Problem 22E

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.

1.1 - Population, Sample, and Procedures Monday, August 28, 2017 7:00 PM Population an investigation will typically focus on a well-defined collection of objects constituting a number of interests Census what desired information is available for all objects in the population Sample a subset of the population that is selected EX: "A sample of 10 students out of 130" Characteristic categorized such as gender, type of malfunction, or numerical in nature Variable any characteristic where the value may change from one object to another in the population Univariate consists of observationof a single variable Data Set EX: "Determining the type of tran