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I'll calculate the sample variance and standard deviation from the given data, and then estimate the population variance and standard deviation with 99% confidence.
\n" ); document.write( "```python?code_reference&code_event_index=3
\n" ); document.write( "import numpy as np
\n" ); document.write( "from scipy.stats import chi2\r
\n" ); document.write( "\n" ); document.write( "# Data
\n" ); document.write( "data = np.array([66.0, 75.8, 70.9, 73.9, 63.4, 68.5, 73.3, 65.9])
\n" ); document.write( "n = len(data)\r
\n" ); document.write( "\n" ); document.write( "# 1. Sample variance and standard deviation
\n" ); document.write( "sample_variance = np.var(data, ddof=1) # Use ddof=1 for unbiased estimate
\n" ); document.write( "sample_std_dev = np.std(data, ddof=1)\r
\n" ); document.write( "\n" ); document.write( "print(\"Sample Variance:\", sample_variance)
\n" ); document.write( "print(\"Sample Standard Deviation:\", sample_std_dev)\r
\n" ); document.write( "\n" ); document.write( "# 2. Chi-square values for 99% confidence interval
\n" ); document.write( "alpha = 0.01
\n" ); document.write( "chi2_lower = chi2.ppf(alpha/2, n-1)
\n" ); document.write( "chi2_upper = chi2.ppf(1-alpha/2, n-1)\r
\n" ); document.write( "\n" ); document.write( "print(\"Chi-square lower value:\", chi2_lower)
\n" ); document.write( "print(\"Chi-square upper value:\", chi2_upper)\r
\n" ); document.write( "\n" ); document.write( "# 3. Estimate population variance and standard deviation
\n" ); document.write( "pop_variance_lower = (n - 1) * sample_variance / chi2_upper
\n" ); document.write( "pop_variance_upper = (n - 1) * sample_variance / chi2_lower
\n" ); document.write( "pop_std_dev_lower = np.sqrt(pop_variance_lower)
\n" ); document.write( "pop_std_dev_upper = np.sqrt(pop_variance_upper)\r
\n" ); document.write( "\n" ); document.write( "print(\"Estimated Population Variance (Lower):\", pop_variance_lower)
\n" ); document.write( "print(\"Estimated Population Variance (Upper):\", pop_variance_upper)
\n" ); document.write( "print(\"Estimated Population Standard Deviation (Lower):\", pop_std_dev_lower)
\n" ); document.write( "print(\"Estimated Population Standard Deviation (Upper):\", pop_std_dev_upper)
\n" ); document.write( "```
\n" ); document.write( "```text?code_stdout&code_event_index=3
\n" ); document.write( "Sample Variance: 19.78696428571428
\n" ); document.write( "Sample Standard Deviation: 4.448254071623414
\n" ); document.write( "Chi-square lower value: 0.9892556831329504
\n" ); document.write( "Chi-square upper value: 20.27773987496262
\n" ); document.write( "Estimated Population Variance (Lower): 6.830581260736055
\n" ); document.write( "Estimated Population Variance (Upper): 140.01309505884856
\n" ); document.write( "Estimated Population Standard Deviation (Lower): 2.613538073328195
\n" ); document.write( "Estimated Population Standard Deviation (Upper): 11.832712920494968\r
\n" ); document.write( "\n" ); document.write( "```
\n" ); document.write( "I'll perform hypothesis testing to check whether the population variance for the percentage rate of home ownership is the same as 5%.
\n" ); document.write( "```python?code_reference&code_event_index=7
\n" ); document.write( "# 1. Setup the hypothesis test
\n" ); document.write( "# Null hypothesis: Population variance = 5%
\n" ); document.write( "# Alternative hypothesis: Population variance != 5%
\n" ); document.write( "pop_variance_0 = 5 \r
\n" ); document.write( "\n" ); document.write( "# 2. Calculate the chi-squared test statistic
\n" ); document.write( "chi2_stat = (n - 1) * sample_variance / pop_variance_0
\n" ); document.write( "print(\"Chi-squared test statistic:\", chi2_stat)\r
\n" ); document.write( "\n" ); document.write( "# 3. Find the p-value (two-tailed test)
\n" ); document.write( "p_value = 2 * min(chi2.cdf(chi2_stat, n-1), 1 - chi2.cdf(chi2_stat, n-1))
\n" ); document.write( "print(\"P-value:\", p_value)\r
\n" ); document.write( "\n" ); document.write( "# 4. Interpret the results
\n" ); document.write( "alpha = 0.01
\n" ); document.write( "if p_value < alpha:
\n" ); document.write( " print(\"Reject the null hypothesis. There is sufficient evidence to suggest that the variation of rates of home ownership in the selected states is different from the population variance of 5%.\")
\n" ); document.write( "else:
\n" ); document.write( " print(\"Fail to reject the null hypothesis. There is not enough evidence to suggest that the variation of rates of home ownership in the selected states is different from the population variance of 5%.\")
\n" ); document.write( "```
\n" ); document.write( "```text?code_stdout&code_event_index=7
\n" ); document.write( "Chi-squared test statistic: 27.701749999999993
\n" ); document.write( "P-value: 0.0004980021213811803
\n" ); document.write( "Reject the null hypothesis. There is sufficient evidence to suggest that the variation of rates of home ownership in the selected states is different from the population variance of 5%.\r
\n" ); document.write( "\n" ); document.write( "```
\n" ); document.write( "The estimated population variance for the percentage rate of home ownership is between 6.83 and 140.01, and the standard deviation is between 2.61 and 11.83, with 99% confidence. Hypothesis testing shows that the variation of rates of home ownership in the selected states is different from the population variance of 5%, with a p-value of 0.0005, which is less than the significance level of 0.01. \n" ); document.write( "