BIOSTATS 540 Fall 2017 Exam 1 Page 1 of 12

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1 BIOSTATS 540 Fall 2017 Exam 1 Page 1 of 12 BIOSTATS Introductory Biostatistics Fall 2017 Examination 1 Units 1&2 Summarizing Data & Data Visualization Due: Tuesday October 10, 2017 Last Date for Submission with Credit: Tuesday October 17, 2017 Rules: This is an open book take-home exam. You are welcome to use any reference materials you wish. You are welcome to use the computer as you wish, too. However, you MUST work this exam by yourself and you may not consult with anyone. Instructions and Checklist 1. Start each problem on a new page. 2. Write your name on every page 3. Make a copy of your exam for safekeeping (sometimes a mailed exam is lost!) 4. Submit a completed signature page (See next page). How to submit your exam: Worcester in-class Section: 1. Bring your completed exam to class on Tuesday October 10, 2017 being sure that you have made a copy for safe keeping; OR 2. Mail your completed exam to me with post-mark October 10, 2017 to my address below. Blackboard Learn Online Section: 1. Upload your completed exam to the ASSIGNMENT tab no later than 11:59 pm on Monday October 10, This must be a single pdf ; OR 2. Mail your completed exam to me with post-mark October 10, 2017 to my address below. Address and Telephone Number for Mailing Carol Bigelow Biostatistics & Epidemiology/402 Arnold House University of Massachusetts/Amherst 715 North Pleasant Street Amherst, MA Tel

2 BIOSTATS 540 Fall 2017 Exam 1 Page 2 of 12 BIOSTATS Introductory Biostatistics Fall 2017 Examination 1 Units 1&2 Summarizing Data & Data Visualization Due: Tuesday October 10, 2017 Last Date for Submission with Credit: Tuesday October 17, 2017 Signature This is to confirm that in completing this exam, I worked independently and did not consult with anyone. Signature: Printed Name: Date:

3 BIOSTATS 540 Fall 2017 Exam 1 Page 3 of (10 points total) The following table lists some variables that might be of interest in your next data analysis. For each variable, complete the associated table indicating whether it is categorical (and if so, is it nominal or ordinal) or numerical (and if so, is it discrete or continuous). Categorical Numerical Variable nominal ordinal discrete continuous Example Eye color X 1a (1 point) Sex 1b (1 point) Number of runs scored in a baseball game 1c (1 point) Profession 1d (1 point) Temperature, measured in Farenheit 1e (1 point) Confidence in one s ability to to statistics as measured by yes/no to the statement: I will do well 1f (1 point) Number of siblings 1g (1 point) Distance an individual can run in five minutes 1h (1 point) Ethnicity 1i (1 point) Number of MD s at UMass/Worcester who also have a PhD 1j (1 point) Lack of coordination as measured by time it takes an individual to complete a certain puzzle.

4 BIOSTATS 540 Fall 2017 Exam 1 Page 4 of (10 points total) Here is a hypothetical situation. In 2015 a program aimed at reducing infant mortality was implemented in two regions, Pepi and Quepi. The following table (this is hypothetical, sorry) shows the numbers of births and infant deaths in two regions (Pepi and Quepi) in each of two years: 2014 and Pepi Quepi Births Infant Deaths Births Infant Deaths , ,000, , ,000, a. (5 points) In which region is there more convincing evidence that the reduction in mortality was caused by the program? 2b. (5 points) If the program can be continued in one region ONLY, which would you choose? In developing your answer, you may assume that the reductions shown were in fact caused by the program.

5 BIOSTATS 540 Fall 2017 Exam 1 Page 5 of (10 points total) The following are some data on some famous statisticians. Yes! Florence Nightingale, among her other talents, was a statistician! Statistician Gender Year of Birth Year of Death Sir Francis Galton Karl Pearson William Sealy Gosset Ronald Aylmer Fisher Harald Cramer Prasanta Mahalanobis Jerzy Neyman Egon S. Pearson Gertrude Cox Samuel S Wilks Florence Nightingale David John Tukey a. (5 points) By any means you like (by hand is just fine), create a stem-and-leaf summary of the data on the variable YEAR OF BIRTH. Display it here. Then use this visual summary to answer questions #3b - #3e below. 3b. (1 point) Are there any outliers (i.e., extreme values) in this distribution? Explain. 3c. (1 point) How would you describe the shape of this distribution? Explain. 3d. (1 point) What is/are the most frequently occurring score(s) in this distribution? How many times does it/do they occur? 3e (1 point) Can we use this stem-and-leaf to obtain the original set of values for this variable? Explain.

6 BIOSTATS 540 Fall 2017 Exam 1 Page 6 of (10 points total) 4a. (3 points) When a distribution is skewed to the right i) TRUE or FALSE: The median is greater than the mean. ii) TRUE or FALSE: The distribution is uni-modal iii) TRUE or FALSE: The majority of observations are less than the mean. 4b. (3 points) The shape of a frequency distribution can be described using: i) TRUE or FALSE: A box and whisker plot. ii) TRUE or FALSE: A table of frequencies iii) TRUE or FALSE: A histogram 4c. (4 points) For the sample 3, 1, 7, 2 and 2: i) TRUE or FALSE: The sample mean is 3 ii) TRUE or FALSE: The sample median is 7 iii) TRUE or FALSE: The range is 1 iv) TRUE or FALSE: The sample variance is 5.5

7 BIOSTATS 540 Fall 2017 Exam 1 Page 7 of (10 points total) The following table shows the numbers of geriatric admissions, each week from May through September, to a certain facility in each of two years, 2012 and Week # Admissions 2012 # Admissions 2013 Week # Admissions 2012 # Admissions a. (5 points) By any means you like (by hand is just fine), summarize these data graphically. Display it here. Then use this visual summary to answer question #5b. 5b. (5 points) Why do you think these two years were different? Note There is no single correct answer here. I will accept any well-reasoned interpretation. I m looking for you to think about what you see!

