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Chapter 8: Q. 8.18 (page 391)

Repeat part (a)of Problem8.2when it is known that the variance of a student鈥檚 test score is equal

to25.

Short Answer

Expert verified

Therefore,

P{X鈮�85}鈮�15=.2.

Step by step solution

01

Given Information.

the variance of a student鈥檚 test score is equal to25.

02

Explanation.

Let Xrepresents the test score of a student taking her final examination. Assume that Xis a random variable with mean 渭=75and variance蟽2=25.

Then, using Corollarylocalid="1649833457509" 5.1. (textbook)a=10炉, we get:


localid="1650364326770" P{X鈮�85}=P{X鈮�75+10}鈮�蟽2蟽2+102=2525+100=15=0.2

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Most popular questions from this chapter

Student scores on exams given by a certain instructor have mean 74 and standard deviation 14. This instructor is about to give two exams, one to a class of size 25 and the other to a class of size 64.

(a) Approximate the probability that the average test score in the class of size 25 exceeds 80.

(b) Repeat part (a) for the class of size 64.

(c) Approximate the probability that the average test score in the larger class exceeds that of the other class by more than 2.2 points.

(d) Approximate the probability that the average test score in the smaller class exceeds that of the other class.

by more than 2.2 points.

(a)Let Xbe a discrete random variable whose possible values are1,2,.... If P[X=k]is nonincreasingk=1,2,..., prove that

P(X=k)鈮�2E[X]k2

(b)Let Xbe a non-negative continuous random variable having a nonincreasing density function. Show thatf(x)鈮�2E[X]x2for allx>0.

Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P{0 < X < 40}?

LetZn, n鈮�1be a sequence of random variables andca constant such that for each

蔚>0,P|Zn鈭�c|>蔚鈫�0asn鈫�q. Show that for any bounded continuous functiong,

E[g(Zn)]鈫�g(c)asn鈫�q.

Let X1,X2,鈥�be a sequence of independent and identically distributed random variables with distributionF, having a finite mean and variance. Whereas the central limit theorem states that the distribution of鈭�i=1nXiapproaches a normal distribution as ngoes to infinity, it gives us no information about how largenneed to be before the normal becomes a good approximation. Whereas in most applications, the approximation yields good results whenevern鈮�20, and oftentimes for much smaller values ofn, how large a value of nis needed depends on the distribution ofXi. Give an example of distribution Fsuch that the distribution鈭�i=1100Xiis not close to a normal distribution.

Hint: Think Poisson.
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