Answer the following questions for the sampling distribution of the sample mean shown in the figure. (a) What is the value of = ? (b) What is the value of 55 70 85 100 115 130 145 X (c) If the sample size is n = 25, what is the standard deviation of the population from which the sample was drawn?
Answer the following questions for the sampling distribution of the sample mean shown in the figure. (a) What is the value of = ? (b) What is the value of 55 70 85 100 115 130 145 X (c) If the sample size is n = 25, what is the standard deviation of the population from which the sample was drawn?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Sampling Distribution of the Sample Mean
#### Analyze the Given Figure
The image shows a normal distribution curve representing the sampling distribution of the sample mean. The x-axis is labeled \(X\) and ranges from 55 to 145 in increments of 15 units. The curve is symmetric, centered at 100, indicating that the mean (\( \mu_{\bar{x}} \)) of the sampling distribution is 100.
#### Questions and Explanations
(a) What is the value of \( \mu_{\bar{x}} \) ?
*Answer: [100]*
- **Explanation**: The mean of the sampling distribution of the sample mean (\( \mu_{\bar{x}} \)) is the same as the population mean. From the figure, it is evident that the distribution is centered at 100, so \( \mu_{\bar{x}} = 100.**
(b) What is the value of \( \sigma_{\bar{x}} \) ?
*Answer: [To be determined]*
- **Explanation**: The standard deviation of the sampling distribution of the sample mean (\( \sigma_{\bar{x}} \)) needs to be calculated. From the graph, we can estimate \( \sigma_{\bar{x}} \) using other given information or required parameters.**
(c) If the sample size is \(n = 25\), what is the standard deviation of the population from which the sample was drawn?
*Answer: [To be determined]*
- **Explanation**: To find the standard deviation of the population (\( \sigma \)), use the formula connecting the standard deviation of the sample mean (\( \sigma_{\bar{x}} \)) and the standard deviation of the population (\( \sigma \)):**
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
Given \(n = 25\), rearrange the formula to solve for \( \sigma \):
\[
\sigma = \sigma_{\bar{x}} \times \sqrt{25} = \sigma_{\bar{x}} \times 5
\]
Thus, we need to know \( \sigma_{\bar{x}} \) from part (b) to determine \( \sigma \).
### Diagram Details
The graph is a typical normal distribution bell curve, symmetrically centered at the mean (100). The labels on the graph (55, 70,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2297071-c636-4616-baf3-32dd51d4726a%2Fa90f4584-beb6-4dd9-a0cd-1bec3a018721%2F6126g9g_processed.png&w=3840&q=75)
Transcribed Image Text:### Sampling Distribution of the Sample Mean
#### Analyze the Given Figure
The image shows a normal distribution curve representing the sampling distribution of the sample mean. The x-axis is labeled \(X\) and ranges from 55 to 145 in increments of 15 units. The curve is symmetric, centered at 100, indicating that the mean (\( \mu_{\bar{x}} \)) of the sampling distribution is 100.
#### Questions and Explanations
(a) What is the value of \( \mu_{\bar{x}} \) ?
*Answer: [100]*
- **Explanation**: The mean of the sampling distribution of the sample mean (\( \mu_{\bar{x}} \)) is the same as the population mean. From the figure, it is evident that the distribution is centered at 100, so \( \mu_{\bar{x}} = 100.**
(b) What is the value of \( \sigma_{\bar{x}} \) ?
*Answer: [To be determined]*
- **Explanation**: The standard deviation of the sampling distribution of the sample mean (\( \sigma_{\bar{x}} \)) needs to be calculated. From the graph, we can estimate \( \sigma_{\bar{x}} \) using other given information or required parameters.**
(c) If the sample size is \(n = 25\), what is the standard deviation of the population from which the sample was drawn?
*Answer: [To be determined]*
- **Explanation**: To find the standard deviation of the population (\( \sigma \)), use the formula connecting the standard deviation of the sample mean (\( \sigma_{\bar{x}} \)) and the standard deviation of the population (\( \sigma \)):**
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
Given \(n = 25\), rearrange the formula to solve for \( \sigma \):
\[
\sigma = \sigma_{\bar{x}} \times \sqrt{25} = \sigma_{\bar{x}} \times 5
\]
Thus, we need to know \( \sigma_{\bar{x}} \) from part (b) to determine \( \sigma \).
### Diagram Details
The graph is a typical normal distribution bell curve, symmetrically centered at the mean (100). The labels on the graph (55, 70,
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