5. Amazon.com invites customers to rate books on a scale of 1 to 5 stars. Suppos

Question

5. Amazon.com invites customers to rate books on a scale of 1 to 5 stars. Suppose we randomly select a reviewer and base our decision whether to buy the book on how many stars that person gave it. Below is the probability distribution of number of stars.

Number of Stars 1 2 3 4 5

Probability 0.290 0.146 0.195 0.206

a) What is probability that the reviewer gave it 3 stars?

b) What is probability that the reviewer gave it at most 2 stars?

c) What is the mean and standard deviation of this distribution?

d) Is it unusual for the reviewer to give a book 5 stars? Explain.

Explanation / Answer

Let Number os stars be X = 1,2,3,4,5

So, X: 1 2 3 4 5

P(X): 0.290 0.146 0.195 0.206 Not given

(a) Prob ( 3 Stars) = 0.195

(b) P( atmost 3) = P ( maximum three ) = P( it may be 1 or 2 or 3) = P(1) + P(2) + P(3 ) = 0.290+0.146+0.195 = 0.631

(c) In order to find the Mean and standard devaition let us complete the distribution first

As sum of the probabilty is = 1

For x = 5 P( x= 5) = 1- { P(1)+P(2)+P(3)+P(4)] 1-0.837 = 0.163

Mean = E(X) = xP(x) = 1*0.290+2*0.146 +3*0.195+4*0.206+5*0.163 = 2.806

Var (X) = E(X2) - (E(X))2 = 10- (2.806)*2.806 = 2.126364

Standard Deviation = 2.126364 = 1.458

(d) It is a comperative issue.

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