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Chapter 7: Q7.1-4E (page 384)

In Example 7.1.6, identify the components of the statistical model as defined in Definition 7.1.1.

Short Answer

Expert verified

The random variables of interest are the observable heights\({X_1}...{X_n}\) the hypothetically observable mean (parameter) \(\mu \) , and the sample mean \({X_n}\). The \({X_i}\) are modelled as normal random variables with mean 渭 and variance 9.

Step by step solution

01

Given information

The heights of men in a certain population follow the normal distribution with mean\(\mu \)and variance 9,

This time, assume that we do not know the value of the mean 渭, but rather we wish to learn about it by sampling from the population. Suppose that we decide to sample

n = 36 men and let \({X_n}\) stand for the average of their heights. Then the interval\(\left( {\overline {{X_n}} - 0.98,\overline {{X_n}} + 0.98} \right)\) computed in Example 5.6.8 has the property that it will contain the value of \(\mu \) with probability 0.95.

02

Defining statistical model.

A statistical model consists of an identification of random variables of interest (both observable and only hypothetically observable),a specification of a joint distribution for the observable random variables, the identification of any parameter of those distribution that are assumed unknown and possibly hypothetically observable, and a specification for a joint distribution for a unknown parameters.

03

Identifying the component of the statistical model

By applying the definition of statistical model to the given information ,The random variables of interest are the observable heights\({X_1}...{X_n}\) the hypothetically observable mean (parameter) \(\mu \) , and the sample mean \({X_n}\). The \({X_i}\) are modelled as normal random variables with mean 渭 and variance 9.

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Most popular questions from this chapter

Show that each of the following families of distributions is an exponential family, as defined in Exercise 23:

a. The family of Bernoulli distributions with an unknown value of the parameter p

b. The family of Poisson distributions with an unknown mean.

c. The family of negative binomial distributions for which the value of r is known and the value of p is unknown

d. The family of normal distributions with an unknown mean and a known variance

e. The family of normal distributions with an unknown variance and a known mean

f. The family of gamma distributions for which the value of 伪 is unknown and the value of 尾 is known

g. The family of gamma distributions for which the value of 伪 is known and the value of 尾 is unknown

h. The family of beta distributions for which the value of 伪 is unknown and the value of 尾 is known

i. The family of beta distributions for which the value of 伪 is known and the value of 尾 is unknown.

Suppose that a random sample is to be taken from a normal distribution for which the value of the mean 胃 is unknown and the standard deviation is 2, the prior distribution of 胃 is a normal distribution for which the standard deviation is 1, and the value of 胃 must be estimated by using the squared error loss function. What is the smallest random sample that must be taken in order for the mean squared error of the Bayes estimator of 胃 to be 0.01 or less? (See Exercise 10 of Sec. 7.3.)

Suppose that the number of defects in a 1200-foot roll of magnetic recording tape has a Poisson distribution for which the value of the mean 胃 is unknown, and the prior distribution of 胃 is the gamma distribution with parameters \(\alpha = 3\) and \(\beta = 1\). When five rolls of this tape are selected at random and inspected, the numbers of defects found on the rolls are 2, 2, 6, 0, and 3. If the squared error loss function is used, what is the Bayes estimate of 胃?

Let 胃denote the average number of defects per 100 feet of a certain type of magnetic tape. Suppose that thevalue of胃is unknown and that the prior distribution of胃is the gamma distribution with parameters伪=2 and

尾=10. When a 1200-foot roll of this tape is inspected,exactly four defects are found. Determine the posteriordistribution of胃.

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the Bernoulli distribution with parameter 胃, which is unknown, but it is known that 胃 lies in the open interval 0 <胃< 1. Show that the M.L.E. of 胃 does not exist if every observed value is 0 or if every observed value is 1.
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