f) Find the probability that in a group of 20 randomly selected adults the total time it takes them to finish is above 600 minutes.
Question 3) Consider the following probability experiment.
I get two fair six sided dice. I roll both of them. If I roll two even numbers I record the largest one. If I roll an even number and an odd number I record the even number. If I roll two odd numbers I record the largest one.
(a) Write down a probability model for this experiment. That means write down all the possible outcomes and their associated probabilities. (Remember: outcomes here would be what I actually record)
(b) Calculate the probability you record an odd number.
(c) Calculate the probability you record a number greater than 3.
(d) Calculate the probability you record an odd number and a number greater than 3. (e) Calculate the probability you record an odd number or a number greater than 3. (f) Calculate the probability you record a 4 given that you rolled a 4 on the first dice.
Question 4) I take a random sample of 12 people at a large company and record their salaries in thousands of dollars per year. To do this random sample I divide the company into its four major departments and randomly sample 3 people from each department. Here are my results. 43,48,53,54,57,65,72,75,81,88,92,175
(a) What type of sampling was used to collect your 12 data points? (b) Give the 5 number summary for this data.
(c) Draw a boxplot for this data. Describe the shape of the distribution. (d) Are there any outliers in this data set? Why or why not?
Question 5) I take a cooler full of soda out to the beach. Inside the cooler I have 14 cans of Pepsi, 10 cans of Sprite, and 9 cans of Mountain Dew. Suppose I randomly pull out 5 cans from the cooler without looking.
(a) What is the probability that I select all Pepsis?
(b) What is the probability that I select 2 cans of Pepsi, 2 cans of Sprite and 1 can of Mountain Dew?
(c) What is the probability that I select exactly 2 cans of Pepsi?
(d) What is the probability that I don’t select any cans of Mountain Dew? (e) What is the probability that I select at least one can of Mountain Dew?
Question 6) The chance that an adult American uses Facebook is 68%. Suppose you take a group of 10 adult Americans. Suppose each adult’s use of Facebook is independent.
(a) Find the chance that exactly 5 of the adults use Facebook. (b) Find the chance that less than 2 of the adults use Facebook. (c) Find the chance that at least 2 of the adults use Facebook.
Question 7) Suppose I’m interested in studying memory. To do this I bring a large group of people and show them a list of 50 words. Then I ask them to memorize the words over the course of 20 minutes. During the next 8 hours I periodically test these people and see how many words they can remember. I get the following regression line wherey represents the amount of words they remember andx represents the number of hours since they first saw the list.
ˆ
y=−6x+ 49 r=−.74
(a) What is the value of the slope? Interpret this value in context of the problem. (b) What is the value of the intercept? Interpret this value in context of the problem. (c) Calculate the coefficient of determination. Interpret this value in context of the problem. (d) How many words would you predict someone could remember 5 hours after seeing the list? Would you trust this prediction?
(e) One person you test is able to remember 12 words from the list after 7 hours. According to your model does this person have an above or below average memory? Why?
Question 8) Suppose you are interested in the relationship between age of a Nissan Versa and its value. To do this you take a random sample of 7 Nissan Versas, record their age in years and their value in dollars. You get the following data:
Age Value (in dollars)
0 16000
1 12800
3 8200
5 5200
6 4200
8 2700
10 1700
(a) Create a scatterplot for this data.
(b) Calculate the correlation coefficient for this data. Is their enough evidence for an association? (c) Construct the least squares regression line.
(d) Interpret the slope and intercept of your LSR.
Question 9) Suppose there is a slot machine that has five wheels. The first four wheels each have one of five images on it, a heart, a club, a spade, a diamond or a dollar sign. The last wheel has only 4 images, 3 X’s and a Jackpot. The has the following pay out system.
4 Dollar Signs and Jackpot, Win 700 dollars 4 Diamonds and Jackpot, Win 500
3 Dollar Signs, 1 Diamond and Jackpot, Win 70 dollars 1 Dollar Sign, 3 Diamonds and Jackpot, Win 50 dollars Otherwise, lose 1 dollar
(a) What is the probability that you win money on any one play of this machine? (b) What is the probability that you will lose on 10 plays of the machine in a row?
