Find 95% confidence interval for variance for SF / variance for NYC

To find the 95% confidence interval for the ratio of variances between San Francisco (SF) and New York City (NYC), we will use the F-distribution. The confidence interval for the ratio of two variances can be calculated as follows:

### Step 1: Define the Problem

Given:
- Sample size for SF: \( n_{\text{SF}} = 400 \)
- Sample variance for SF: \( s_{\text{SF}}^2 = 12000^2 = 144000000 \)
- Sample size for NYC: \( n_{\text{NYC}} = 900 \)
- Sample variance for NYC: \( s_{\text{NYC}}^2 = 9000^2 = 81000000 \)

We want to find the 95% confidence interval for the ratio of variances, \( \frac{\sigma_{\text{SF}}^2}{\sigma_{\text{NYC}}^2} \).

### Step 2: Calculate the F-Statistic

The F-statistic is given by:

\[
F = \frac{s_{\text{SF}}^2}{s_{\text{NYC}}^2} = \frac{144000000}{81000000} \approx 1.7778
\]

### Step 3: Determine the Confidence Interval

The confidence interval for the ratio of variances can be determined using the following formula:

\[
\text{CI} = \left[ \frac{F}{F_{0.975, \text{df}_1, \text{df}_2}}, \frac{F}{F_{0.025, \text{df}_1, \text{df}_2}} \right]
\]

Where:
- \( \text{df}_1 = n_{\text{SF}} - 1 = 400 - 1 = 399 \)
- \( \text{df}_2 = n_{\text{NYC}} - 1 = 900 - 1 = 899 \)
- \( F_{0.975, \text{df}_1, \text{df}_2} \) and \( F_{0.025, \text{df}_1, \text{df}_2} \) are the critical values from the F-distribution table.

For \( \text{df}_1 = 399 \) and \( \text{df}_2 = 899 \):
- \( F_{0.025, 399, 899} \approx 0.753 \)
- \( F_{0.975, 399, 899} \approx 1.33 \)

### Step 4: Calculate the Confidence Interval

Now, we can plug in the values:

\[
\text{CI} = \left[ \frac{1.7778}{1.33}, \frac{1.7778}{0.753} \right]
\]

\[
\text{CI} = \left[ 1.3375, 2.3612 \right]
\]

### Conclusion

The 95% confidence interval for the ratio of the variances \( \frac{\sigma_{\text{SF}}^2}{\sigma_{\text{NYC}}^2} \) is approximately \( [1.34, 2.36] \).

This interval suggests that the variance in the cost of living in San Francisco could be between 1.34 and 2.36 times that in New York City.

Would you like more details or have any questions? Here are some related questions:

1. How does the F-distribution depend on the degrees of freedom?
2. What is the relationship between the confidence interval for variances and standard deviations?
3. How would the confidence interval change if the sample sizes were different?
4. How does the choice of confidence level (e.g., 99% vs. 95%) affect the interval?
5. What are some limitations of using the F-test for comparing variances?

**Tip**: When interpreting confidence intervals for variance ratios, remember that a wider interval indicates more uncertainty about the ratio, while a narrower interval indicates more precision.

Learn how to calculate the 95% confidence interval for the ratio of variances between San Francisco (SF) and New York City (NYC) using the F-distribution. Detailed step-by-step solution provided with sample sizes and variances. Suitable for advanced statistics enthusiasts.

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