F21Exam3

Exam 3 STAT 100 Fall 2021 EXAM 3: Statistics 100 READ THE DIRECTIONS BELOW TWICE! This test is ALL multiple choice. Circle all answers on this exam and fill in the corresponding bubble on your orange scantron . All questions have exactly one answer. If you circle/bubble in more than one answer, you will automatically be marked wrong. Make sure to circle the answers on this test and fill out your scantron. If you don’t do both, you will get a 0. Make sure you have all 8 pages including the normal table (64 questions). There is NO CLASS on Friday! Scores will be posted on Canvas by Monday at noon and exams will be returned in class next week. Online students may pick up their exam in 0060 Siebel Center for Design during office hours next week. WARNING - The exams look alike but you are sitting next to people who have a different version than you. Copying from anyone is equivalent to giving a signed confession. All cheating including being caught with a non-permissible calculator or anything else you aren’t allowed to have will result in a 0 and an academic integrity violation on your University record. By filling out this page of the exam, you are agreeing to follow the rules. Cover Sheet Questions 1) What’s your name ? _______________________________ ___________________________ (Last name) (First name) 2) What’s your net ID (email)? ___________________ 3) Which section are you in? Circle one: i) L1 (Karle Flanagan In Person) iii) ONL (Karle Flanagan Online) SCANTRON Form Directions • Print and bubble in your last name with no spaces in the YOUR LAST NAME box. • Print and bubble in the first letter of your first name in the FIRST NAME INI box. • Print and bubble in your University Identification Number (UIN) in the STUDENT NUMBER box. • Print and bubble in your netID with no spaces in the NETWORK ID box. • You do NOT need to fill out the SECTION bubbles or TEST FORM bubble parts of the scantron! • Write Stat 100 on the COURSE line. • Write your instructor’s name on the INSTRUCTOR line. • Write your section on the SECTION line. Either L1 (in person) or ONL (online). • Sign your name on the STUDENT SIGNATURE line. READ THIS: Failure to fill out your scantron correctly will actually result in a loss of 2 points on your exam! At this point, everyone should be doing this correctly!!!!

Exam 3 STAT 100 Fall 2021 1 For questions 1-6, fill in the blanks to make the chances equally likely in both scenarios: First, let’s look at flipping a coin 56 times vs. 2,016 times. 1. The number of tosses (n) is increasing by a factor of _______? 28 ± 4 heads in 56 tosses is about as likely as ____ ± _____ heads in 2,016 tosses. Fill in the first blank with the new EV and the second blank with the new SE that will make the chances equal. 2. The first blank, or the new EV, is ______. 3. The second blank, or the new SE, is _____. Next, we are looking at flipping a coin 100 times vs. 6,400 times. 4. The number of tosses (n) is increasing by a factor of ____? 50% ± 8% heads in 100 tosses is about as likely as _____% ± _____% heads in 6,400 tosses. Fill in the blank with the new EV and the second blank with the new SE that will make the chances equal. 5. The first blank, or the new EV, is ______. 6. The second blank, or the new SE, is _____. The following situation pertains to questions 7-22: Fill in the first blank with the number of draws, the second blank with either “with” or “without”, and the third blank with the letter corresponding to the appropriate box model. Choose from the box models below. Use each box model exactly once. A fair coin is tossed 27 times and the number of heads is counted. This corresponds to drawing ______ times ______ replacement from Box ____. 7. How many times are we drawing? 8. Is it with or without replacement? 9. From which box? A die is rolled 250 times and the number of 1’s is counted. This corresponds to drawing _____ times ______ replacement from Box ____. 10. How many times are we drawing? 11. Is it with or without replacement? 12. From which box? a) With b) Without a) With b) Without a) A b) B c) C d) D a) A b) B c) C d) D a) 36 b) 0.028 c) 6 d) 1960 a) 168 b) 50 c) 1,008 d) 28 a) 144 b) 24 c) 0.67 d) 0.11 a) 64 b) 0.016 c) 8 d) 640 a) 400 b) 3200 c) 6.25 d) 50 a) 0.125 b) 64 c) 1 d) 8 a) 27 b) 2 c) 13.5 d) 100 a) 100 b) 6 c) 250 d) 1

