- The sample mean is approximately \(72.14\) and the sample variance is approximately \(4.31\).
- Using the Chi-Square distribution with \(df=6\) and \(\alpha=0.01\), we find the critical values and calculate the confidence interval for the variance.
- The 99% confidence interval for the population variance is \(\boxed{1.39 < \sigma^2 < 38.27}\).
- Taking the square root of these bounds provides the 99% confidence interval for the population standard deviation, which is \(\boxed{1.18 < \sigma < 6.19}\).
Explanation
- Calculate the sample statistics
To estimate the population variance \(\sigma^2\) and population standard deviation \(\sigma\) for the home ownership rates, we first summarize the provided sample data:
Data: \(X = \{74.3, 71.5, 72.9, 69.4, 73.7, 73.8, 69.4\}\)
Sample size \(n = 7\).
We calculate the sample mean \(\bar{x}\):
\(\bar{x} = \frac{74.3 + 71.5 + 72.9 + 69.4 + 73.7 + 73.8 + 69.4}{7} = \frac{505}{7} \approx 72.14\).
The sample variance \(s^2\) is calculated using the formula \(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\):
\(s^2 = \frac{(74.3-72.14)^2 + (71.5-72.14)^2 + (72.9-72.14)^2 + (69.4-72.14)^2 + (73.7-72.14)^2 + (73.8-72.14)^2 + (69.4-72.14)^2}{6} \approx 4.31\).
- Determine the confidence interval for the population variance
The confidence interval for the population variance \(\sigma^2\) at a 99% confidence level uses the Chi-Square distribution with \(df = n - 1 = 6\) degrees of freedom. For a confidence level of 0.99, \(\alpha = 0.01\), we look for critical values \(\chi^2_{0.005, 6}\) and \(\chi^2_{0.995, 6}\).
The values are:
\(\chi^2_{0.995, 6} \approx 0.676\)
\(\chi^2_{0.005, 6} \approx 18.548\)
The confidence interval for the population variance is:
\(\frac{(n-1)s^2}{\chi^2_{0.005, 6}} < \sigma^2 < \frac{(n-1)s^2}{\chi^2_{0.995, 6}}\)
\(\frac{6 \times 4.3095}{18.5476} < \sigma^2 < \frac{6 \times 4.3095}{0.6757}\)
\(1.39 < \sigma^2 < 38.27\).
- Determine the confidence interval for the population standard deviation
To find the confidence interval for the population standard deviation \(\sigma\), we take the square root of the bounds of the variance interval:
\(\sqrt{1.3941} < \sigma < \sqrt{38.2657}\)
\(1.18 < \sigma < 6.19\).
Thus, the 99% confidence interval for the population standard deviation is approximately \((1.18, 6.19)\).
Examples
This method is widely used in economics and public policy to estimate the reliability of survey data. For example, if a government wants to understand the diversity of homeownership across states, calculating the variance helps them determine the degree of inequality or spread in ownership rates, allowing for more targeted policy interventions.