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Chapter 6: Q. 35 (page 378)

The length in inches of a cricket chosen at random from a field is a random variable Xwith mean 1.2inches and standard deviation of 0.25 inches. Find the mean and standard deviation of the length Y of a randomly chosen cricket from the field in centimeters. There are 2.54 centimeters in an inch.

Short Answer

Expert verified

The mean and standard deviation are:

渭Y=3.048

蟽Y=0.635

Step by step solution

01

Given Information

Given

渭X=1.2

蟽X=0.25

2.5cm=1inch

Since 1inch is also 2.54cm:

Y=2.54X

02

Explanation

Properties mean and standard deviation:

渭aX+b=a渭X+b

蟽aX+b=a蟽X

Then we obtain for Y=2.54X:

localid="1649911083440" 渭Y=渭2.54X=2.54渭X=2.54(1.2)=3.048

localid="1649911096910" 蟽Y=蟽2.54X=2.54蟽X=2.54(0.25)=0.635

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Most popular questions from this chapter

54. The Tri -State Pick 3Refer to Exercise 42. Suppose 365
(a) Find the mean and standard deviation of your total winnings. Show your work.
(b) Interpret each of the values from (a) in context .

Refer to the previous Check Your Understanding (page 390 ) about Mrs. Desai's special multiple-choice quiz on binomial distributions. We defined X=the number of Patti's correct guesses.

1. Find 渭X. Interpret this value in context.

Exercises 47 and 48 refer to the following setting. Two independent random variables Xand Yhave the probability distributions, means, and standard deviations shown.

48. Difference Let the random variable D=X-Y.

(a) Find all possible values of D. Compute the probability that Dtakes each of these values. Summarize the probability distribution ofD in a table.
(b) Show that the mean of Dis equal to渭X-渭Y.
(c) Confirm that the variance of Dis equal to 蟽X2+蟽Y2.Find all possible values of D. Compute the probability that takes each of these values. Summarize the probability distribution of in a table.

A deck of cards contains 52cards, of which 4are aces. You are offered the following wager: Draw one card at random from the deck. You win \(10if the card drawn is an ace. Otherwise, you lose \)1. If you make this wager very many times, what will be the mean amount you win?

(a) About 鈭�\(1, because you will lose most of the time.

(b) About \)9, because you win \(10but lose only \)1.

(c) About 鈭�\(0.15; that is, on average you lose about 15cents. (d) About \)0.77; that is, on average you win about 77cents.

(e) About $0, because the random draw gives you a fair bet.

49. Checking independence In which of the following games of chance would you be willing to assume independence of Xandlocalid="1649903939419" Y in making a probability model? Explain your answer in each case.
(a) In blackjack, you are dealt two cards and examine the total points localid="1649903945891" Xon the cards (face cards count localid="1649903950298" 10points). You can choose to be dealt another card and compete based on the total pointslocalid="1649903956461" Y on all three cards.
(b) In craps, the betting is based on successive rolls of two dice. localid="1649903966379" Xis the sum of the faces on the first roll, and localid="1649903971075" Yis the sum of the faces on the next roll.
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