Exam 3 STAT 100 Fall 2021 3 Questions 23-31 pertain to the following scenario: 25 draws are made at random with replacement from the box containing these 5 tickets: 2, 3, 4, 5, 6. 23. The smallest the sum of the 25 draws could possibly be is ____? 24. The largest the sum of the 25 draws could possibly be is _____? 25. What is the expected value (EV) for the sum of the draws? 26. What is the standard error (SE) for the sum of the draws? The SD of the box is 1.4. Now suppose you draw at random with replacement from the same box above, but this time you’re only interested in the percent of 6’s that you get. What is the EV and SE of the percent of 6’s in 25 draws? Hint: draw a new box. 27. Your new box should have _____ ticket(s) marked 1. 28. Your new box should have ____ ticket(s) marked 0. 29. What is the expected value of the percent of 6’s in 25 draws? 30. What is the SD of your new box? 31. What is the SE for the percent of 6’s in 25 draws? The following situation pertains to questions 32-36. Suppose you randomly draw 81 marbles with replacement from a bag that contains 2 red marbles, 5 blue marbles, and 3 green marbles. If a red marble is drawn you win $1, if a blue marble is drawn you lose $7, if a green marble is drawn you win $5. Draw the appropriate box model by labeling the 3 tickets inside the box below with the correct numbers and writing how many of each ticket there are in the blanks above them. Assume the red marble tickets are on the far left, the blue marble tickets are in the middle, and the green marble tickets are on the far right. The tickets are the 3 rectangles. 32. What numbers go above the tickets? a) 1, -7, 5 b) 2, 5, 3 c) 2, -35, 15 d) 81, 3, 10 33. What numbers go inside the tickets? a) 1, -7, 5 b) 2, 5, 3 c) 2, -35, 15 d) 81, 3, 10 34. What is the average of the box? 35. What is the expected value of the sum of 81 draws (your net gain/loss)? 36. The SD of the box is $2.22. What is the SE of the sum of your winnings? a) 25 b) 50 c) 100 d) 150 a) 150 b) 20 c) 500 d) 100 a) 6.25 b) 4 c) 100 d) 20 e) 25 a) 7 b) 0.28 c) 37.5 d) 28 e) 35 a) 5 b) 0 c) 4 d) 1 a) 1 b) 4 c) 5 d) 0 a) 0.2% b) 4% c) 20% d) 25% e) 50% a) 0.4 b) 0.16 c) 0 d) 0.5 a) 1.33% b) 2% c) 4.67% d) 8% e) 28% a) -0.2 b) -1.8 c) -9 d) -18 e) -31.5 a) -90 b) -729 c) -18 d) -2551.5 e) -145.8 a) 0.25 b) -16.2 c) 19.98 d) -145.8 e) 179.82

Exam 3 STAT 100 Fall 2021 4 Questions 37-39 pertain to the follow situation: The 3 histograms below (in scrambled order) are the probability histograms for the sum of 4, 11, and 21 random draws with replacement from a box that has 12 tickets, 2 marked “0” and 10 marked “1”. 37. Which histogram depicts 4 draws? 38. Which histogram depicts 21 draws? 39. Which histogram depicts 11 draws? The following scenario pertains to questions 40-47. A gambler plays roulette 49 times betting $1 on 6 numbers, 1-6, each time. If the ball lands on any of those 6 numbers, the gambler wins $5, if the ball lands on any of the other 32 numbers, the gambler loses $1. The roulette wheel has 38 slots numbered 1-36, 0, and 00. 40. Describe the appropriate box model. a) The box has 49 tickets with 5 marked “1” and the rest marked “0” b) The box has 49 tickets with 6 marked “5” and the rest marked “-1” c) The box has 38 tickets with 6 marked “49” and the rest marked “0” d) The box has 38 tickets with 6 marked “5” and the rest marked “-1” e) The box has millions of tickets with 6% of them marked “5” and the rest marked “-1” 41. How many draws are made from this box? 42. Are the draws made with or without replacement? 43. What is the average of this box? a) 1/2 b) 5/38 c) -2/38 d) -2/49 e) None of the above 44. What is the SD of this box? a) 0.36 b) 0.5 c) 1.82 d) 2.19 e) Impossible to calculate Use the normal approximation and estimate the EV to be about $-2 and the SE to be about $15 (approximately) to figure out the chance that the gambler will win more than $10 in 49 plays. 45. First, calculate the z-score a) 0.2 b) 0.5 c) 0.8 d) 1 e) None of the above 46. Now figure out the chance that the gambler will win more than $10 in 49 plays. I’d recommend drawing a normal curve, marking the z-score, and shading the area that represents winning more than $10. What is that chance closest to? a) 21% b) 31% c) 38% d) 42% e) 58% Suppose we were interested in how many times we’d expect the gambler to win playing 49 times (instead of how many dollars we’d expect him to win. 47. What is the appropriate box model? a) The box has 38 tickets: 6 marked “5” and 32 marked “0” b) The box has 38 tickets: one each of 1, 2, 3 … 36, 0, and 00 c) The box has 38 tickets: 6 marked “1” and 32 marked “0” a) Histogram 1 b) Histogram 2 c) Histogram 3 a) Histogram 1 b) Histogram 2 c) Histogram 3 a) Histogram 1 b) Histogram 2 c) Histogram 3 a) With b) Without a) 49 b) 38 c) 6 d) 32

