Question

Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 41 minutes and standard deviation 22 minutes. A researcher observed 41 students who entered the library to study. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. If one randomly selected student is timed, find the probability that this student's time will be between 39 and 44 minutes.
5. For the 41 students, find the probability that their average time studying is between 39 and 44 minutes.
6. Find the probability that the randomly selected 41 students will have a total study time less than 1845 minutes.
7. For part e) and f), is the assumption of normal necessary? YesNo
8. The top 20% of the total study time for groups of 41 students will be given a sticker that says "Great dedication". What is the least total time that a group can study and still receive a sticker? minutes

Answer 1

a)X ~ N(41 ,22)

b) xbar ~ N(41 ,3.4358)

c) ~ N(1681 , 140.8687)

d)

probability =P(39<X<44)=P((39-41)/22)<Z<(44-41)/22)=P(-0.09<Z<0.14)=0.5542-0.4638=0.0905

e)

probability =P(39<X<44)=P((39-41)/3.436)<Z<(44-41)/3.436)=P(-0.58<Z<0.87)=0.8087-0.2802=0.5285

f)

probability =P(X<1845)=(Z<1845-1681)/140.869)=P(Z<(1.1642)=0.8778

g)

No

h)

for 80th percentile critical value of z=		0.842
therefore corresponding value=mean+z*std deviation=			1799.5581