1 STAT 350 Exam 1 (Version 1) Spring 2015 Name (Print) : Instructor (circle one): Ayu Eagan Findsen Sellke Troisi Class Time: 11:30 AM 12:30 PM 1:30 PM Instructions: 2:30 PM 3:30 PM 4:30 PM ONLINE 1. You are expected to uphold the honor code of Purdue University. It is your responsibility to keep your work covered at all times. Anyone cheating on the exam will automatically fail the course, and will be reported to the Office of Dean of Students. 2. Please alert proctors if you observe any cheating during the exam. We highly appreciate it. 3. It is strictly prohibited to smuggle this exam outside. Your exam will be returned to you after it is graded. 4. You may have one double-sided 8.5 in x 11 in crib sheet to take this test. The crib sheet can be handwritten or typed. 5. The only materials that you are allowed are your calculator, writing utensils and erasers and your crib sheet. If you bring any other papers in to the exam, you will get a zero on the exam. We will provide scratch paper if you need more room. 6. Leave all your belongings except those permitted for the exam in the front of the room. We are not responsible for any loss. 7. If you share your calculator or use a cell phone, you will get a zero on the exam. 8. Breaks (including bathroom breaks) during the exam are not allowed. If you leave the exam room, you must turn in your exam and you will not be allowed to come back. 9. You must show ALL your work to obtain full credit. An answer without showing any work may result in zero credit. 10. All numeric answers should have two decimal places except answers from the z-table should have four decimal places. 11. If your work is not readable, it will be marked wrong. 12. After you complete the exam, please turn in your exam as well as your crib sheet, tables and any scrap paper. Please be prepared to show your Purdue picture ID. Your exam is not valid without your signature below. STUDENT: I attest here that I have followed the instructions above honestly while taking this test and that work submitted is my own, produced without assistance from books, other people, notes other than my own crib sheets, or other aids. Signature of Student:
2 Points Earned Problem 1 (15 points) Problem 2 (30 points) Problem 3 (10 points) Problem 4 (20 points) Problem 5 (15 points) Problem 6 (10 points) Problem 7 (5 points) Total (105 / 100 )
3 1. (15 points) The boxplots below show the display case prices (in dollars) of varieties of wine produced by vineyards along three different lakes. Keuka a. (3 pts) Which distribution of wine prices has the largest median? b. (6 pts) Which distribution of wine prices has the largest spread? Justify your answer. Seneca, b/c min = min_s AND max = max_s Keuka c. (3 pts) Which distribution of wine prices is clearly skewed to the left? d. (3 pts) Which distribution includes the most expensive wine prices? Seneca
4 2. (30 points) Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.0 in and a standard deviation of 1.0 in. a. (4 pts) Sketch the Normal Curve. Be sure to label the mean and the standard deviation. b. (8 points) If one man is randomly selected, what is the probability that his head breadth is less than 6.2 in? P(X < 6.2) = P(Z < (6.2-6)/1 = 0.2) = 0.5793 c. (8 points) ACME motorcycle company is making a new adjustable helmet. You must design a helmet that will fit all but largest 5% of male head breadths. What is the largest size male head breadth that your new helmet will fit? P(X > x) = 1 - P(X <= x) = 0.05 => P(X <= x) = 1-0.05 = 0.95 => P(Z <= z) = 0.95 => z =_{TABLE} 1.645 = (x - 6)/1 = (x - \mu)/\sigma => x = \mu + z\sigma = 6 + 1.645*1 = 7.645 d. (10 points) If 16 men are randomly selected, find the probability that their mean head breadth is less than 6.2 in. Xbar ~ N(\mubar = \mu = 6, \sigmabar = \sigma/n^0.5 = 1/16^0.5 = ¼ = 0.25) => P(Xbar < 6.2) = P(Z < (6.2-6)/0.25 = 0.8) = 0.7881
