Find the minimum sample size needed to be 95%
 confident that the sample's standard deviation is within 30%
 of the population's standard deviation.

To determine the minimum sample size needed for a 95% confidence level where the sample standard deviation is within 30% of the population standard deviation, we use the concept of the **chi-squared distribution**. The formula relates the sample size, confidence level, and allowable error (in this case, 30% of the population standard deviation).

### Key Variables:
- **Confidence level**: 95%
- **Allowable margin of error**: 30% of the population standard deviation
- **Population standard deviation (\(\sigma\))**: unknown, but our goal is to make the sample's standard deviation close to this value
- **\(\alpha\)**: Significance level, which is \(1 - 0.95 = 0.05\)
- **Degrees of freedom**: \(n - 1\), where \(n\) is the sample size
- **Chi-squared distribution values**: We'll use \(\chi^2_{\alpha/2}\) and \(\chi^2_{1-\alpha/2}\) based on the degrees of freedom and the desired confidence interval.

### Formula:

The formula for the confidence interval of the population standard deviation is:

\[
\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}
\]

Where:
- \(s\) is the sample standard deviation
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size
- \(\chi^2_{\alpha/2}\) and \(\chi^2_{1-\alpha/2}\) are chi-squared critical values for the given confidence level and degrees of freedom.

To ensure that the sample standard deviation is within 30% of the population standard deviation, the inequality:

\[
0.7 \leq \frac{s}{\sigma} \leq 1.3
\]

must hold, which leads us to solve for the sample size \(n\).

### Steps to Calculate the Sample Size:

1. **Set confidence level**: We have a 95% confidence level, so we use \(\alpha = 0.05\), meaning \(\alpha/2 = 0.025\).
2. **Find chi-squared values**: We need to find the chi-squared critical values for \(\alpha/2 = 0.025\) and \(1 - \alpha/2 = 0.975\) for different sample sizes.
3. **Apply margin of error**: Use the requirement that the sample standard deviation is within 30% of the population standard deviation.
4. **Iterate to find the smallest \(n\)** that satisfies this condition.

The required sample size \(n\) can be found through iterative methods, typically by checking values of \(n\) until the relationship holds, using tables or a statistical calculator to compute the chi-squared critical values for different sample sizes.

### Approximation Method:
For practical purposes, the approximate sample size \(n\) can be calculated using the following simplified rule-of-thumb formula:

\[
n \approx \left(\frac{Z_{\alpha/2} \times (1 + M)}{M}\right)^2
\]

Where:
- \(Z_{\alpha/2}\) is the z-score corresponding to the desired confidence level (1.96 for 95%)
- \(M\) is the margin of error (30% or 0.30 in this case).

Plugging the values in:

\[
n \approx \left(\frac{1.96 \times (1 + 0.30)}{0.30}\right)^2 = \left(\frac{1.96 \times 1.30}{0.30}\right)^2 = \left(\frac{2.548}{0.30}\right)^2 \approx 72
\]

### Conclusion:
The minimum sample size needed to be 95% confident that the sample's standard deviation is within 30% of the population's standard deviation is approximately **72**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How does changing the confidence level affect the sample size requirement?
2. What happens if the allowable margin of error changes to 20%?
3. How can we calculate sample size for different types of data distributions?
4. What is the role of chi-squared distribution in confidence intervals for variance?
5. How does the sample size affect the accuracy of the sample standard deviation?

**Tip**: Always consider the trade-off between confidence level and margin of error—the more precise your estimate (smaller error), the larger your required sample size!

Find the minimum sample size needed to be 95% confident that the sample's standard deviation is within 30% of the population's standard deviation.

Learn how to calculate the minimum sample size required to ensure a 95% confidence level that a sample's standard deviation is within 30% of the population's standard deviation. This problem involves key statistical concepts such as chi-squared distribution, confidence intervals, and margin of error. Suitable for undergraduate statistics students.

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