Descriptive Statistics Self-Test Answers

Descriptive Statistics Self-Test 1. 2. Giv en t he dat a: 5 .7 a. Find t he m ean. c. Find t he m ode. M ATH 3 1 0 3 .2 4 .6 5 .5 b. Find t he m edian. d. Find t he range. 6 .3 4 .6 e. 5 .9 6 .6 6 .2 6 .4 Est im at e t he st andard dev iat ion. For t his dat a (lif espans, in y ears, of eight een t ort oises of spec ies “ G” ): 138 110 a. b. c. d. e. f. g. 142 148 144 138 138 148 99 149 114 145 162 169 134 140 94 136 Draw a st em -and-leaf diagram f or t he dat a, using classes as indicat ed in part b. Classif y t he dat a, using int erv als of w idt h 1 0 , in such a w ay t hat 9 9 .5 w ill be one of t he boundaries. Draw t he hist ogram f or t he classif ied dat a. Find t he m edian using t he raw dat a. Opt ional: Use t he hist ogram t o est im at e t he m edian. Find t he m ean of t he raw dat a Find t he int erquart ile range. Draw a box plot of t he dat a. 3. Giv e an ex am ple of t w o set s of dat a w it h t he sam e m ean but dif f erent st andard dev iat ions. Giv en t he dat a: 2 5 2 7 2 8 3 2 has m ean 2 8 and st andard dev iat ion Ñ 2 .6 * (h-o had 8 .6 7 , needs square root ), f ind dat a w it h a m ean of 1 3 and t he sam e st andard dev iat ion. (* s = 2 .9 4 ) 4. a. b. c. d. 5. St at e w het her eac h of t he f ollow ing sit uat ions is possible, and, if so, giv e an ex am ple. Can a st andard dev iat ion of a set of dat a be: a. zero? b. negat iv e? c. larger t han t he m ean of t he dat a? 6. a. Giv e an ex am ple of dat a w here t he m edian is bet t er t han t he m ean t o represent “ t y pical v alue” of a populat ion. b. Giv e an ex am ple of a sit uat ion w here w e m ight be m ore int erest ed in t he m ode t han in t he m edian or m ean. 7. On a t est , M r J ones’ class of 4 0 av eraged 6 8 , & M s. Sm it h’ s 2 0 st udent s av eraged 7 4 . Find t he com bined m ean. 8. Tw o hundred kindergart ners w ere asked t heir f av orit e color. Eight y responded " red" ; f if t y " green" ; f ort y " blue" ; six " y ellow " ; and t w ent y -f our nam ed ot her colors. Display t hese result s in a pie chart . 9. In t he f ollow ing pict ograph, each - st ands f or 1 0 0 0 housing unit s. A ccording t o t he graph, how m any housing unit s are: a. included in t he dat a? b. under t en y ears old? c. Est im at e t he m edian age of t he housing unit s. A ges of Housing Unit s A ge (y rs) Num ber of houses Sc ores of 2 5 0 st udent s in BV USD on t he NM A T 0 - 9 -------------1 0 -1 9 -----• • 2 0 -4 9 ------76 82 86 93 97 50+ ------ If ev ery v alue in a set of dat a is t ripled, how are t he m ean and st andard dev iat ion af f ect ed? If t en is added t o ev ery v alue in a set of dat a, how are t he m ean and st andard dev iat ion af f ect ed? If ev ery v alue in a set of dat a is doubled, t hen decreased by f iv e, how are t he m ean and st d. dev . af f ect ed? The m ean and t he st andard dev iat ion of a set of " x " dat a (x 1 , x 2 , x 3 , et c) are 4 2 and 1 0 respect iv ely , w hat are t he m ean and st andard dev iat ion of t he new " y " dat a obt ained by let t ing each y be (x – 4 2 )/ 1 0 f or each x v alue? (That is y 1 = (x 1 – 4 2 )/1 0 and y 2 = (x 2 – 4 2 )/1 0 and so on.) 10. Using t he box plot abov e right , f ind t he range and t he int erquart ile range. How m any of t he BV USD st udent s scored bet w een 8 2 and 9 3 on t he NM A T? 11. J onat hin earned 8 5 point s on t he f irst t w o t est s, and 6 8 on t he t hird. W hat score does he need on t he upcom ing f ourt h t est so t hat his av erage on t he t est s w ill be at least a B (8 0 and abov e)? Should he st udy his head of f , in an ef f ort t o earn an A ? Susan has an av erage of 8 6 f or her f irst t hree t est s. W it h a score of 1 0 0 on t he f ourt h t est , w hat w ill her av erage be? 12. The dist ribut ion at right appears as t hough it could hav e a st andard dev iat ion c lose t o: a. – 2 0 b. 0 c. 2 0 d. 4 0 e. 6 0 – 20 – 10 r v 1 s t a t _ x x re v s 8 * * * SEE A LSO THE PRA CTICE FINA L SECTION B. * * * 0 10 20 30 40 Descript iv e St at ist ics Self -Test A nsw ers 1. 