Descriptive Statistics Self-Test

De script ive S ta tist ics Se lf- Te st M AT H 3 1 0 F6 1. Given t he dat a: 5 .7 a. Find the mean. d. Find t he range. 2. For th is dat a (lifes pans, in y ears, o f eig ht een t ort oises of speci es “ G” ): 138 110 a. b. c. d. e. f. g. 142 148 144 138 3 .2 4 .6 5.5 6.3 4.6 5.9 6.6 b. Find the median. c. Find t he mo de. e. Estimate the standard deviation. 138 148 99 149 114 145 162 169 134 140 6.2 6. 4 94 136 Draw a stem-and-leaf diagram for the data, using classes as indicated in part b. Classify th e dat a, usin g int ervals of w idt h 1 0, in suc h a w ay t hat 99 .5 w ill be one o f t he bou ndaries. Draw th e hist ogram fo r th e classif ied dat a. Find the median using the raw data. Optional: Use the histogram to estimate the median. Find t he mean of th e raw dat a Find t he int erquart ile range. Draw a box plot of th e dat a. 3. Give an ex amp le of tw o set s of dat a w ith th e same m ean but dif feren t st andard deviat ions. Given t h e d at a : 2 5 2 7 28 3 2 has mean 28 and standard deviation . 8.67 , find data wit h a mean of 13 and the same standard deviation. 4. a. b. c. d. If every value in a set of data If ten is added to every value If ev ery v alue in a set of dat a affected? If t h e m ea n an d t h e s t an d ar d and st andard deviat ion of th e is tripled, how are the mean and standard deviation affected? in a set of data, how are the mean and standard deviation affected? is doub led, t hen dec reased by fiv e, how are the mean and standard deviation de v ia t io n o f a s et o f " x " dat a are 42 and 1 0 resp ect ively , w hat are th e mean new " y" dat a obt ained by lett ing eac h y b e (x– 42 )/1 0 f or each x v alue? 5. S t at e whet h e r e ac h o f th e fo l lo w i n g si tu ations is possi bl e, and, if so, give an example. Can a st and ard dev iat ion of a set of dat a be: a. ze ro? b. n egat ive? c. larg er th an t he mean of th e dat a? 6. a . Gi v e an example of data w here the median is better than the mean to represent “t ypical value” of a population. b. Give an example of a situation w here w e might be more interested in the mode than in the median or mean. 7. On a test , M r Jon es’ c lass of 30 averaged 70, & Ms. Smith’ s 20 students averaged 75. Find th e com bined mean. 8. Tw o hun dred kindergart ners w ere asked their fav orite co lor. Eight y responded " red" ; fif ty " green" ; fo rty " blue" ; six " yellow " ; and t w enty -four n amed ot her colors. Display these results in a pie chart . 9. In the following pictograph, each - stands for 100 0 housing units. According to the graph, how many housing u ni t s are: a. inc luded in th e dat a? b. un der t en years o ld? c. Estim ate t he med ian age of th e housin g unit s. Ag e (yrs) 0 - 9 10-19 20-49 50+ Num ber of houses -------------------------------------- Scores of 25 0 st udent s in BMGUSD on t he NM SAT 76 82 86 93 97 10. Us ing t he bo x pl o t a bo v e rig h t, fi n d th e range, and the i nterquarti le range. How many of the BMGUSD students scored betw een 82 and 93 on the NMSA T? 11. Jonathon earned 85 points on the first tw o tests, and 68 on the third. What score does he need on the upcoming f o ur t h test so t hat his averag e on t he t ests w ill be at least a B (80 and above)? Should he stu dy h is head of f, in an effort to earn an A ? Susan has an av erage of 86 fo r her f irst t hree tests. With a score of 10 0 on the fourth test, w hat w ill her average be? 12. The distribution at right appears as though it could have a standard deviation c loses t t o: a. – 2 0 b . 0 c. 20 d . 40 e. 60 – 20 – 10 0 10 20 30 40 Descript ive St atist ics Self -Test An sw ers 1. 2a. a. 5.5 b. 5.8 c . 4.6 d . 3.4 F6 e . F Ñ 1 . 05 , L if e sp an s o f 1 8 2 b . 