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.

EVIDENCE-BASED PUBLIC HEALTH The PERIE Approach ● Problem ● Etiology ● Recommendations ● Implementation ● Evaluation How can we describe a health problem ● The burden of disease ○ Morbidity = disability ○ Mortality = death ● Course of disease ○ How often it occurs ○ How likely it is to be present currently ○ What happens once it occurs ● Distribution of disease ○ Who ○ When ○ Where How can understanding the distribution of disease help us generate hypotheses about disease causation ● Finds group associations or patterns in the freq