https://file.aiquickdraw.com/mathbot/file/1728162748767lva8hmgf.jpg

The problem shown in the image is a sample size determination problem for estimating a population proportion with a given level of confidence and margin of error. Here’s the breakdown of the problem:

**Question:**
A researcher wishes to estimate the number of households with two tablets. What sample size should be obtained in order to be 95% confident that the sample proportion will not differ from the true proportion by more than 5%? A previous study indicates that the proportion of households with two tablets is 18%. Round up to the nearest whole number.

### Solution:

We use the formula for the sample size \( n \) when estimating a proportion:

\[
n = \left( \frac{Z_{\alpha/2}^2 \cdot p \cdot (1 - p)}{E^2} \right)
\]

Where:
- \( Z_{\alpha/2} \) is the z-score corresponding to the confidence level (for 95% confidence, \( Z_{\alpha/2} = 1.96 \)),
- \( p \) is the estimated proportion (here, \( p = 0.18 \)),
- \( E \) is the margin of error (5% or 0.05),
- \( n \) is the required sample size.

### Step-by-Step Calculation:

1. **Identify the values**:
   \[
   Z_{\alpha/2} = 1.96, \quad p = 0.18, \quad E = 0.05
   \]

2. **Substitute into the formula**:
   \[
   n = \left( \frac{(1.96)^2 \cdot 0.18 \cdot (1 - 0.18)}{(0.05)^2} \right)
   \]

3. **Simplify the expression**:
   \[
   n = \left( \frac{3.8416 \cdot 0.18 \cdot 0.82}{0.0025} \right)
   \]
   \[
   n = \left( \frac{0.56619552}{0.0025} \right)
   \]
   \[
   n = 226.478208
   \]

4. **Round up to the nearest whole number**:
   \[
   n \approx 227
   \]

### Conclusion:
The researcher should obtain a sample size of **227** households to be 95% confident that the sample proportion will not differ from the true proportion by more than 5%.

---

Would you like more details or further clarifications? Here are some related questions to consider:

1. How is the z-score determined for a confidence level?
2. Why do we use the margin of error in the sample size formula?
3. What happens if the estimated proportion \( p \) is unknown?
4. How does changing the confidence level affect the sample size?
5. How is this formula modified for a finite population?

**Tip:** Always round up the sample size to ensure the desired confidence and precision are maintained.

A researcher wishes to estimate the number of households with two tablets. What sample size should be obtained in order to be 95% confident that the sample proportion will not differ from the true proportion by more than 5%? A previous study indicates that the proportion of households with two tablets is 18%. Round up to the nearest whole number.

Learn how to calculate the required sample size for estimating the proportion of households with two tablets with 95% confidence. This problem involves key statistical concepts like confidence intervals, Z-scores, and proportion estimation. Suitable for undergraduate students or advanced high school learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation