Question 1194437: The percentage rates of home ownership for 8 randomly selected states are listed below. Estimate
the population variance and standard deviation for the percentage rate of home ownership with 99%
confidence. Test whether the variation of rates of home ownership in the selected states were the
same to the population variance of 5%.
66.0 75.8 70.9 73.9 63.4 68.5 73.3 65.9
Answer by parmen(42)   (Show Source):
You can put this solution on YOUR website! 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.
```python?code_reference&code_event_index=3
import numpy as np
from scipy.stats import chi2
# Data
data = np.array([66.0, 75.8, 70.9, 73.9, 63.4, 68.5, 73.3, 65.9])
n = len(data)
# 1. Sample variance and standard deviation
sample_variance = np.var(data, ddof=1) # Use ddof=1 for unbiased estimate
sample_std_dev = np.std(data, ddof=1)
print("Sample Variance:", sample_variance)
print("Sample Standard Deviation:", sample_std_dev)
# 2. Chi-square values for 99% confidence interval
alpha = 0.01
chi2_lower = chi2.ppf(alpha/2, n-1)
chi2_upper = chi2.ppf(1-alpha/2, n-1)
print("Chi-square lower value:", chi2_lower)
print("Chi-square upper value:", chi2_upper)
# 3. Estimate population variance and standard deviation
pop_variance_lower = (n - 1) * sample_variance / chi2_upper
pop_variance_upper = (n - 1) * sample_variance / chi2_lower
pop_std_dev_lower = np.sqrt(pop_variance_lower)
pop_std_dev_upper = np.sqrt(pop_variance_upper)
print("Estimated Population Variance (Lower):", pop_variance_lower)
print("Estimated Population Variance (Upper):", pop_variance_upper)
print("Estimated Population Standard Deviation (Lower):", pop_std_dev_lower)
print("Estimated Population Standard Deviation (Upper):", pop_std_dev_upper)
```
```text?code_stdout&code_event_index=3
Sample Variance: 19.78696428571428
Sample Standard Deviation: 4.448254071623414
Chi-square lower value: 0.9892556831329504
Chi-square upper value: 20.27773987496262
Estimated Population Variance (Lower): 6.830581260736055
Estimated Population Variance (Upper): 140.01309505884856
Estimated Population Standard Deviation (Lower): 2.613538073328195
Estimated Population Standard Deviation (Upper): 11.832712920494968
```
I'll perform hypothesis testing to check whether the population variance for the percentage rate of home ownership is the same as 5%.
```python?code_reference&code_event_index=7
# 1. Setup the hypothesis test
# Null hypothesis: Population variance = 5%
# Alternative hypothesis: Population variance != 5%
pop_variance_0 = 5
# 2. Calculate the chi-squared test statistic
chi2_stat = (n - 1) * sample_variance / pop_variance_0
print("Chi-squared test statistic:", chi2_stat)
# 3. Find the p-value (two-tailed test)
p_value = 2 * min(chi2.cdf(chi2_stat, n-1), 1 - chi2.cdf(chi2_stat, n-1))
print("P-value:", p_value)
# 4. Interpret the results
alpha = 0.01
if p_value < alpha:
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%.")
else:
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%.")
```
```text?code_stdout&code_event_index=7
Chi-squared test statistic: 27.701749999999993
P-value: 0.0004980021213811803
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%.
```
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.
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