Distribution of the original population is 
Sample Size = 
Mean of the Sample Mean = 
Standard Deviation of the Sample Mean = 


The purpose of this assignment is to show you that assupmtion of normal for orginal population is not neccessary when the sample size () is large. In this class, that is when   30.     What are the answers

To understand and answer your question, let’s break it down step by step. The problem revolves around the **Central Limit Theorem (CLT)**, which states that when the sample size is large enough (typically 30 or more), the sampling distribution of the sample mean will be approximately normal, even if the original population distribution is not normal. 

Here’s how we address each part:

### 1. **Distribution of the Original Population**
   - **Answer:** The distribution of the original population is **unknown**. However, this question suggests that the assumption of normality is not required because the sample size will be large enough (≥30).

### 2. **Sample Size**
   - **Answer:** The sample size is denoted as **n**, and the problem mentions that the sample size is "large," meaning typically **n ≥ 30**.

### 3. **Mean of the Sample Mean (μₓ̄)**
   - **Answer:** The mean of the sample mean is equal to the mean of the original population, denoted by **μ**. This is a direct consequence of the CLT.
   
   \[
   \text{Mean of Sample Mean (μₓ̄)} = \mu
   \]

### 4. **Standard Deviation of the Sample Mean (σₓ̄)**
   - **Answer:** The standard deviation of the sample mean (also called the **standard error**) is given by:
   
   \[
   \sigmaₓ̄ = \frac{\sigma}{\sqrt{n}}
   \]
   where:
   - **σ** = standard deviation of the original population
   - **n** = sample size
   
   The standard error decreases as the sample size increases.

---

### Purpose of the Assignment
The purpose of this assignment is to demonstrate that the original population does not need to follow a normal distribution for the sample means to be approximately normal when the sample size is large enough (n ≥ 30).

---

Do you need further clarification or more details on any of these points?

Here are 5 related questions to expand your understanding:
1. What happens to the standard error as the sample size increases?
2. How does the Central Limit Theorem apply when the population distribution is skewed?
3. How would the results differ if the sample size were less than 30?
4. Can you calculate the standard deviation of the sample mean without knowing the population standard deviation?
5. How does the Central Limit Theorem apply to proportions?

**Tip:** Always ensure to know whether the population standard deviation is available when applying the Central Limit Theorem, as this impacts how you calculate the standard error.

The distribution of the original population is unknown. Sample Size = n, Mean of the Sample Mean = μ, Standard Deviation of the Sample Mean = σ / √n. The purpose of this assignment is to show that the assumption of normality for the original population is not necessary when the sample size (n) is large (typically n ≥ 30). What are the answers?

This guide explains how the Central Limit Theorem applies when the sample size is large (n ≥ 30), showing that the original population distribution need not be normal. It covers key concepts such as the mean and standard deviation of the sample mean, and the relationship between sample size and standard error.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation