Economics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
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1 Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls (1) How do you classify the data by type ad by level of measuremet? What is the sample size? Maually sort the data i ascedig order (A good method that helps sort data maually is the stem-ad-leaf display o page 19 of the textbook). Aswer By type: umerical; by level of measuremet: ratio level of measuremet. Sample size = 24. The ascedig sorted array is (2) Costruct frequecy distributio tables with 3 classes, 6 classes, ad the 10 classes. Draw histograms for each table ad compare the histograms. Aswer (i) The umber of classes is already specified. (ii) We thus start from step two: determiig iterval width. iterval width = max mi umber of classes rouded. For 3 classes, the ratio is (29 1)/3 = ad we roud it up to 9.5 or 10, say. For 6 classes, the ratio is (29 1)/6 = ad we roud it up to 5. For 10 classes, the ratio is (29 1)/10 = 2.8 ad we roud it up to 3. You may choose other umbers for these of course. (iii) Step three: determie class itervals. Just keep i mid the three requiremets: itervals must be of equal legth (usually), iclusive, ad o-overlappig. For 3 classes, we may set the itervals as (> 0, 10), (> 10, 20), (> 20, 30). For 6 classes, we may set the itervals as (> 0, 5), (> 5, 10), (> 10, 15), (> 15, 20), (> 20, 25), (> 25, 30). For 10 classes, we may set the itervals as (> 0, 3), (> 3, 6), (> 6, 9), (> 9, 12), (> 12, 15), (> 15, 18), (> 18, 21), (> 21, 24), (> 24, 27), (> 27, 30). (Sice the data are all itegers, oe might fid it somewhat more coveiet to set the class boudaries as.5 umbers. Thus, for example, for 3 classes, the itervals could be set as (> 0.5, 10.5) 1
2 (> 10.5, 20.5) (> 20.5, 30.5). Either way is fie.) (iv) Step four: cout the frequecies i each classes ad draw the frequecy distributio table. We have them below. Miutes Frequecy >0, >10, 20 7 >20, 30 3 Miutes Frequecy >0, 5 5 >5, 10 9 >10, 15 3 >15, 20 4 >20, 25 2 >25, 30 1 Miutes Frequecy >0, 3 2 >3, 6 4 >6, 9 4 >9, 12 5 >12, 15 2 >15, 18 2 >18, 21 2 >21, 24 2 >24, 27 0 >27, 30 1 The histograms are at the ed of the aswer. As you may see, the patter of the data is ot obvious ( hidde i the wide class itervals ) i the 3-class case. I the 10-class case, the patter is fie as compared with that i the 6-class case, but there is a isolated bar eve though the observatio is ot quite a outlier. Overall, the 6-class oe is better. (3) Costruct a relative cumulative frequecy distributio table from the frequecy distributio table with 6 classes, ad superimpose a ogive o the histogram with 6 classes. it Aswer From the frequecy distributio table for the 6 class case, we directly get the relative frequecy distributio as i the table o the left below (with some umbers rouded up or dow). From this, we ca get the relative cumulative frequecy distributio table as the table o the right below. Miutes Relative Frequecy >0, % >5, % >10, % >15, % >20, % >25, % The ogive is show at the ed of the aswer. Bis Cumulative % % % % % % % (4) Fid the mea, media, ad mode of these legths. Locate them o the horizotal axis of the histogram with 6 classes. Describe the skewess of the 2
3 data ad its relatio with the relative magitude of the mea ad the media for the data. Aswer Mea X = Media= 24+1 th positio value = = Mode= 10, sice its frequecy is 4, the highest amog all observatios. These are marked i the histogram at the ed of the aswer. From the shape of the histogram, we ca see the the tail exteds to the right. Thus, these data are positively skewed. As oe ca see, the mea is also larger tha the media, cofirmig the patter show i the histogram. (5) Fid the mea absolute deviatio, variace, stadard deviatio, coefficiet of variatio, ad iterquartile rage of the data (Note that this is a sample rather tha a populatio). Draw a Box-ad-Whisker plot. Aswer Sice the sample mea is X = 11.75, the deviatios from the mea are (for the ascedig sorted array) Takig absolute values of these, we get the mea absolute deviatios is i deviatio i 24 = The sample variace is equal to 24 (x i X) = The sample stadard deviatio is s = Coefficiet of variatio is CV = s X 100% = % = 63.50%. 3
4 The secod quartile is the 0.25 ( + 1) = 6.25th positio value i the ascedig ordered array. The third quartile is the 0.75 ( + 1) = 18.75th positio value. Thus Q 2 = 6.5, ad Q 3 = 18. Cosequetly, iterquartile rage is IQR = Q 3 Q 1 = = The Box-ad-Whisker plot is at the ed of the aswer. (6) Suppose you do ot kow the data set i detail ad are oly give a frequecy distributio with 6 classes as you costructed i part (2). How would you estimate the sample mea ad the sample stadard deviatio? Very briefly state how they compare with the true sample mea ad sample stadard deviatio. Aswer We take the middle poit of each class iterval as a estimate of the observatio values for that class. These middle poits are m 1 = 2.5, m 2 = 7.5, m 3 = 12.5, m 4 = 17.5, m 5 = 22.5, m 6 = The correspodig frequecies are still f 1 = 5, f 2 = 9, f 3 = 3, f 4 = 4, f 5 = 2, f 6 = 1. There are K = 6 classes. Usig the formulae, we get the sample mea ad the sample variace X = K f i m i = K f i (m i X) 2 1 = (1) Show that (x i X) 2 = x 2 i X 2, where X = 1 x i. This result immediately implies that the sample variace (x i X) 2 1 4
5 ca be writte as x 2 i X 2. 1 Aswer Please see the lecture ote of Jauary 16. (2) The show that ad, thus, X 2 = ( x i ) 2 x 2 i ( x i) 2 1. Aswer X 2 = ( x i ) 2 = ( x i ) 2 = ( x i ) 2. 2 The rest the follows. (3) Suppose you oly kow the followig iformatio about a sample = 25, x i = 609, x 2 i = Calculate the sample mea. Calculate the sample stadard deviatio usig the formula above. Aswer We may use the latter two formulae. The last oe is actually more coveiet. (609)2 s = = Suppose the telephoe call legths i questio 1 are represete by variable X, ad the charges (i cets) for these phoe calls are represeted by aother variable Y, which is related to X by Y = 5 + 3X. Fid the mea ad sample stadard deviatio of the charged usig the formulae for them for liear trasformatios. 5
6 Aswer Ȳ = a+b X = 5+3 X = = 40.25, s Y = b s X = = Write out the Z-score of the observatio 5 i the data set i questio (1). Aswer Z i = x i X = s X = There are three products, labeled 1, 2, ad 3; each of them may be of good quality or defective. (1) The basic outcomes with respect to the quality of the three products are thus 1 is good, 2 is good, 3 is good, 1 is good, 2 is good, 3 is defective,. Complete the list of the 2 3 = 8 basic outcomes. This is the sample space i the questio. Aswer We use g to deote that a product is good ad d to deote that a product is defective. The list of all basic outcomes (or the sample space) is the 1g, 2g, 3g 1g, 2g, 3d 1g, 2d, 3g 1d, 2g, 3g 1g, 2d, 3d 1d, 2d, 3g 1d, 2g, 3d 1d, 2d, 3d (2) Determie whether the evets i each of the statemets below are mutually exclusive, collectively exhaustive, or complemets. Each pair of evets may potetially be described by more tha oe of these three terms. (a) The evet that all three are good ad the evet that exactly oe of them is defective. (b) The evet that all three are good ad the evet that at least oe of them is defective. (If you fid the reasoig u-ituitive, you may write out the basic outcomes i each evet first.) Aswer (a) These two evets are mutually exclusive, i that they caot happe simultaeously. They are ot collectively exhaustive, as there are basic outcomes ot i ay of these evets. Cosequetly, they are ot complemets. (b) These two evets are mutually exclusive ad collectively exhaustive, ad they are complemets. 6
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