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Chapter 7: Q7.1-5E (page 384)

In Example 7.1.6, identify any statistical inference mentioned.

Short Answer

Expert verified

Interval \(\left( {\overline {{X_n}} - 0.98,\overline {{X_n}} + 0.98} \right)\) has probability 0.95 of containing \(\mu \) is an inference

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

Identifying the inference from the given information.

From the given information, the statement that the interval \(\left( {\overline {{X_n}} - 0.98,\overline {{X_n}} + 0.98} \right)\) has probability 0.95 of containing \(\mu \) is an inference.

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Most popular questions from this chapter

Suppose that a single observation X is to be taken from the uniform distribution on the interval \(\left[ {{\bf{\theta - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{,\theta + }}\frac{{\bf{1}}}{{\bf{2}}}} \right]\), the value of 胃 is unknown, and the prior distribution of 胃 is the uniform distribution on the interval [10, 20]. If the observed value of X is 12, what is the posterior distribution of 胃?

The uniform distribution on the integers\(1,2,3...\theta \), as defined in Sec. 3.1, where the value of\(\theta \)is unknown\(\left( {\theta = 1,2...} \right);T = \max \left( {{X_{1,}}...{X_n}} \right)\).

Suppose that a random sample of 100 observations isto be taken from a normal distribution for which the valueof the mean 胃is unknown and the standard deviation is2, and the prior distribution of胃is a normal distribution.Show that no matter how large the standard deviationof the prior distribution is, the standard deviation of theposterior distribution will be less than 1/5.

Question: Prove that the method of moments estimator of the mean of a Poisson distribution is the M.L.E.

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?
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