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Chapter 6: Q.119 (page 432)

Standard deviations (6.1) Continuous random variables A, B, and C all take values between 0 and 10 . Their density curves, drawn on the same horizontal scales, are shown here. Rank the standard deviations of the three random variables from smallest to largest. Justify your answer.

Short Answer

Expert verified

the standard deviations from smallest to largest is B, C, A.

Step by step solution

01

Given Information

The figures are

02

Simplification

The standard deviation calculates the average departure of the data values from the mean.渭.

This then means that if more number of data values lie closest to 渭, then the standard deviation would be smaller.

Because the peak of the density curve is largest at, the standard deviation of B is the least. 渭and therefore the most of the data values lie very close to the means 渭.

The standard deviation of A is largest, because of the peak of the density curve is the highest at the points furthest from the mean 渭and therefore the most of the data values lie far away from the mean 渭.

As a result, the standard deviations from lowest to greatest are B,C,A.

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Most popular questions from this chapter

Taking the train Refer to Exercise 80 . Would you be surprised if the train arrived on time on fewer than 4 days? Calculate an appropriate probability to support your answer.

Fire insurance Suppose a homeowner spends \(300 for a home insurance policy that will pay out \)200,000 if the home is destroyed by fire in a given year. Let P = the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is 0.0002.

(a) Make a table that shows the probability distribution of P.

(b) Calculate the expected value of P. Explain what this result means for the insurance company.

(c) Calculate the standard deviation of P. Explain what this result means for the insurance company.

Working outExercise 10 described a large sample survey that asked a sample of people aged 19 to 25 years, 鈥淚n the past seven days, how many times did you go to an exercise or fitness center or work out?鈥� The response Y for a randomly selected survey respondent has the probability distribution shown here. From Exercise 10, E(Y)=1.03. Find the standard deviation of Y. Interpret this value.

Class is over! Mr. Shrager does not always let his statistics class out on time. In fact, he

seems to end class according to his own 鈥渋nternal clock.鈥� The density curve here models

the distribution of Y, the amount of time after class ends (in minutes) when Mr. Shrager

dismisses the class on a randomly selected day. (A negative value indicates he ended class

early.)

a) Find and interpret P(鈭�1鈮�Y鈮�1).

b) What is 渭Y ? Explain your answer.

c)Find the value of k that makes this statement true: localid="1654015283453" P(Y鈮�k)=0.25

Life insurance If four 21-year-old men are insured, the insurer鈥檚 average income is

V=X1+X2+X3+X44=0.25X1+0.25X2+0.25X3+0.25X4

where Xiis the income from insuring one man. Assuming that the amount of income earned on individual policies is independent, find the mean and standard deviation of V. (If you compare with the results of Exercise 57, you should see that averaging over more insured individuals reduces risk.)
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