# Question

There are many problems, particularly in industrial applications, in which we are interested in the proportion of a population that lies between certain limits. Such limits are called tolerance limits. The following steps lead to the sampling distribution of the statistic P, which is the proportion of a population (having a continuous density) that lies between the smallest and the largest values of a random sample of size n.

(a) Use the formula for the joint density of Y1 and Yn shown in Exercise 8.52 and the transformation technique of Section 7.4 on page 215 to show that the joint density of Y1 and P, whose values are given by

Is

(b) Use the result of part (a) and the transformation technique of Section 7.4 to show that the joint density of P and W, whose values are given by

Is

For w > 0, p > 0, w + p < 1, and φ(w,p) = 0 elsewhere

(c) Use the result of part (b) to show that the marginal density of P is given by

This is the desired density of the proportion of the population that lies between the smallest and the largest values of a random sample of size n, and it is of interest to note that it does not depend on the form of the population distribution.

(a) Use the formula for the joint density of Y1 and Yn shown in Exercise 8.52 and the transformation technique of Section 7.4 on page 215 to show that the joint density of Y1 and P, whose values are given by

Is

(b) Use the result of part (a) and the transformation technique of Section 7.4 to show that the joint density of P and W, whose values are given by

Is

For w > 0, p > 0, w + p < 1, and φ(w,p) = 0 elsewhere

(c) Use the result of part (b) to show that the marginal density of P is given by

This is the desired density of the proportion of the population that lies between the smallest and the largest values of a random sample of size n, and it is of interest to note that it does not depend on the form of the population distribution.

## Answer to relevant Questions

Use the result of Exercise 8.58 to show that, for the random variable P defined there, What can we conclude from this about the distribution of P when n is large? For random samples from an infinite population, what happens to the standard error of the mean if the sample size is (a) Increased from 30 to 120; (b) Increased from 80 to 180; (c) Decreased from 450 to 50; (d) decreased ...A random sample of size 100 is taken from a normal population with σ = 25. What is the probability that the mean of the sample will differ from the mean of the population by 3 or more either way? Consider the sequence of independent random variables X1, X2, X3, . . . having the uniform densities Use the sufficient condition of Exercise 8.7 to show that the central limit theorem holds. Find the probability that in a random sample of size n = 3 from the beta population of Exercise 8.77, the largest value will be less than 0.90.Post your question

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