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Example 9.6: What are the standard errors of the sample mean X and the sample proportion p?

Example 9.6: What are the standard errors of the sample mean X and the sample proportion p?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Example 9.6

**Theorem 9.2:** Let \( X_1, X_2, \ldots, X_n \) be a random sample from a normal distribution with mean \( \mu \) and standard deviation \( \sigma \). Then the estimator \( \hat{\mu} = \overline{X} \) is the MVUE for \( \mu \). **Reporting a Point Estimate - The Standard Error:** - Besides reporting the value of a point estimate, some indication of its precision should be given. The usual measure of precision is the **standard error** of the estimator. **Definition:** The **standard error** of an estimator \( \hat{\theta} \) is its standard deviation \( \sigma_{\hat{\theta}} = \sqrt{\sigma^2_{\hat{\theta}}} = \sqrt{\text{Var}(\hat{\theta})} \). If the standard error itself involves unknown parameters whose value can be estimated, we can obtain an **estimated standard error**, which is typically denoted by \( \hat{\sigma}_{\hat{\theta}} \). **Example 9.6:** What are the standard errors of the sample mean \( \overline{X} \) and the sample proportion \( \hat{p} \)?
Transcribed Image Text:**Theorem 9.2:** Let \( X_1, X_2, \ldots, X_n \) be a random sample from a normal distribution with mean \( \mu \) and standard deviation \( \sigma \). Then the estimator \( \hat{\mu} = \overline{X} \) is the MVUE for \( \mu \). **Reporting a Point Estimate - The Standard Error:** - Besides reporting the value of a point estimate, some indication of its precision should be given. The usual measure of precision is the **standard error** of the estimator. **Definition:** The **standard error** of an estimator \( \hat{\theta} \) is its standard deviation \( \sigma_{\hat{\theta}} = \sqrt{\sigma^2_{\hat{\theta}}} = \sqrt{\text{Var}(\hat{\theta})} \). If the standard error itself involves unknown parameters whose value can be estimated, we can obtain an **estimated standard error**, which is typically denoted by \( \hat{\sigma}_{\hat{\theta}} \). **Example 9.6:** What are the standard errors of the sample mean \( \overline{X} \) and the sample proportion \( \hat{p} \)?
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