2 a. a. 5.5 b. 5.8 ( s = { (2.3 2 + .9 2 + @@@ + 1.1 2 )/9 } 1/2 Ñ 1 .1 1 ) e. F Ñ 1 .0 5 c . 4.6 d. 3.4 Lif espans of 1 8 2 b . 1 8 t ort oises of spec ies G spec ies G t ort oises Lif espan (y rs. ) f 16 29 8 9 .5 - 9 9 .5 2 15 9 9 . 5 -1 0 9 . 5 0 14 0245889 1 0 9 . 5 -1 1 9 . 5 2 13 46888 1 1 9 . 5 -1 2 9 . 5 0 12 1 2 9 . 5 -1 3 9 . 5 5 11 04 Leg en d : 1 3 9 . 5 -1 4 9 . 5 7 10 9 49 1 4 9 . 5 -1 5 9 . 5 0 9 49 represent s 1 5 9 . 5 -1 6 9 . 5 2 9 4 y rs & 9 9 y rs. 2 d. M edian f rom t he raw dat a: 1 3 9 (av erage 1 3 8 & 1 4 0 ) 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 (If y ou did not hav e t he raw dat a, y ou c ould est im at e m ean f rom hist ogram : 9 4 .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 T he sm all dif f erenc e in t he result is due t o t he inac c urac ies int roduc ed by using t he m id-v alues f or eac h c lass in p lac e o f t h e ac t u al d at a v alu es. ) 2f. IQ R = Q 3 – Q 1 = 1 ½ IQ Rs = 2 1 148 – 134 = 14 134 – 21 = 113 .. . m ak ing 9 4 , 9 9 & 1 1 0 all out liers. Lif espans of 1 8 G t ort oises, in y ears: 2 g. t 94 t 99 t C 110 114 C 134 139 148 169 Dat a set : 1 0 ,2 0 ,3 0 has m ean 2 0 , F Ñ 2 .6 (s Ñ 3 .1 ). Dat a set : 1 9 ,2 0 ,2 1 has m ean 2 0 , F Ñ.8 (s = 1 ). Dat a set : 1 0 , 1 2 , 1 3 , 1 7 w ill hav e m ean 1 3 (15 less than 28) and st andard dev iat ion unchanged! 4 . a. V alues A ND dist ances are t ripled! so m ean and s are bot h t ripled. b. V alues inc rease, dist anc es bet w een values do not change. St d Dev is unchanged, m ean increases by 1 0 . c . M ean is doubled, less f iv e; s doubles. d. 0 & 1 . 5 . a. Yes. { 4 , 4 , 4 , 4 , 4 } b. No. ( q% $ 0 ) c. Y es, t hey are independent { -2 0 , 0 , 2 0 } 6 a. The m edian is a b et t er indic at or of a " t y pic al" v alue w hen som e ex t rem ely high or ex t rem ely low v alues are inv olv ed; f or inst anc e, in a c lass of t w ent y st udent s, m ost receiv e an allow ance bet w een $ 1 and $ 5 per w eek, but one child receiv es $ 1 5 0 /w eek—t hat one high v alue w ould raise t he m ean f rom som ew here bet w een $ 1 and $ 5 t o bet w een $ 8 and $ 1 2 , y et only one child receiv es ov er $ 5 . (More discussion: see DSB-6.) 6 b. I am going t o m anu f ac t ure hat s in one size only . I w ant t o c hoose t he size t hat w ill f it t he great est num ber of w om en. Suppose, f or ex am ple, 4 5 % of w om en w ear size " S" , 3 0 % " M " , 2 0 % " L" , and 5 % " X L" . By m anuf act uring size " S" , m y hat s w ill f it 4 5 % of w om en. If S is 1 8 " , M is 1 9 " , L is 2 1 " and X L is 2 3 " , t he m edian is 1 9 " (M ) and t he m ean is 1 9 .1 5 " (M )—size M hat s w ould f it only 3 0 % . (In reality, I’ d make 19" hats, labelled “ S” !) 7 . 7 0 . ( How : (40x66+ 20x76)/60 . Not e t here are t w ice as m any in J ones’ class, and t he m ean is t w ice as close t o 8 . See right . 6 8 as it is t o 7 4 . ) (e.g. Red: 8 0 /2 0 0 = 4 0 % 4 0 % of 3 6 0 E = 1 4 4 E) 9 . a. (1 4 + 6 + 7 + 6 ) x 1 0 0 0 housing unit s = 3 3 0 0 0 housing unit s. OTHER ú b. 1 4 0 0 0 unit s are under 1 0 y ears old. c. The m edian age is about half w ay f rom YELLOW ú 1 0 t o 1 9 .. . w hich is roughly 1 5 y ears old. 3% 12% 3. 10. Range = 9 7 – 7 6 = 2 1 IQR = 9 3 – 8 2 = 1 1 Perc ent age w it hin m iddle t w o quart iles = 5 0 % BLUE 20% 11. J onat hin needs 4 x 8 0 = 3 2 0 pt s t ot al. He already has 2 3 8 , so he needs 8 2 (or bet t er). For an A , he needs 4 x 9 0 = 3 6 0 pt s t ot al. 3 6 0 – 2 3 8 = 1 2 2 pt s. Not possible. Susan’ s new av erage w ill be (1 0 0 + (3 x 8 6 )) /4 = 8 9 .5 (Susan needs som e ex t ra bonus pt s. t o get an A .) RED 40% GREEN 25% 1 2 . 2 0 is t he best answ er. 4 0 is a lit t le t oo high; not ice t he ent ire range is 6 0 . a,b,e are obv iously im possible! 2 c. # of tortoises Lif espans of 1 8 t ort oises of species G 7 6 5 4 3 2 1 0 90—99 100— 109 110— 119 120— 129 130— 139 140— 149 150-159 160— 169 Lif espans (in y ears) * * * SEE A LSO THE PRA CTICE FINA L SECTION B. * * *