Stats Chapter 7 Homework
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6.2 Real Applications of N
ormal Distributions What to read and understand in this section? This section involves nonstandard normal distributions, where the center is not zero but equals the actual mean value for that particular problem. However, it is rare to find other real applications of a standard normal distribution. We now extend the methods of the previous section so that we can work with any nonstandard normal distribution (with a mean different from O and/or a standard deviation different from 1). The formula below relates x and z. z is the z-score and x is the individual data value. The key is a simple conversion (Formula below) that allows us to "standardize" any normal distribution so that x values can be transformed to z scores; then the methods of the preceding section can be used. Formula ><-A ----
6 Procedure for Finding Areas with a N
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ormal Distribution \; slltc~ ()\._ . Vl<:::.r:f"\.~~ ('..0-VV,:I \ C,\,b C ~+k c"\cc-.v--. G.v..0-
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-.'.>c,<u'C The area in any normal distribution bounded by some score x (as in Figure below) is the same as the area bounded by the corresponding z score in the standard normal distribution 8
p X (a) Nonstandard Normal Distribution f"' lz= --
0 z (b) Standard Normal. Distribution Find the area of the shaded region in the examples below. Assume that the data normally distributed with a mean of 100 and a standard deviation of 15. Example \,n 61 vV1 o--
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\\Ct:zi Q: Mickey Mouse Disney World requires that people employed as a Mickey Mouse character must have a height between 56 in. and 62 in. Note that Men's heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in. Women's heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in. 10
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a) Find the percentage of men meeting the height requirement. What does the re-
sult suggest about Lhe genders of the people who are employed as Mickey Mouse characters? 7 -r -
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b) If the height requirements are changed to exclude the tallest 50% of men and the shortest 5% of men, what are the new height requirements? X 5°tv l-c: ;< v~l l..\.(_ i > Inv' )\Jovi/"\ (o.0s-
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a) If she cunres by adding 15 to each grade, whnt is lhe new mean and st,1ndard deviation? l.o 'v , I J l , ' x ' t' )( t. ! >, -i 'r ... -
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c) If the grades are curved so that grades of Bare given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B. I} N r
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h.4 Central Limit Theorem 1h11. is the 0nh lhnorcm used in thi~ courqc, nnrl thi<-
is on r,tr\-,.,ncly import.1nl idea from n thmn pcmpect1Vl"'. Thi" theorem allow!-, U'-
to u"c tht' Normal di~tribution for ,;omr 1mportnnt .. 1
,r,ltc i1ll(lns Given any population Wlth ony d1~tribution (unifom1, 6kcwcd, '"'-'hntt.-VC't) the <listrihution of snmplc means can be approximated hy a normal distrihution ~
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hen the snmplcs nre large enough with n > 30. For o ll snmpl~ of the same size n with n > 30 the sampling distribution of f can he l,e.rrox1matPla. by a normal distribution with mean 1' and standard deviation u I ,f,i. Key elements 1 , 1 ( .) , -1--
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Con1ideralions for Practical Problem Solving l• t r \ \ t I l (" I\ V'\ ._ ~U \ \ , "I\ I ~, i ' I.I c-.v, 1.;.1 .....
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x PX 6.4.1 Applications of Central Limit Theorem Many practical problems can be solved with the central limit theorem. Example below is a good illustration of the central limit theorem because we can see the difference between working with an individual value in part (a) and working with the mean for a sample in part (b). Q: American Airlines uses Boeing 737 jets with 126 seats in the main cabin. In an attempt to create more room, an engineer is considering a reduction of the seat width from 16.6 in. to 16.0. in. Adult males have hip widths that are normally distributed with a mean of 14.3 in. and a standard deviation of 0.9 in. (based on data from Applied Ergonomics). 15
n) l md tlw p1ohab1lity 1h:il i1 1;1ndnmly 5el1><.t1 d ,1<lult 1111111 h;i~ ,1 i ip \\ idth gr1 Jt r th.rn llw h•'•'' width 111 lh () 111 1,l \ I \ (( . • , II y V\ ,',,\ 1.J{ J I l,. { f \(. v '1 (1 <\q<t (U ,, I l) b) Find the probability that 126 main cabin seats are all occupied by males with a mean hip width greater than the seat width of 16.0 in. ~lv,1..,, WL o.V-c. v\l',J111~ v ... (L, V"-\1v-•\ v~ (l :St•""'fll0 0 ' h ,_ \ I \ • ' J • J l . .lv"I c, l, we. LA.~ e '\ ll...o ~~V-
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L_ c) For the design of the aircraft seals, which is more relevant: The result from part (a) or the result from part (b )? Why? What do the results suggest about the reduction of the seat width to 16.0 in.? f' \
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\/\ IG,CJ (._ ~~{Al (.Q1,.lJ \cc..J +o 'i;S~;~IC~~+ ('~C. \~C."'St.j Q: Water Taxi Safety Passengers died when a water taxi sank in Baltimore's Inner Har-
bor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 "Body Data" in Appendix). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb. a) Given that the water ta.xi that sank was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers? x~ ·
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r b) If the water taxi is filled with 25 randornJy selected men, what is the probabiJityl that their mean wcight exceeds the value from part (a)? ()\ '.) I ,,, ) 1" \ Vt.,\ l l •. ('\I L I \; \.._ ( ._, C I\., )u\ .. -:. l 'f ,, \ L V\--
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1 '1 c) After the water taxi sank, the weight assumptions were revised so that the new capacity became 20 passengers. If the water taxi is filled with 20 randomly selected men, what is the probability that their mean weight exceeds 175 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb? s I vi s G:.V'\v U\, l I ,"\ '. ,i., _\-
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