BUSI 410 FINAL EXAM Fall 2020 This exam consists of three short cases Case Curious Faculty Boston Housing Uber Pages Points 2 5 6 9 10 14 Total 40 30 30 100 Answer the questions after each case Show all of your work on the exam paper The final page of the exam has the Durbin Watson table The cases questions are not ordered in difficulty If you are stuck try the next question This exam is open book note and laptop However no communication of any kind is allowed verbal written or electronic Disable your laptop s Wi Fi Internet connectivity Turn off and put away your cell phone Name Please Print Section circle start time Honor Code Pledge I will neither give nor receive unauthorized aid during this exam Signature 11 00 12 30 9 30 2 00 DO NOT TURN TO THE NEXT PAGE UNTIL YOU ARE ASKED TO Page 1 of 15 Curious Faculty A UNC faculty member who teaches an undergraduate business analytics course would like to do some statistical analysis to better understand the distribution of grades for the midterm exam of his class He teaches two sections with 50 students enrolled in each section The following is a snapshot of the grades for the two sections with the summary statistics presented at the bottom Section A is at 9 30 and Section B is at 2 00 you can assume that these are samples of the whole population of morning students Section A and afternoon students Section B 1 Indicate a function in Excel that you would use to perform a statistical test to determine whether the mean scores of morning and afternoon students are different use the 5 significance level for the test Be sure to indicate how you would use the output of your function to make your conclusion 4 pts Let and denote the mean scores of morning and afternoon students respectively H1 H0 p value T TEST B2 B51 C2 C51 2 3 if p value 0 05 we conclude that the mean scores of morning and afternoon students are statistically different otherwise we cannot reject the fact that the mean scores of morning and afternoon students are statistically the same Page 2 of 15 2 Construct an approximate 95 confidence interval for the mean difference between the scores of morning and afternoon students and interpret the confidence interval in practical terms 7 pts Pooled SE Approximate 95 CI 3 42 5 12 1 70 8 54 Average difference 83 46 80 04 3 42 Approximate 95 MoE 2 2 56 5 12 2 56 We are roughly 95 confident that the mean score of afternoon students is at most 1 70 below and at most 8 54 above the mean score of morning students In order to get a better sense of the overall performance of his students the instructor combined the two samples and ordered the scores from lowest to highest as depicted in Column A of the following snapshot with the summary statistics presented at the bottom 3 Construct a conservative approximate 95 confidence interval for the percentage of students who would get a score higher than 70 4 pts The sample proportion of students with scores higher than 70 is 100 21 100 79 The conservative approximate 95 MoE is 10 Thus the CI is 69 89 Page 3 of 15 pts 4 The instructor would like to test and see whether the mean score for all students is lower than a constant for some 80 What can you say about the p value and conclusion of that test 4 H1 H0 for some constant 80 Because the sample mean is 81 75 the p value of the test would be The test hypotheses are as follows higher than 0 5 which means that we cannot reject the null hypothesis 5 The instructor would also like to test and see whether the mean score for all students is lower than 85 Please formally state the null and alternative hypothesis that he should test perform the test at the 5 significance level and state your conclusion in practical terms 8 pts H1 85 H0 85 9 9 9 2 53 t stat p value 1 T DIST RT 2 53 99 0 007 Because p value is less than the significance level 5 we accept the alternative hypothesis and conclude that the mean score is below 85 Page 4 of 15 6 If the median score turns out to be 86 what can you say about the skewness of the distribution of grades 3 pts When the median is higher than mean the distribution is negatively skewed 7 Construct an exact 90 confidence interval for the mean score for all students 4 pts Critical t T INV 2T 0 1 99 1 66 SE 12 85 10 1 285 MoE critical t SE 2 13 90 CI sample mean MoE 81 75 2 13 79 62 83 88 8 Using this year s results give a rough estimate for the probability that the average score next year would be lower than 79 3 pts The answer is 2 5 The sample mean i e average score is normally distributed based on the central limit theorem We know that for a normal distribution 95 of the distribution is within 2 SD from the mean Because 79 is roughly 2 standard deviation i e 1 285 below the mean score of 81 75 there is roughly 2 5 chance that the average score will be below 79 9 One can note that as the sample size and therefore degrees of freedom increases the critical t value decreases How can you explain this 3 pts T distribution looks very similar to a standard normal distribution with the difference that it has heavier tails As the degrees of freedom increases the T distribution looks more and more similar to a standard normal distribution with lighter tails therefore the critical t value decreases Page 5 of 15 Boston Housing In an attempt to understand what drives variability in real estate pricing within a given metropolitan area you have collected data on over 500 communities think of a community as larger than a neighborhood but smaller than a town in the greater Boston area The data consists of the median home price in each community as well as a number of factors that could potentially impact home prices Please see below for the list of variables Per capita crime rate Indicating a community on the Charles River Nitric Oxide level pp10M Average number of rooms per dwelling Proportion of owner occupied units built prior to 1940 Weighted distances to five Boston employment centers Square feet of median sized owner occupied home CRIM CHAS NOX RM AGE DIS SF PTRATIO Pupil teacher ratio MEDV Median value of owner occupied homes in 000s Part of the data is presented below Page 6 of 15 1 Do you suspect that this data might present a multi collinearity problem If so please list any pairs of variables that you suspect might cause multi collinearity and provide a very brief justification for your suspicion 2 pts RM …