Exam 3 STAT 100 Fall 2021 6 Questions 52-54 pertain to the following scenario: In pre-election polls in close races, pollsters often want a small margin of error. These races are extremely hard to predict because all polls using samples have a certain amount of error. Let’s say that back in 2020, we wanted to take a poll in both Nevada and Pennsylvania the day before the election to see who people planned on voting for. Both states wanted a small margin of error of 1%. The population in Pennsylvania is about 13 million and the population in Nevada is about 3 million. 52. Other things being equal, in order to obtain the same accuracy in the two polls, the number of people you’d have to poll in Pennsylvania is ___________ the number of people you’d have to poll in Nevada. a) Larger than b) Smaller than c) Equal to d) Unable to answer because there was some fraudulent voting that occurred in Nevada 53. How many people would you have to poll to get a 95% confidence interval with a margin of error of 1% in Nevada? Assume SD = 0.48. a) 96 b) 19,200 c) 10,000 d) 100 e) 9,216 54. Say Nevada changed their minds and wanted an even smaller margin of error, only 0.5%. How should they adjust their sample size to get a smaller margin of error? (Still assume SD = 0.48). a) Increase it b) Decrease it c) Keep it the same Questions 55-61 pertain to this situation: Last exam, I graded a random sample of 64 of the 1,500 exams to see how students would do. The average of the 64 exams was 83 with an SD of 10. 55. Which most closely resembles the relevant box model? a) The box has 1,500 tickets marked with “1”s and “0”s b) The box has 64 tickets marked with “1”s and “0”s. The exact percentages are unknown, but estimated from the sample c) The box has 64 tickets with an average of 83 and an unknown SD, estimated to be around 10 d) The box has 1,500 tickets with an average of 83 and a SD of 10 e) The box has 1,500 tickets, marked with numbers ranging from 0 to 100, the exact average and SD of the box are unknown and are estimated from the sample 56. How many draws from the box? 57. Are the draws made with or without replacement? 58. The best estimate for the average exams scores of all 1,500 students is ______. a) 83 b) 10 c) 64 d) Cannot be determined 59. What is the SE of the sample average? a) 0.258 b) 0.5 c) 1.1 d) 1.25 e) Unable to calculate because we don’t know how many 1’s and 0’s are in the box. 60. A 95% confidence interval for the average of all 1,500 exams is about _______ to ________. a) 61.5, 66.5 b) 63, 103 c) 80.5, 85.5 d) 81.5, 84.5 e) 82.5, 83.5 61. Suppose we also computed a 68% CI for the average of all 1,500 exams. Which interval would be larger? a) The 68% CI b) The 95% CI c) They’d be the same width d) Impossible to tell a) With b) Without c) Either is fine a) 1,500 b) 10 c) 83 d) 100 e) 64

Exam 3 STAT 100 Fall 2021 7 The following situation pertains to questions 62-64. Suppose a government survey organization took a simple random sample of 1,000 people in Illinois and computed the SE% and a 95% confidence interval in order to estimate the percentage of all voters in Illinois who support making public colleges and universities tuition free. 62. If they decided to increase the sample size to 9,000, the new SE% would…. a) Stay the same b) Be multiplied by 3 c) Be multiplied by 9 d) Be divided by 3 e) Be divided by 9 63. If the sample size was increased to 9,000, the width of the new 95% confidence interval would…. a) Stay the same b) Be multiplied by 3 c) Be multiplied by 9 d) Be divided by 3 e) Be divided by 9 Suppose 100 pollsters each randomly sampled 1,000 Illinois voters asking the same question. All 100 pollsters computed 85% confidence intervals to estimate the percentage of all Illinois voters who support making public colleges and universities tuition free. 64. About how many of the 100 confidence intervals would miss the true population percentage? a) 0 b) 15 c) 85 d) 100 e) None of the above Exam 3 Formulas EVsum = n*average of box SEsum = sqrt(n) * SD of box EVavg = average of box SEavg = SD of box / sqrt(n) EV% = percent in box SE% = [SD of box / sqrt(n)] * 100% SD Shortcut Formula = |a – b| * sqrt(fraction of “a” tickets * fraction of “b” tickets) Z = (Value – EV) /SE