1 8 tort oises of sp ecies G species G to rtoises Lifes pan (y rs.) f 16 29 89.5 - 99.5 2 15 99.5 -109.5 0 14 0245889 109 .5-119 .5 2 119 .5-129 .5 0 13 46888 12 129 .5-139 .5 5 11 04 Legend: 139 .5-149 .5 7 10 9 49 149 .5-159 .5 0 9 49 r ep re se nt s 159 .5-169 .5 2 94 yrs & 99 yrs. s = { (2.3 2 + .9 2 + @@@ + 1.1 2 ) /9 } 1/2 Ñ 1 . 1 1 2 d . Med ian from the raw d at a : 1 3 9 (average 1 38 & 1 40 ) 2 e . raw : (9 4 + 9 9 + 1 1 0 + 1 1 4 + . .. + 1 6 9 ) /1 8 = 2 4 4 8 / 1 8 = 1 3 6 ( Es t i m a t e d f r o m h i s t o g r a m w o u l d b e . . . ( 94 . 5 x 2 + 1 0 4 . 5 x 0 + 1 1 4 . 5 x 2 + . . . + 1 6 4 . 5 x 2 )/ 1 8 . 1 3 5 . 1 The sm all diff erence is due t o the in accurac ies i n t r o d u c e d b y c l a ss i f y i n g t h e d a t a (e s s en t i a l l y re p l a c in g a c t u a l v a l u es b y t h e m i d - v a l u es f o r e a c h c l as s . ) 2f. IQR = Q3 – Q1 = 1½ IQRs = 21 13 4 – 148 – 134 = 14 21 = 11 3 .. . mak ing 94 , 99 & 1 10 all outliers. Lifespans o f 1 8 G t ortoises, in years: 2g. t 94 t 99 t C 110 114 C 134 139 148 169 Dat a set : 10 ,2 0 ,3 0 h a s m ea n 2 0 , F Ñ 2 . 6 (s Ñ 3.1). D ata set: 19,20,21 has mean 20, F Ñ .8 (s = 1). Data set: 10 , 12, 13, 1 7 w ill have mean 13 ( 15 less than 28) and standard deviation unchanged! 4 . a. V alues & dist ances are t ripled ! s o mean and s are both tri pled. b. V alues in c re as e , d i sta n c e s d o n o t. Std D ev i s unchanged, mean i ncreases by ten. c . M ean is d o u bl ed , l es s fi v e ; s d o u b le s. d. 0 & 1. 5 . a. Y es . { 4, 4 , 4 , 4 , 4 } b . N o . (o $ 0) c. Y es, they are independent { -20, 0, 20} 6 a. The median is a bett er indic ato r of a " ty pical" value w hen some extremel y hi gh or ext remely low values are invo lved; fo r inst ance, in a class of tw ent y st udent s, m ost receiv e an all ow ance betw een $1 a nd $5 per week, but one c hild rec eives $ 15 0/ w eek—th at o ne high value w ould r aise the mean from somewhere betw een $1 and $5 t o bet w e en $ 8 a n d $ 1 2 , y e t o n l y o n e chi ld receives over $5. (More discussion: see bottom of page S4.) 6b. I am going t o m a n u fa c tu re h ats i n on e si ze only. I w ant to ch oose t he size that w ill fit the greatest number of w omen. Suppo se, f or exam ple, 4 5% of w om en w ear size " S " , 3 0 % " M " , 20% " L" , and 5% " XL". By manufacturing s iz e " S " , my h a ts w i l l fit 45% of w omen. If S i s 18" , M i s 19" , L is 21 " and X L is 23 " , th e median is 19" ( M ) an d th e me an i s 1 9 .1 5 " (M )—size M hats w oul d fi t onl y 30% . (In reality, I’d make 19" hats, labelled “S”!) 7 . 72. ( How: (30x70+ 20x75)/50 . N o te th e resul t i s 2/5 of the w ay betw een 70 and 75.) 8 . See right. ( e. g . Re d: 8 0 / 2 0 0 = 4 0 % 4 0 % o f 3 6 0 ° = 1 4 4 °) 9 . a. (1 7 + 13 + 8 + 3) x 10 00 housin g unit s OTHER ú = 41 00 0 h ousing unit s. 12% b. 1700 0 un i ts are un d e r 1 0 y e a rs o l d . YELLOW ú c. The median age is about (4/13) of the w ay from 3% 10 t o 19.. . w hich is about 13 y ears old. 1 0 . Ra ng e = 9 7 – 7 6 = 2 1 IQ R = 9 3 – 8 2 = 1 1 BLUE RED Percent age w ith in mid dle t w o quar tiles = 50 % 20% 40% 3. 11. J onat hon nee d s 4 x 8 0 = 3 2 0 p t s to ta l. He already has 2 38 , so h e needs 8 2 (o r bet ter). F or an A , he n ee d s 4 x 9 0 = 3 6 0 p t s to t al. 36 0 – 23 8 = 12 2 p ts. Not possible. S usan’s new a v e ra g e w i l l b e (1 0 0 + (3 x 86)) /4 = 89.5 ( S us an need s ex tra b o n us p ts . to g et a n A .) GREEN 25% 1 2 . 20 is th e best answ er. 4 0 is a lit tle t oo hig h. a, b,e are im possible! Number of tortoises 2 c. Lifesp ans of 18 to rto ises of species G 7 6 5 4 3 2 1 0 89.5 99.5 109.5 119.5 129.5 Lifesp ans (in y ears) 139.5 149.5 159.5 169.5