### Understanding Sampling Distribution of Sample MeansA population has a mean ($ \mu $) of 83 and a standard deviation ($ \sigma $) of 24. We are tasked with finding the mean and standard deviation of a sampling distribution of sample means with a sample size ($ n $) of 249.**Problem Details:**- Population Mean ($ \mu $): 83- Population Standard Deviation ($ \sigma $): 24- Sample Size ($ n $): 249To determine the parameters of the sampling distribution of the sample means, we use the following formulas:1. **Mean of the Sampling Distribution ($ \mu_{\bar{X}} $)**: The mean of the sampling distribution of the sample means is equal to the mean of the population. Therefore: \[ \mu_{\bar{X}} = \mu = 83 \]2. **Standard Deviation of the Sampling Distribution ($ \sigma_{\bar{X}} $)**: The standard deviation of the sampling distribution of the sample means (also known as the standard error) is calculated using the population standard deviation and the square root of the sample size: \[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{249}} \] Calculating the value: \[ \sigma_{\bar{X}} = \frac{24}{\sqrt{249}} \approx \frac{24}{15.77} \approx 1.522 \]So, the standard deviation of the sampling distribution of the sample means is approximately 1.522, rounded to three decimal places.**Summary:**- **Mean of the Sampling Distribution** ($ \mu_{\bar{X}} $) = 83- **Standard Deviation of the Sampling Distribution** ($ \sigma_{\bar{X}} $) = 1.522These concepts are crucial for understanding how sampling distributions behave, especially in the context of inferential statistics.**Formulas:**- \[ \mu_{\bar{X}} = 83 \quad \text{(Simplify your answer.)} \]- \[ \sigma_{\bar{X}} = 1.522 \quad \text{(Type an integer or decimal rounded to

Introduction: The sampling distribution of the sample mean, X̄, based on a sample of size n, taken…

A population has a mean u= 83 and a standard deviation o = 24. Find the mean and standard deviation of a sampling distribution of sample means with sample size n= 249, H; = (Simplify your answer.) (Type an integer or decimal rounded to three decimal places as needed.)

A population has a mean u= 83 and a standard deviation o = 24. Find the mean and standard deviation of a sampling distribution of sample means with sample size n= 249, H; = (Simplify your answer.) (Type an integer or decimal rounded to three decimal places as needed.)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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Question

### Understanding Sampling Distribution of Sample Means

A population has a mean ($ \mu $) of 83 and a standard deviation ($ \sigma $) of 24. We are tasked with finding the mean and standard deviation of a sampling distribution of sample means with a sample size ($ n $) of 249.

**Problem Details:**

- Population Mean ($ \mu $): 83
- Population Standard Deviation ($ \sigma $): 24
- Sample Size ($ n $): 249

To determine the parameters of the sampling distribution of the sample means, we use the following formulas:

1. **Mean of the Sampling Distribution ($ \mu_{\bar{X}} $)**:
The mean of the sampling distribution of the sample means is equal to the mean of the population. Therefore:

\[
\mu_{\bar{X}} = \mu = 83
\]

2. **Standard Deviation of the Sampling Distribution ($ \sigma_{\bar{X}} $)**:
The standard deviation of the sampling distribution of the sample means (also known as the standard error) is calculated using the population standard deviation and the square root of the sample size:

\[
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{249}}
\]

Calculating the value:

\[
\sigma_{\bar{X}} = \frac{24}{\sqrt{249}} \approx \frac{24}{15.77} \approx 1.522
\]

So, the standard deviation of the sampling distribution of the sample means is approximately 1.522, rounded to three decimal places.

**Summary:**

- **Mean of the Sampling Distribution** ($ \mu_{\bar{X}} $) = 83
- **Standard Deviation of the Sampling Distribution** ($ \sigma_{\bar{X}} $) = 1.522

These concepts are crucial for understanding how sampling distributions behave, especially in the context of inferential statistics.

**Formulas:**
- \[
\mu_{\bar{X}} = 83 \quad \text{(Simplify your answer.)}
\]
- \[
\sigma_{\bar{X}} = 1.522 \quad \text{(Type an integer or decimal rounded to

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