8 BIOSTATS 540 Fall 2017 Exam 1 Page 8 of (10 points total) 6a. (2 points) You read that the median income of U.S. households in 2010 was $49,455. In 1-2 sentences at most, explain in plain language what the median income is. 6b. (2 points) The Census Bureau website gives several choices for average income in its historical income data. In 2010, the median income of American households was $49,455. The mean household income was $67,530. The median income of families was $60,395, and the mean family income was $78,361. The Census Bureau says, Households consist of all people who occupy a housing unit. The term family refers to a group of two or more people related by birth, marriage, or adoption who reside together. In at most 5 sentences, explain carefully why mean incomes are higher than median incomes and why family incomes are higher than household incomes. 6c. (2 points) A January 2012 magazine article reported that the average income for readers of the business magazine Forbes was $217,000. In your opinion, is the median wealth of these readers greater or less than $217,000? In at most 1-2 sentences, explain your reasoning. 6d. (2 points) The distribution of individual incomes in the United States is strongly skewed to the right. In 2008, the mean and median incomes of the top 1% of Americans were $558,726 and $1,137,680. Which of these numbers is the mean and which is the median? In at most 1-2 sentences, explain your reasoning. 6e. (2 points) By any means you like (by hand is fine) which of the following two data sets is more spread out? Show your work. In at most 1-2 sentences, explain your reasoning. Data set A : Data set B :

9 BIOSTATS 540 Fall 2017 Exam 1 Page 9 of (20 points total) A box plot is the graph of a five number summary. The central box spans the quartiles. The line in the box mark the median. The size of the box is a measure of spread. The lines extending out from the box give an indication of extremes, if any. Side-by-side box plots are useful for comparing two distributions. As an example, consider the following table. It lists the average month s temperature (Farenheit) of Springfield, Massachusetts and San Francisco, California. Month Ave Temp (F) Springfield Month Ave Temp (F) San Francisco January 32 January 49 February 36 February 52 March 45 March 53 April 56 April 55 May 65 May 58 June 73 June 61 July 78 July 62 August 77 August 63 September 70 September 64 October 58 October 61 November 45 November 55 December 36 December 49 7a. (5 points) Obtain the five number summary for the average monthly temperatures, separately for each data set, Springfield versus San Francisco. Use these values to complete the following table. Minimum Q1 Q2 = median Q3 Maximum Springfield San Francisco 7b. (10 points) By any means you like (by hand is fine), produce a side-by-side box and whisker plot of the two distributions of average monthly temperatures. You will use this visual to answer question #7c. 7c. (5 points) i) (1 point): Are the 2 cities similar in their typical (median) average temp? ii) (3 points): Are the 2 cities similar in terms of temperature spread? Explain iii) (1 point): Which city requires owning a larger wardrobe of clothes?

10 BIOSTATS 540 Fall 2017 Exam 1 Page 10 of (20 points total) This last exercise gives you practice working with the fundamentals of calculations of the sample mean, the sample variance and the sample standard deviation. It also gives you practice producing and interpreting a histogram. On the next page is a table of data on X = blood glucose levels (mmol/l) obtained from a simple random sample of n=40 first year medical students. The students are indexed using a subscript i that ranges from i=1 to i=40. 8a. (1 points)) First calculate the sample mean. To do this, obtain the sum of the individual blood glucose values and divide this by the sample size. 40 i) x i = i=1 ii) n = iii) Sample mean = 40 x i i=1 n fill in = fill in = 8b. (3 points)) Next, calculate the individual squared values of individual blood glucose levels. In developing your answer complete the entries to the 3 rd column of the table. All done? Now obtain the sum of the squared values of the individual blood glucose levels. Enter this total at the bottom. 8c. (2 points) Next, calculate the individual squared values of the deviations of the individual blood glucose levels about the sample mean. In developing your answer complete the entries to the 4 th and 5 th columns of the table. All done? Now obtain the sum of the individual squared values of the deviations of the individual blood glucose values about the sample mean. Enter this total at the bottom of the 5 th column.

11 BIOSTATS 540 Fall 2017 Exam 1 Page 11 of 12 i x i x i 2 (x i x) (x i x) Total of column

12 BIOSTATS 540 Fall 2017 Exam 1 Page 12 of 12 8d. (2 points) Calculate the sample variance using the appropriate column totals in TWO ways. Show your work. Tip - You should get the same answer, thus illustrating a shortcut when doing calculations by hand and clarifying the confusion you might have encountered when encountering more than one formula for this calculation. i) s 2 = 40 i=1 (x i x) 2 (n 1) ii) s 2 = 40 2 x i i=1 n (n 1) [ ] x 2 8e. (2 points)) Finally, calculated the sample standard deviation. 8f. (5 points) By any means you like (by hand is fine), produce a histogram of these data. 8g. (2 points) Calculate the mean + 1 standard deviation and the mean + 2 standard deviations. Indicate these points on your histogram. 8h. (3 points) What term best describes the shape of the distribution of blood glucose in this sample: symmetrical, skewed to the right, or skewed to the left?