(c) Calculate the expected value of this slot machine. Interpret this value. Is this a good or bad game for you to play?
(d) Interpret the expected value for this game if you played the game 1000 times.
(e) What should be the pay out for 3 Dollar Signs and a Jackpot for the game to be fair?
Question 10) You work for your favorite drink company. You are in charge of redesigning their logo. After making some changes you want to check that a majority of people like the new logo. You create a focus group of 2000 people and aks them if they like the redesign. 1080 say they like the redesign.
(a) Carry out a hypothesis test at α = .05 to decide if the majority of consumers will like the redesign. Make sure to confirm all necessary conditions.
(b) Build and interpret a 95% confidence interval for the true proportion of people who like the redesigned logo.
(c) Describe what it would mean to make a Type I error in this situation.
Question 11) Suppose you work for an advertising company that is creating an ad for the re-lease of a new product. Your company creates two ads, one comedic ad and one dramatic ad. You want to know if people respond differently to these two ads. To do this you create two focus groups each of 200 randomly chosen adults. To construct these focus groups you make sure that you sample 50 adults from age 18−25, 50 adults from age 25−45, 50 adults from age 50−65 and 50 adults age 65+. You show the first group the comedic ad and 134 of them report that they would buy your new product. You show the other 200 randomly chosen adults the dramatic ad and 112 of them report that they would buy your new product.
(a) Carry out a hypothesis test at α = .05 to test the claim that people respond differently to the two ads. Make sure to confirm all necessary conditions. State your conclusion in context of the problem. (b) If you were worried about falsely claiming there was a difference between the two
Question 12) Suppose I want to study the online homework scores for large calculus lectures at a local university. For these online homeworks (which are graded on a scale from 0 to 100), the students have as much as time as needed and can resubmit their answers as many times as they want. This means students most of the time get very high scores. After some research you find that the average homework score is 92 with a standard deviation of 21
(a) What shape do you expect this distribution to have? Why?
(b) Suppose now I take a sample of size 50. Describe the distribution of the sample mean.
(c) What is the probability that in my sample the average homework score was between 90 and 95? (d) What is the probability that in my sample the average homework score was below 85?
(e) If an individual reported that they had a homework score of 70 would you consider this likely, unlikely, impossible, or you can’t determine from the information given?
Question 13) Suppose that a car company claims their new compact car gets 30 miles per gal-lon. A consumer advocacy group worries that this new compact car gets less miles per gallon than this. They take a random sample of 15 of these cars (note: the company has produced around 2000 of these new cars) and test their fuel efficiency. They get the following data, in miles per gallon. 27.4,27.5,28.2,28.3,28.8,29.0,29.4,29.5,29.7,29.8,30.0,30.1,30.2,30.6,31.3
(a) Carry out a hypothesis test at α = .05 to see if there is evidence for the advocacy group’s claim. Make sure to confirm all necessary conditions.
(b) Build and interpret a 95% confidence interval for the average miles per gallon that this type of car gets.
(c) If the advocacy group is worried about allowing the car company to continue falsely advertising that their cars get more than 30 miles per gallon should they retest at α=.01 orα =.1?
Question 14) Suppose you are interested in measuring the accuracy of a kicker in the NFL for extra point attempts. You watch the kicker attempt 46 extra point attempts (you may assume the attempts are random and that there are many attempts to choose from) and the kicker makes 40 of these attempts.
(a) Build a 90% confidence interval for the true proportion of how often this kicker makes extra point attempts. Make sure to use an apporpriate method and confirm all necessary conditions. (b) Using your interval would you consider it be likely, unlikely, or impossible that the true pro-portion is 93%?
