251y0112 10 08 01 Part I ECO251 QBA1 FIRST HOUR EXAM OCTOBER 2 2001 Name key SECTION MWF 10 11 TR 11 12 30 10 points 1 Indicate whether the following are Nominal Data Ordinal Data Interval Data Continuous Ratio Data or Discrete Ratio Data 3 a Price to earnings ratio of your stock Ans Continuous Ratio b Number of customers who said that service was unsatisfactory in a survey Ans Discrete Ratio c The Likert Scale rates customer satisfaction with your firm s service on a one to five scale where 1 is exceptional and 5 is unsatisfactory Ans Ordinal Note See text p 13 for most of this discrete continuous was defined in class 2 D 68 Which of the following explains the shape of a distribution best 1 a Mean b Median c Box Plot d Stem and leaf plot e Mode Note See text pp 39 44 3 Make a diagram of a table and show where the field is 1 4 The accompanying box plot shows the sale prices of homes in thousands in a Pennsylvania town 0 30 60 70 80 110 140 a What percent of home prices fall between 60 thousand and 80 thousand why 2 Ans Since 60 is the first quartile and has 25 below it and 80 is the third quartile with 25 above it 50 must be between them b If the mean price is 69 thousand is the data skewed to the left or right 1 Ans Since for data that is skewed to the left Mean median mode because the diagram shows that the median is 70 and the mean is lower it must be skewed to the left 5 Which of the following is a graph that shows cumulative frequency 2 a Histogram b Ogive c Frequency Polygon d Pie chart e None of the above 1 251y0112 10 08 01 Part II Compute an appropriate answer showing your work 15 Points a A distribution of 89 home sale prices has a mean of 67500 a median of 72500 and a standard deviation of 10000 What is the maximum number of homes that have prices that could be below 37500 2 x 37500 67500 3 Ans Since 37500 is 3 standard deviations below the mean z 10000 according to Chebyschev there could only be 1 k2 1 32 1 9 above 97500 this is less than 10 homes b Assume that the distribution above is symmetrical and unimodal Give a rough answer to the question in a and explain your reasoning 2 Ans Since 37500 is 3 standard deviations below the mean the Empirical rule says that there will be almost none below 37500 c The smallest selling price in the distribution above was 25 000 and the largest was 146 000 Note correction If these data are to be presented in five classes what intervals would you use Explain your reasoning using an appropriate formula and use it to fill in the table below 3 146000 25000 24200 so use 25000 This is only a suggestion Any number Ans 5 somewhat above 24200 will work as long as you cover the range Class A B C D E From 25000 50000 75000 100000 125000 to 49999 74999 99999 124999 149999 d WIM technology weighs and measures trucks driving at highway speeds Trucks are classified in a report as follows A WIM gross weight above 70 000 lbs B WIM gross weight 70 00 lbs or less C WIM total length above 60 ft D WIM total length no more than 60 ft Which of the following classes are mutually exclusive Circle 1 5 A and C C and D A B and C Which of the following classes are collectively exhaustive Circle 1 5 A and C C and D A B and C Note This was grade at 0 5 for each item correctly marked or not marked 2 251y0112 10 08 01 there is e For the numbers 3 103 203 303 and 403 compute the i Root mean square ii Harmonic mean iii Geometric mean 2 5 each Solution Note that x 1015 This is not used in any of the following calculations and no reason why you should have computed it i The Root Mean Square 1 1 1 2 x rms x 2 3 2 1032 2032 3032 4032 9 10609 41209 91809 162409 n 5 5 1 1 306045 61209 So x rms x 2 61209 247 404 5 n ii The Harmonic Mean 1 1 xh n xh 1 1 1 1 1 1 1 1 0 353749899 0 070749979 So 5 1 1 n 1 x 5 3 103 203 303 403 5 0 333333333 0 009708738 0 0049261008 0 00 1 x 1 14 13427965 0 070749979 iii The Geometric Mean 1 x g x1 x 2 x 3 x n n n 7659531243 0 2 x 5 3 103 203 303 403 5 7659531243 7659531243 94 8070 Or ln x g 1 n 1 ln x 5 ln 3 ln 103 ln 203 ln 303 ln 403 1 1 0986 4 6347 5 3132 5 7137 5 9989 1 22 7522 4 55684 So 5 5 x g e 4 55684 94 8070 Or log x g 1 n 1 log x 5 log 3 log 103 log 203 log 303 log 403 1 0 47712 2 01284 2 30750 2 48144 2 60531 1 9 88420 1 97684 5 5 1 97684 94 8070 So x g 10 Notice that the original numbers and all the means are between 3 and 403 3 1 5 251y0112 10 08 01 Part III Do the following problems 25 Points 1 I have the following data for sales clerk work hours at a sample of 8 stores 310 254 180 170 116 100 96 320 Compute the following a The Median 1 b The Standard Deviation 4 c The 3rd Decile 2 Solution Compute the Following Note that x is in order Index x x x x x 2 x2 1 96 9216 97 25 9457 6 2 100 10000 93 25 8695 6 3 116 13456 77 25 5967 6 4 170 28900 23 25 540 6 5 180 32400 13 25 175 6 6 254 64516 60 75 3690 6 7 310 96100 116 75 11630 6 8 320 102400 126 75 16065 6 1546 356988 0 00 58223 5 Note that to be reasonable the mean median and 3 rd decile must fall between 96 and 320 n 8 x 1546 x 2 356988 x x 0 00 x x 2 58223 5 a Just put the numbers in order and average the middle numbers x 5 Or formally position p n 1 a b 5 9 4 5 x1 p x 4 x 5 170 180 175 2 2 x a b x a 1 x a so x1 5 x 5 x 4 0 5 x 5 x 4 170 0 5 180 170 175 b x x 1546 193 25 n x x 8 s2 x 2 nx 2 n 1 356988 8 193 25 2 8317 64 or 7 …
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