Describe the sampling distribution of p. Assume the size of the population is 20,000. n = 200, p = 0.2
Describe the sampling distribution of p. Assume the size of the population is 20,000. n = 200, p = 0.2
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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Describe the sampling distribution of p. Assume the size of the population is 20,000.
n = 200, p = 0.2
![### Sampling Distribution of Proportion
**Context:**
We are examining the sampling distribution of the sample proportion \(\hat{p}\). The given conditions include a sample size \(n = 200\) and a population proportion \(p = 0.2\). The population size is 20,000.
**Question:**
Choose the phrase that best describes the shape of the sampling distribution of \(\hat{p}\):
1. **A.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
2. **B.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) < 10\).
3. **C.** Not normal because \(n \leq 0.05N\) and \(np(1-p) < 10\).
4. **D.** Not normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
**Solution Steps:**
1. **Sample Size Condition:**
Check if the sample size \(n\) is less than or equal to 5% of the population size \(N\):
\[
n = 200, \quad N = 20,000 \quad \Rightarrow \quad n \leq 0.05 \times 20,000
\]
Therefore, \(200 \leq 1,000\), which satisfies the condition.
2. **Normal Approximation Condition:**
Check if the product \(np(1-p) \geq 10\):
\[
np(1-p) = 200 \times 0.2 \times (1 - 0.2) = 200 \times 0.2 \times 0.8 = 32
\]
Clearly, \(32 \geq 10\), which satisfies the condition for a normal approximation.
**Answer:** The correct choice is **A.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
**Determine the Mean and Standard Deviation of \(\hat{p}\):**
- **Mean of the Sampling Distribution (\(\mu_{\hat{p}}\)):**
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Transcribed Image Text:### Sampling Distribution of Proportion
**Context:**
We are examining the sampling distribution of the sample proportion \(\hat{p}\). The given conditions include a sample size \(n = 200\) and a population proportion \(p = 0.2\). The population size is 20,000.
**Question:**
Choose the phrase that best describes the shape of the sampling distribution of \(\hat{p}\):
1. **A.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
2. **B.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) < 10\).
3. **C.** Not normal because \(n \leq 0.05N\) and \(np(1-p) < 10\).
4. **D.** Not normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
**Solution Steps:**
1. **Sample Size Condition:**
Check if the sample size \(n\) is less than or equal to 5% of the population size \(N\):
\[
n = 200, \quad N = 20,000 \quad \Rightarrow \quad n \leq 0.05 \times 20,000
\]
Therefore, \(200 \leq 1,000\), which satisfies the condition.
2. **Normal Approximation Condition:**
Check if the product \(np(1-p) \geq 10\):
\[
np(1-p) = 200 \times 0.2 \times (1 - 0.2) = 200 \times 0.2 \times 0.8 = 32
\]
Clearly, \(32 \geq 10\), which satisfies the condition for a normal approximation.
**Answer:** The correct choice is **A.** Approximately normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
**Determine the Mean and Standard Deviation of \(\hat{p}\):**
- **Mean of the Sampling Distribution (\(\mu_{\hat{p}}\)):**
\
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