Student's Printed Name: Solutions & Grading Guidelines Lecture Instructor: CUID: Lecture Section # : You are permitted to use a calculator on all portions of this test. You are not allowed to use any notes, textbooks, cell phones, or laptops on this test. All devices that can send and receive information (including smart watches) must be turned off and put away while you are in the testing room. During this test, any communication with any person (other than the instructor or test proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No part of this test may be removed from the testing room. Read each question very carefully. In order to receive full credit for the free response portion of the test, you must: 1. Show legible and logical (relevant) justification which supports your final answer. 2. Use complete and correct mathematical notation. 3. Include proper units, if necessary. You have 90 minutes to complete the entire test. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student's Signature: Do not write below this line. Free Response Problem Points Possible Points Earned 1 10 2 10 3 10 4 9 Free Response Total 39 Multiple Choice 60 Correct Scantron 1 Test Total 100 Page 1 of 12
Part I: Multiple Choice. There are 20 multiple choice questions. Solve each question using the available space for scratch work. Decide which is the best of the choices given and fill in the corresponding oval on the provided scantron using a #2 pencil. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. Each multiple choice question is worth 3 points. 1. A relative frequency bar graph of the class standing for the 568 respondents to the STAT 2300 Student Survey is shown below. Approximately how many more sophomores than juniors are in STAT 2300 this semester? (A) 30 (B) 170 (C) 300 (D) Cannot be determined from the given information 2. A distributor of industrial lubricants is hiring three salespeople from a pool of 28 applicants for the three areas of the country where they operate. The three areas, in order of sales volume, are Northeast, Southeast, and Midwest. How many ways can the company fill these three positions? (A) 21,952 (B) 19,656 (C) 3,276 (D) 784 Page 2 of 12
3. The health center at a large university is concerned about the amount of time students have to wait before seeing a doctor when they walk in without an appointment. The staff recorded the wait times for the 50 walk-in patients one week and then computed the sample mean wait time for these 50 observations. The two largest observations in the data set were 75 minutes and 120 minutes. It was later determined that there was an error recording the data and the observation of 120 minutes should have been 12 minutes. How would correcting this error affect the sample mean? (A) The corrected sample mean will be smaller (B) The correction will not change the value of the sample mean (C) The corrected sample mean will be larger (D) The corrected sample mean will be smaller only if the original sample mean (computed with 120 as one of the observations) was greater than 12 4. A team of agricultural researchers conducted a randomized experiment to determine whether two fertilizers, A and B, give different yields of tomatoes. A total of 33 tomato plants were grown; 16 using fertilizer A, and 17 using fertilizer B. At the conclusion of the experiment, the number of tomatoes grown per plant was recorded. What are the treatments in this experiment? (A) The yields of the 33 tomato plants (B) The two fertilizers (A and B) (C) The 33 tomato plants (D) The agricultural researchers 5. One of the questions on the STAT 2300 Student Survey asked students, "To the nearest quarter of an hour, how many hours of sleep did you get last night?" The responses had a distribution that is symmetric and unimodal with a mean of 6.96 hours and a standard deviation of 1.56 hours. Which of the following best describes the meaning of the standard deviation? (A) Students reported sleep times that differed from 6.96 by about 1.56 hours, on average (B) About 50% of the sleep times were in the interval 6.96 ± 1.56 (C) Sleep times more than 1.56 hours away from 6.96 are considered outliers (D) The longest reported sleep time was 1.56 hours above 6.96 Page 3 of 12
6. The Mars candy company claims that 24% of the plain M&M's it produces are blue. Based on a sample of 5000 randomly selected plain M&M's, it was estimated that the true proportion of blue candies is 20% with a margin of error of 3%. Which of the following is a correct statement? (A) The statement that 20% ± 3% of plain M&M's are blue is an example of descriptive statistics (B) The value 20% is a parameter because it is a numerical summary of a sample (C) The value 24% is a statistic because it is a numerical summary of a population (D) The value 24% is a parameter because it is a numerical summary of a population 7. Sandy has an average bowling score of 180 and bowls in a league where the average for all bowlers is 150 and the standard deviation is 20. Chantel has an average bowling score of 190 and bowls in a league where the average is 160 and the standard deviation is 15. Who is the better bowler relative to the rest of her league, Sandy or Chantel? (A) Chantel, because her average bowling score of 190 is higher than Sandy's 180 (B) Sandy and Chantel are equally impressive relative to their rest of their leagues, because both are 30 points above the mean (C) Chantel, because of her higher z-score (D) Sandy, because of her higher z-score 8. A stem-and-leaf plot of the amount of fat (in grams) for McDonald's beef sandwiches is below. 0 9 1 2 1 9 2 3 4 2 6 6 7 9 9 3 1 3 4 3 Key: 4 3 = 43 grams of fat Find the IQR for this data set. (A) 8 grams (B) 10 grams (C) 12 grams (D) 14.4 grams Page 4 of 12
