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# Applet Exercisea Use the applet Chi-Square Probabilities ISBN: 9780495110811 47

## Solution for problem 23E Chapter 7

Mathematical Statistics with Applications | 7th Edition

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Problem 23E

Problem 23E

Applet Exercise

a Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y )] when Y has χ 2 distributions with 10, 40, and 80 df.

b What did you notice about P[Y > E(Y )] as the number of degrees of freedom increases as in part (a)?

c How does what you observed in part (b) relate to the shapes of the χ 2 densities that you obtained in Exercise 7.22?

Reference

Applet Exercise As we stated in Definition 4.10, a random variable Y has a χ 2 distribution with ν df if and only if Y has a gamma distribution with α = ν/2 and β = 2.

a Use the applet Comparison of Gamma Density Functions to graph χ 2 densities with 10, 40, and 80 df.

b What do you notice about the shapes of these density functions? Which of them is most symmetric?

c In Exercise 7.97, you will show that for large values of ν, a χ 2 random variable has a distribution that can be approximated by a normal distribution with μ = ν and σ = √ 2ν. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the χ 2 random variable Y ?

d Refer to the graphs of the χ 2 densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the χ 2 distribution can be approximated with a normal distribution. Does this surprise you? Why?

Reference

Let X1, X2, . . . , Xn be independent χ 2-distributed random variables, each with 1 df. Define Y as b A machine in a heavy-equipment factory produces steel rods of length Y, where Y is a normally distributed random variable with mean 6 inches and variance .2. The cost C of repairing a rod that is not exactly 6 inches in length is proportional to the square of the error and is given, in dollars, by C = 4(Y μ)2. If 50 rods with independent lengths are produced in a given day, approximate the probability that the total cost for repairs for that day exceeds \$48.

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##### ISBN: 9780495110811

This full solution covers the following key subjects: distribution, random, applet, exercise, variable. This expansive textbook survival guide covers 32 chapters, and 3350 solutions. This textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. The answer to “Applet Exercisea Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y )] when Y has ? 2 distributions with 10, 40, and 80 df.b What did you notice about P[Y > E(Y )] as the number of degrees of freedom increases as in part (a)?c How does what you observed in part (b) relate to the shapes of the ? 2 densities that you obtained in Exercise 7.22?ReferenceApplet Exercise As we stated in Definition 4.10, a random variable Y has a ? 2 distribution with ? df if and only if Y has a gamma distribution with ? = ?/2 and ? = 2.a Use the applet Comparison of Gamma Density Functions to graph ? 2 densities with 10, 40, and 80 df.b What do you notice about the shapes of these density functions? Which of them is most symmetric?c In Exercise 7.97, you will show that for large values of ?, a ? 2 random variable has a distribution that can be approximated by a normal distribution with ? = ? and ? = ? 2?. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the ? 2 random variable Y ?d Refer to the graphs of the ? 2 densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the ? 2 distribution can be approximated with a normal distribution. Does this surprise you? Why?ReferenceLet X1, X2, . . . , Xn be independent ? 2-distributed random variables, each with 1 df. Define Y as b A machine in a heavy-equipment factory produces steel rods of length Y, where Y is a normally distributed random variable with mean 6 inches and variance .2. The cost C of repairing a rod that is not exactly 6 inches in length is proportional to the square of the error and is given, in dollars, by C = 4(Y ? ?)2. If 50 rods with independent lengths are produced in a given day, approximate the probability that the total cost for repairs for that day exceeds \$48.” is broken down into a number of easy to follow steps, and 361 words. The full step-by-step solution to problem: 23E from chapter: 7 was answered by , our top Statistics solution expert on 07/18/17, 08:07AM. Since the solution to 23E from 7 chapter was answered, more than 306 students have viewed the full step-by-step answer. Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811.

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Applet Exercisea Use the applet Chi-Square Probabilities