Question 15) Suppose you are interested in studying if caffeine decreases reaction time. To do this you take a random sample of 10 people. You bring them to your lab one day and give them a reflex test. The reflex test is done by putting two buttons in front of them, one that says red and one that says blue. A color then flashes on the screen and the subject has to press the correct button. You then bring all the participants back the next day and give them caffeine pills. You wait twenty minutes for the pills to take effect and then give them the test again. You get the following results (note all values are in milliseconds):
Subject Day 1 (Control) Day 2 (Caffeine)
1 270 240
2 350 200
3 400 350
4 360 300
5 290 190
6 520 400
7 450 320
8 370 330
9 560 340
10 400 220
(a) Carry out a hypothesis test at α=.01 to test the researcher’s claim. Make sure to confirm all necessary conditions.
(b) Build a 99% confidence interval for the difference in reaction time. Make sure to write a sen-tence interpreting your interval.
(c) If someone criticized the study by saying that every person has a different reaction time, some are slower and some are faster, would this be a legitimate criticism of your experiment? If it is, how could you correct it?
(d) If someone criticized the study by saying that people may have improved their reaction time by simply getting to take the test a second time, rather than the caffeine, is this a legitimate criticism of your experiment? If iti s, how could you correct it?
Question 16) Suppose you own a coffee company with two different store locations. One is outside a university, the other next to a theater. You are interested in studying if the two locations get different amounts of customer traffic. To study this you randomly choose 30 times to visit the university location and watch the number of customers that come in for an hour and randomly choose 40 times to visit the theater location and watch the number of customers that come in for an hour. You get an average of 56 customers per hour with a standard deviation of 17 customers per hour for the university location and an average of 43 customers per hour with a standard deviaton of 25 customers per hour for the theater location.
(a) Carry out a hypothesis test at α = .05 to test this claim. Make sure to cofirm all necessary conditions.
Question 17) Suppose you are interested in studying if there is a relationship between region and favorite sport (out of the Big 3). You take a sample of 400 adults from across the US and record where they lived and their favorite sport. You get the following data:
Basketball Baseball Football
East Coast 32 51 48
West Coast 44 23 45
Midwest 38 49 70
(a) If you randomly chose someone from your study what is the chance they lived on the West Coast?
(b) If you randomly chose someone from your study what is the chance that lived in the Midwest and Baseball was their favorite sport?
(c) If you randomly chose someone from your study what is the chance that they lived on the East Coast or that Football was their favorite sport?
(d) If you randomly choose someone from your study what is the chance that Basketball was their favorite sport given that they were from the West Coast?
(e) Which region had the highest rate of choosing Football as their favorite sport?
Question 18) For this question use the data from Question 17
(a) Carry out a hypothesis test at α = .05 to test if there is a relationship between region and favorite sport. Make sure to check all necessary conditions and state your conclusion in context of this situation.
(b) Describe what it would mean to make a Type I error here. (c) Describe what it would mean to make a Type II error here.
Question 19) Suppose you work for a large company. You are interested studying customer sat-isfaction. From previous research in 2012 you had the following breakdown in terms of customer satisfaction.
Very Satisfied 35%, Satisfied 40%, Neutral 15%, Not Satisfied 10%
To study this, you take a sample by emailing out a survey to all of your customers (your com-pany has around 50000 customers) and ask them how satisfied they are with your comcom-pany. You get 200 responses.
Very Satisfied: 58, Satisfied: 88, Neutral: 38, Not Satisfied: 16
(a) Test at α = .05 to see if customer satisfaction with your company has changed since 2012. Make sure to check all necessary conditions and state your conclusion in the context of this situa-tion.
(b) What are two forms of bias with this study?
Question 20) Suppose that a food truck owner is investigating whether different locations around Los Angeles earn him different average amounts of money per day. To do this he parks his truck 10 times in each of the following locations and records his profit in that day. He gets the following results:
Near the Beach: 430,540,630,650,700,720,720,830,870,940
Near a University: 440,550,570,600,610,650,680,730,780,820
Near Hollywood: 640,730,760,800,820,850,900,910,940,1030
(a) Test at α = .05 to see if there is evidence that at least one of the locations earns him a different average profit than the other locations. Make sure to check all necessary conditions and state your conlcusion in the context of this situation.