9. A study was conducted to determine the proportion of Clemson students who look both ways before entering a campus crosswalk. Twenty different crosswalk locations were randomly selected. At each location, observers counted the total number of students using the crosswalk during a two hour time period and the number of these students who looked both ways before entering the crosswalk. This is an example of what type of sampling method? (A) Simple Random Sampling (B) Stratified Sampling (C) Convenience Sampling (D) Cluster Sampling 10. Consider the boxplot for a data set of n = 50 observations given below. 10 15 20 25 30 35 40 45 50 55 60 65 70 If the maximum value of 65 is changed to 70, would the value of 70 be considered an outlier? (A) Yes, because 70 is beyond the upper fence (B) No, because 70 is not beyond the upper fence (C) Yes, because 70 is more than 3 standard deviations above the mean (D) Cannot be determined from the given information 11. According to the 2010 U.S. census, 49.2% of Americans are male and 50.8% are female. Also, according to Wikipedia, 8% of males are color blind and.5% of females are color blind. If an American is randomly selected, what is the probability that the person is a color blind male? (A).0025 (B).0394 (C).0406 (D).5326 Page 5 of 12
12. Consider the following question and answer choices included on a survey conducted by CCIT: How satisfied were you with the service provided by CCIT? (1) Not Satisfied (2) Somewhat Satisfied (3) Very Satisfied The variable "Satisfaction Rating" is at what level of measurement? (A) Nominal (B) Ordinal (C) Interval (D) Ratio 13. Suppose P(A) =.5 and P(B) =.6. Which of the following must be true? (A) A and B are mutually exclusive (B) A and B are not mutually exclusive (C) A and B are independent (D) A and B are not independent 14. One of the questions on the STAT 2300 Student Survey asked students, "In a typical week, how many hours do you participate in sports or other physical exercise?" A dotplot of the results for the 19 students in lab section 14 is given below. Based on the dotplot above, which of the following measures of spread should be reported when describing the amount of variability in this data set? (A) Interquartile Range (B) Range (C) Standard Deviation (D) Variance Page 6 of 12
15. Researchers want to study the effects of air pollution on health. Six U.S. cities were randomly selected and all residents of the cities were asked to record their health over the next 20 years. This is an example of: (A) An experiment (B) Stratified sampling (C) An observational study (D) Convenience sampling 16. A researcher uses four different sampling methods to select samples of size 10 from a population of 200 values with a population mean of 70. For each sampling method, she selects 500 different samples from the population, computes the average of each sample, and then creates a histogram of the 500 sample averages. The resulting histograms are displayed below. After evaluating the histograms, the researcher determines that one of the four sampling methods is biased. Select the histogram which corresponds to the biased sampling method. (A) (B) (C) (D) Page 7 of 12
17. Parking services wants to conduct a study to determine the average length of time necessary for commuting students to find a parking spot. Parking services plans to randomly select 500 students with a commuter parking pass and ask them to record the length of time it takes for them to find a parking spot the next time they travel to campus. The variable of interest in this study is: (A) Length of time to find a parking spot (B) Average amount of time the 500 students spend looking for a parking spot (C) Average amount of time commuting students spend looking for a parking spot (D) Whether or not students were able to find a valid parking spot 18. The Clinton campaign wants to determine the proportion of Clemson residents who will vote for Hillary Clinton in the upcoming election. Volunteers from the Clinton campaign plan to set up a booth decorated with Clinton signs on the library bridge of Clemson's campus during 5 randomly selected time slots. The volunteers at the booth will attempt to interview everyone crossing the bridge during these time frames. This survey is likely to contain: (A) Sampling bias (B) Nonresponse bias (C) Both (A) and (B) (D) Neither (A) nor (B) Use the following information to answer questions 19 20. Based on exit interviews with graduating students, the university estimates that 92% of Clemson seniors would choose Clemson again if they could start over as freshmen. Suppose that two Clemson seniors are selected at random, such that their opinions are independent of one other. 19. What is the probability that both of these two seniors would choose Clemson again? (A).0064 (B).8464 (C).9936 (D) 1.840 20. What is the probability that at least one of these two seniors would choose Clemson again? (A).0064 (B).0736 (C).9200 (D).9936 Page 8 of 12
Part II: Free Response. Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. Answers with no justification will receive no credit. 1. The lengths of female great white sharks follows a distribution that is roughly symmetric and unimodal with a mean of 15.6 feet and a standard deviation of 2.6 feet. (a) Use the Empirical Rule to draw a sketch of this distribution. Clearly label the horizontal axis with appropriate values. (2 pts) 1 pt for drawing a bell-shaped curve 1 pt for correctly labeling horizontal axis (b) Approximately what percentage of female great white sharks are between 10.4 feet long and 20.8 feet long? (2 pts) According to the Empirical Rule, approximately 95% of female great white shark lengths are within two standard deviations of the mean, that is, between 10.4 feet and 20.8 feet long. 2 pts for identifying the correct percentage (c) Approximately what percentage of female great white sharks are more than 20.8 feet long? (3 pts) From the Empirical Rule we have that 100% 95% = 5% are either less than 10.4 feet or more than 20.8 feet. By symmetry, this means that ½(5%) = 2.5% are more than 20.8 feet. 1 pt for subtracting percentage given in (b) from 100% 2 pts for dividing the result by 2 Note: Subtract 1 pt if no work is shown but a correct answer is given. (d) Approximately what percentage of female great white sharks are between 13 feet long and 20.8 feet long? (3 pts) From the Empirical Rule we have that 100% 68% = 32% are either less than 13 feet or more than 18.2 feet. By symmetry, this means that ½(32%) = 16% are less than 13 feet. Combined with the fact that 2.5% are more than 20.8, we have that 100% 16% 2.5% = 81.5% are between 13 feet and 20.8 feet. 2 pts for implementing a correct method with supporting work shown 1 pt for an answer consistent with parts (b) and (c) Page 9 of 12
