e m p i r i c a l r u l / c h e b y s h e v s t h e o r e m w o r k s h e e...

Empiri ca l Rul e /Ch e by s h e v ' s Th e or e m Work s h eet 1) Adul t IQ scor es h a v e a bell - sh a p e d di st ribu t ion wi t h a mea n of 100 a nd a sta nda rd d e vi at ion of 15. U se t h e Empiri ca l Rul e t o find t h e p e rce n ta ge of a dul ts wi t h scor es bet w ee n 70 and 130. 2) L e ngt hs of pr egn a nci es of huma ns a r e norma lly dist ribu t e d wi t h a mea n of 265 d a ys a nd a sta nd ard d e vi at ion of 10 d a y s. U se t he Empiri ca l Rul e t o d et e rmin e t h e p e rce n ta ge of wome n whose pr egn a nci es ar e bet w ee n 255 and 275 d a ys. 3) A comp et e ncy t est h as scor es wi t h a mea n of 82 and a sta nda rd de vi at ion of 2. A hi st ogra m of t h e d ata show s t h at t h e di st ribu t ion is norma l . Bet w ee n wh at t wo v alu es do a bou t 99.7% of t h e v a lu es li e? 4) A pl ace me n t e xa m for e n t r a nce in to a mat h cl ass yi e ld s a me a n of 80 a nd a sta nd a rd de vi at ion of 10. Th e di st ribu t ion of t he scor es i s roughly be ll - sh a p e d. Use t h e Empiri ca l Rul e to find t h e pe rce n ta ge of scor es t h at li e b et w ee n 60 a nd 80. 5) S AT ve rbal scor es a r e norma lly di st ribu t e d wi t h a me a n of 433 a nd a sta nda rd d e vi at ion of 90. Use t h e Empiri ca l Rul e t o d et e rmin e wh at p erce n t of t h e scor es li e bet w ee n 433 and 523. 6) The a v e r a ge IQ of st ud e n ts in a pa r t i cul a r cal culus cl ass i s 110, wi t h a sta nd ard d e vi at ion of 5. The di st ribu t ion i s roughly b ell - sh a p e d. Use t h e Empiri ca l Rul e to find t h e pe rce n ta ge of st ud e n ts wi t h an IQ a bov e 120. 7) A t a te nni s t ourn a me n t a stat ist ici a n k ee ps t r ac k of e v e ry se rv e. The stat i st i ci a n r e por t e d t h at t h e me a n serv e s p ee d of a p a r t ic ul a r pl a ye r w as 97 mil es p e r hour (mph) a nd t he sta nd ard d e vi at ion of t h e serv e s pee ds w as 10 mph. If not hing i s known a bou t t h e sh a p e of t h e di st ribu t ion, giv e an in t e rv a l t h at will con ta in t h e sp ee ds of at l eas tt hr ee - fourt hs of t h e pl a y e r's serv es. 8) H e igh ts of a dul t wome n h a v e a mea n of 63.6 in. a nd a sta nd a rd d e vi at ion of 2.5 in. Wh at does Ch eby sh e v 's Th eor em sa y a bout t h e p e rce n ta ge of wome n wi t h h eigh ts bet w ee n 56.1 in. a nd 71.1 in.? 9) A st udy w as d esign e d to inv est igat e t h e eff ects of t wo v a ri abl es - (1) a st ud e n t's l e v e l of mat h e mat i cal a nxi et y and (2) t eaching met hod - on a st ud e n t's achi e v e me n t in a mat h e mati cs course. St ud e n ts who h a d a low l e v e l of mat h e mat i cal a nxi et y w er e ta ugh t using t h e t r a di t ion a l e xposi t ory met hod. Th ese st ud e n ts obta in e d a mea n scor e of 250 wi t h a sta nd a rd d e vi at ion of 50 on a sta nd a rdiz e d test. A ss uming no informat ion conce rning t h e sh a p e of t h e di st ribu t ion i s known, wh at pe rce n ta ge of t h e st ud en ts scor e d bet w ee n 150 and 350? 1

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Page 1: E m p i r i c a l R u l / C h e b y s h e v s T h e o r e m W o r k s h e e tmrcasalinuovo.weebly.com/uploads/1/0/4/4/104457819/... · 2019-12-10 · B e t w e e n w h a t t w o v

E m p i r ica l R u le/ C heb yshev 's T heorem W or ksheet

1) A d ult I Q scores have a bell - shaped distribution w ith a mean of 100 an d a stan dard dev iation of 15. Use theE m pirical Rule to fin d the percentage of ad ults w ith scores betw een 70 an d 130.

2) Lengths of pregnancies of hu mans are normally distributed w ith a mean of 265 days and a standard dev iationof 10 days. Use the E m pirical Rule to determine the percentage of w omen w hose pregnancies are betw een 255an d 275 days.

3) A com petency test has scores w ith a mean of 82 and a stan dard dev iation of 2. A histogram of the data sho wsthat the distribution is normal. Betw een w hat tw o val ues do about 99.7% of the values lie?

4) A placement exam for entrance into a math class yields a mean of 80 an d a standard deviation of 10. Thedistribution of the scores is roughly bell - shaped. Use the E mp irical Rule to fin d the percentage of scores thatl ie betw een 60 and 80.

5) S A T verbal scores are normall y distributed w ith a mean of 433 an d a stan dard dev iation of 90. Use theE m pirical Rule to determine w hat percent of the scores lie betw een 433 and 523.

6) T he average I Q of stu dents in a particular calculus class is 110, w ith a stan dard dev iation of 5. T he distributionis roughl y bell - shaped. Use the E mp irical Rule to fin d the percentage of stu dents w ith an I Q above 120.

7) A t a tennis tournament a statistician keeps track of every serve. T he statistician reported that the mean ser vespeed of a particular player was 97 miles per hour (m p h) and the stan dard dev iation of the ser ve speeds w as10 m ph. If nothing is k no w n about the shape of the distribution, gi ve an interval that w ill contain the speedsof at least three - fourths of the player's ser ves.

8) H eights of ad ult w omen have a mean of 63.6 in. an d a stan dard de viation of 2.5 in. W hat does C hebyshev'sT heorem say about the percentage of women w ith heights betw een 56.1 in. an d 71.1 in.?

9) A stu d y was designed to in vestigate the effects of tw o variables - (1) a stu dent's level of mathematical anxietyan d (2) teaching method - on a student's achievement in a mathematics course. Stu dents w ho had a low levelof mathematical anxiety w ere taught using the traditional expository method. T hese stu dents obtained a meanscore of 250 w ith a stan dard dev iation of 50 on a standardized test. A ssu ming no information concerning theshape of the distribution is k no w n, w hat percentage of the stu dents scored betw een 150 and 350?

Page 2: E m p i r i c a l R u l / C h e b y s h e v s T h e o r e m W o r k s h e e tmrcasalinuovo.weebly.com/uploads/1/0/4/4/104457819/... · 2019-12-10 · B e t w e e n w h a t t w o v

Which of the following data sets is MOST LIKELY to be normally distributed? For other choices explain which you believe they would NOT be normally distributed. A) The hand span measured from the tip of the thumb to the tip of the pinky of a random sample of high school seniors B) The annual salaries of all employees of a large shipping company C) The annual salaries of a random sample of 50 CEOs of major company, 25 women and 25 men D) The dates of 100 pennies taken from a cash drawer in a convenience store.