5 3. (10 points) The current in a certain circuit as measured by an ammeter has the following density function: f (x) = 0.075x + 0.2 for 3 < x < 5 0 elsewhere a) (5 pts) What is the mean of the current in this circuit? E[X] =_{Defn.} \int_{-\infty}^{\infty} xf_x(x)dx =_{X} \int_3^5 x(0.075x + 0.2)dx =[(0.075/3)x^3 + (0.2/2)x^2]_3^5 = [0.025*5^3 + 0.1*5^2] - [0.025*3^3 + 0.1*3^2] = 4.05 b) (5 pts) What is the median of current in this circuit? F_X(x) = \int_3^x 0.075t + 0.2dt = [(0.075/2)t^2 + 0.2t]_3^x = [0.0375x^2 + 0.2x] - [0.0375*3^2 + 0.2*3] = 0.0375x^2 + 0.2x - 0.9375 = 0.0375(x - 3)^2 + 0.2(x - 3) = (x - 3)[0.0375(x - 3) + 0.2] => F_X(x) = 0.0375x^2 + 0.2x - 0.9375 =_{SET} 0.5 => 0.0375x^2 + 0.2x - 1.4375 = 0 =>_{Quadratic Formula} x = (-0.2 + [0.2^2-4*0.0375(-1.4375)]^(½))/(2*0.0375) \approx -8/3 + 6.741249472 = 4.074582805
6 4. (20 points) To investigate whether or not sending text messages while driving impacts driving ability, we had 100 participants drive an obstacle course under one of the following conditions: 1) no texting while driving, 2) sending five text messages while driving, or 3) sending 10 text messages while driving. We measured the accuracy with which the subjects drove the obstacle course from a scale of 1 to 10 (1 = poor and 10 = excellent). a. (2 pts) What type of study is this? (please circle one) i. an observational study ii. an experiment iii. a matched-pairs study b. (2 pts) What are the experimental subjects? 100 Participants (assuming people over minimum driving age within U.S.A.) c. (2 pts) What are the treatments? 1) Send 0 text messages while driving 2) Send 5 text messages while driving 3) Send 10 text messages while driving d. (2 pts) What is the response variable? Accuracy with which the subjects drove the obstacle course on a 1 to 10 scoring scale. e. (2 pts) This study includes which of the following? (circle one) i. blinding ii. control f. (2 pts) Why is a control necessary in this experiment? (circle one) i. The control is used for the subjects who do not know how to send text messages. ii. The control helps determine if women are better drivers than men. iii. The control helps control for the lurking variables. g. (8 pts) Use a diagram to outline a completely randomized design for the study.
7 5. (15 points) Statistician Date Night: Jeremy and his date, Fantasia, cannot decide where to eat Friday night. Jeremy wants to go to McGraw s while Fantasia wants to go to Olive House. To be fair, Fantasia flips a fair coin to decide where to eat. On a Friday, the wait for a table is at McGraws is uniformly distributed between 0 and 60 minutes. The wait for a table at Olive House is uniformly distributed between 0 and 20 minutes. a. (5 pts) What is the probability that Jeremy and Fantasia ate at McGraw s and waited between 10 and 20 minutes? P(M AND 10 < T < 20) =_{Multiplication Rule} P(M)P(10 < T < 20 M) = 0.5*(20-10)/(60-0) = 1/12 = 0.08\bar{3} b. (5 pts)what is the probability that Jeremy and Fantasia wait between 10 and 20 minutes for their table? P(10 < T < 20) =_{TP} P(M AND 10 < T < 20) + P(H)P(10 < T < 20 H) =_{a.} 1/12 + 0.5*(20-10)/(20-0) = 1/12 + ¼ = ⅓= 0.\bar{3} c. (5 pts) The pair waited for a table between 10 and 20 minutes. What are the chances the date took place at McGraw s? P(M 10 < T < 20) =_{Bayes Rule} P(M AND 10 < T < 20)/P(10 < T < 20) =_{a.-b.} (1/12)/(⅓) = ¼ = 0.25
8 6. (10 points) In February, the city of Boston was ravaged by record breaking snowfall. John is waiting for the street in front of his house to be plowed. He is familiar with statistics and realized his waiting time for the plows to come once the snowfall has ceased has an exponential distribution with a mean of 2 hours. f (x) = 0.5 e 0.5x when x > 0 0 elsewhere a) (2 pts) Draw a graph of the density curve(remember the support and label the y-intercept). X ~ Exp(\lambda = 0.5) => y-intercept = 0.5, monotonically decreasing when x > 0. b) (8 pts) What is the probability John waits between half-an-hour and 4 hours after the snowfall ceases? Also, shade the corresponding area on your graph of part a. P(0.5 < X < 4) = e^(-0.5*0.5) - e^(-0.5*4) = e^(-¼) - e^(-2) \approx 0.6434655 7. (5 points) What are the fun things you plan to do over the Spring Break? Brown County/Nashville, TN/Smoky Mountains