Homework Sourse5 The distribution of the amount of money undergraduate stud
5. The distribution of the amount of money undergraduate students spend on books for a term is Normal with a mean of $480 and a standard deviation of $70.
a) If a student is selected at random, what is the probability that this student spends less than $470 on books in a term?
b) If an SRS of 100 undergraduates is selected, what is the probability that their average amount of money spent on books this term is more than $470?
c) The middle 95% of students spend between ____ and ____ dollars on textbooks. d) If an SRS of 100 undergraduates is measured, the middle 95% of averages will fall between ____ and _____ dollars.
d) If an SRS of 100 undergraduates is measured, the middle 95% of averages will fall between ____ and _____ dollars.
Solution
5. let X be the random variable denoting the amount of money that the undergraduate students spend on books for a term.
given that X~N(480,702)
a)here we need to find P[X<470]=P[(X-480)<(470-480)/70]=P[Z<-0.14285]=0.443204 where Z~N(0,1) [answer]
b) let Xbar be the average amount of money spent on books by 100 undergraduate students.
now Xbar~N(480,702/100)
so Standard deviation of Xbar is 70/sqrt(100)=7
here we need to find P[Xbar>470]=P[(Xbar-480)/7>(470-480)/7]=P[Z>-1.4285]=1-P[Z<-1.4285]=1- 0.0765740=0.923426 [answer] [ Z~N(0,1) ]
c) here alpha=0.05
we know P[-taoalpha/2=0.025<Z<taoalpha/2=0.025]=0.95
now tao0.025=1.96 and as noram distribution is symmetric -tao0.025=-1.96
so let a and b are such that P[a<X<b]=0.95
or P[(a-480)/70<Z<(b-480)/70]=0.95
hence (a-480)/70=-1.96 and (b-480)/70=1.96
hence a=342.8 b=617.2
hence The middle 95% of students spend between 342.8 and 617.2 dollars on textbooks
d) lwt c and d are such that P[c<Xbar<d]=0.95
or, P[(c-480)/7<Z<(d-480)/7]=0.95
hence by similar logic from part c
(c-480)/7=-1.96 and (d-480)/7=1.96
or c=466.28 d=493.72
hence If an SRS of 100 undergraduates is measured, the middle 95% of averages will fall between 466.28 and 493.72 dollars [answer]