2. One of the questions on the STAT 2300 Student Survey asked students to report their height in inches. In the JMP output below, a histogram and selected summary statistics for the distribution of heights for the female students is shown first followed by a histogram and selected summary statistics for the distribution of heights for the male students. (a) Write a sentence to compare the shapes of the two distributions. (4 pts) The height distributions for both male and female students are roughly symmetric and unimodal. 2 pts for describing female heights as roughly symmetric 2 pts for describing male heights as roughly symmetric (b) Write a sentence to compare the centers of the two distributions. (3 pts) Male students are typically taller than female students, as the mean height for males (xx = 71 in.) is greater than the mean height for females (xx = 65 in.). 1 pt for identifying which distribution has the lower (or higher) center 1 pt for including the value of a measure of center 1 pt for using an appropriate measure of center for the shapes given in (a) (c) Write a sentence to compare the amount of variability in the two distributions. (3 pts) There is a little bit more variability in male heights, as the standard deviation for males (s = 3.02 in.) is slightly larger than the standard deviation for females (s = 2.76 in.). 1 pt for identifying which distribution has more (or less) variability 1 pt for including the value of a measure of spread 1 pt for using an appropriate measure of spread for the shapes given in (a) Page 10 of 12
3. The table below breaks down one section of STAT 2300 according to gender and political affiliation. Democratic Party Republican Party Libertarian Party Green Party Male 7 15 7 1 30 Female 22 25 5 2 54 Total 29 40 12 3 84 Total One student will be randomly selected from this section of STAT 2300. For the questions below, label your answers with appropriate probability notation, using the first letter of each category name in the table to denote the event the randomly selected person is in that category. Answer with a fraction or a decimal rounded to three decimal places. (a) What is the probability that the selected student is a Male or a Republican? (2 pts) P(M or R) = P(M) + P(R) P(M and R) = 30 84 + 40 84 15 84 = 55 84.655 1 pt for appropriate probability notation 1 pt for correct answer (fraction or decimal form) (b) Are the categories Male and Republican mutually exclusive? Explain. (2 pts) No, because P(M and R) 0. That is, a person can be both Male and Republican. 1 pt for concluding they are NOT mutually exclusive 1 pt for correct justification No credit for incorrect reasoning or for not providing a justification. (c) What is the probability that the selected student is a Democrat? (2 pts) P(D) = 29 84.345 1 pt for appropriate probability notation 1 pt for correct answer (fraction or decimal form) (d) If a Female is selected, what is the probability that the selected student is a Democrat? (2 pts) P(D F) = 22 54.407 1 pt for appropriate probability notation 1 pt for correct answer (fraction or decimal form) (e) Are the categories Female and Democrat independent? Explain mathematically. (2 pts) No, because P(D F) P(D). Knowing that the selected student is female makes it more likely that the selected student is a Democrat. 1 pt for concluding they are not independent 1 pt for correct justification Notes: Can also justify by showing P(F D) P(F) or P(F and D) P(F) P(D). No credit for incorrect reasoning or for not providing a justification. Page 11 of 12
4. A box contains 5 small bags of snacks: one bag of Doritos, two bags of Cheetos, and two bags of Lays. A bag is selected without looking in the box and given to a child, then a second bag is selected and given to a second child. (a) List all of the possible outcomes for the types of snacks that the two children get using the letters D, C, and L. (3 pts) 1 st child 2 nd child Outcome D C L D DC DL CD C C CC L D CL LD L C LC L LL S = {DC, DL, CD, CC, CL, LD, LC, LL} Subtract 1 pt for not listing final outcomes Subtract 1 pt for each missing or incorrect outcome (b) What is the probability that the second child gets Doritos if we know that the first child got Cheetos? Label your answer with appropriate probability notation. (2 pts) P(D second C first) = 1 4 =.25 1 pt for appropriate probability notation 1 pt for correct answer (fraction or decimal form) (c) What is the probability that one child gets Doritos and the other gets Lays? Label your answer with appropriate probability notation. (4 pts) P(DL or LD) = P(DL) + P(LD) = 1 5 2 4 + 2 5 1 4 = 2 20 + 2 20 1 pt for appropriate probability notation 1 pt for finding P(DL) correctly 1 pt for finding P(LD) correctly 1 pt for adding P(DL) and P(LD) to get final answer Note: If the outcomes are assumed to be equally likely, then no = 4 = 1 =.20 credit will be awarded for finding P(DL) or P(LD) correctly. 20 5 Did you correctly fill in your scantron? (1 pt) Page 12 of 12