Question content area top
Part 1
Describe the sampling distribution of ModifyingAbove p with caretp. Assume the size of the population is 15 comma 00015,000.
nequals=500500​, pequals=0.20.2
Question content area bottom
Part 1
Choose the phrase that best describes the shape of the sampling distribution of ModifyingAbove p with caretp below.
A.
Not normal because n less than or equals 0.05 Upper Nn≤0.05N and np left parenthesis 1 minus p right parenthesis less than 10.np(1−p)<10.
B.
Not normal because n less than or equals 0.05 Upper Nn≤0.05N and np left parenthesis 1 minus p right parenthesis greater than or equals 10.np(1−p)≥10.
C.
Approximately normal because n less than or equals 0.05 Upper Nn≤0.05N and np left parenthesis 1 minus p right parenthesis greater than or equals 10.np(1−p)≥10.
D.
Approximately normal because n less than or equals 0.05 Upper Nn≤0.05N and np left parenthesis 1 minus p right parenthesis less than 10.np(1−p)<10.

To describe the sampling distribution of \(\hat{p}\), we need to consider two main conditions:

1. **Sample Size Condition (\(n \leq 0.05N\))**: This condition checks whether the sample size \(n\) is no more than 5% of the population size \(N\). It ensures that the sample size is small enough relative to the population to treat the sampling as independent.
   
   - Here, \(n = 500\) and \(N = 15,000\).
   - Calculate \(0.05N = 0.05 \times 15,000 = 750\).
   - Since \(n = 500\) is less than 750, this condition is satisfied.

2. **Normality Condition (\(np(1-p) \geq 10\))**: This condition checks whether the product of the sample size, the probability of success \(p\), and the probability of failure \((1-p)\) is at least 10. It ensures that the sampling distribution of \(\hat{p}\) is approximately normal.

   - Here, \(p = 0.2\).
   - Calculate \(np(1-p) = 500 \times 0.2 \times 0.8 = 80\).
   - Since 80 is greater than or equal to 10, this condition is satisfied.

Since both conditions are satisfied, the sampling distribution of \(\hat{p}\) is approximately normal.

### Correct Answer:  
**C. Approximately normal because \(n \leq 0.05N\) and \(np(1-p) \geq 10\).**

Would you like more details or have any questions on this?

---

Here are 5 related questions to explore further:

1. What is the mean of the sampling distribution of \(\hat{p}\)?
2. How do you calculate the standard error of \(\hat{p}\)?
3. Why is the \(np(1-p) \geq 10\) condition important for normality?
4. How would the sampling distribution change if \(p\) were closer to 0 or 1?
5. What role does the population size play in determining the independence condition?

**Tip:** Always check both conditions for sample size and normality to ensure you are using the right approximation method.

Describe the sampling distribution of p̂. Assume the size of the population is 15,000. n = 500, p = 0.2.
Choose the phrase that best describes the shape of the sampling distribution of p̂ below:
A. Not normal because n ≤ 0.05N and np(1−p) < 10.
B. Not normal because n ≤ 0.05N and np(1−p) ≥ 10.
C. Approximately normal because n ≤ 0.05N and np(1−p) ≥ 10.
D. Approximately normal because n ≤ 0.05N and np(1−p) < 10.

Explore the sampling distribution of p̂ for a sample size of 500 from a population of 15,000 with p = 0.2. Learn why the distribution is approximately normal by verifying the sample size and normality conditions using np(1−p) ≥ 10. A clear breakdown for students studying statistics and probability at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation