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A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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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 \( \hat{p} \)?](https://content.bartleby.com/qna-images/question/121a245b-4265-470c-906a-aeb7dc29a223/854bf9a7-d345-484d-85f2-b5713902e34c/o6p187i_thumbnail.jpeg)
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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