The Bear Population In Northern New Jersey Is Becoming A Pro
The Bear Population In Northern New Jersey Is Becoming Problematic For
The bear population in northern New Jersey is becoming problematic for some homeowners. Find a 90% confidence interval for the mean number of bears in northern New Jersey, given that in 2014, an average of 124 bears were observed with a standard deviation of 12, and the sample size was 34. Assume normality. Additionally, calculate the margin of error (ME) for a 95% confidence interval based on a sample of size n = 50, where the mean is unknown, but the standard deviation is 0.289.
Paper For Above instruction
Understanding bear population dynamics is crucial for wildlife management and addressing human-wildlife conflicts, especially in regions like northern New Jersey where increasing bear populations have raised concerns among residents. Accurate statistical estimates of the mean bear population can inform policy decisions, promote coexistence strategies, and mitigate conflicts. This paper explores the calculation of confidence intervals for the mean bear population using sample data, emphasizing the importance of statistical inference in wildlife ecology.
Context and Data Overview
The initial data provided indicates that in 2014, the average number of bears observed was 124, with a standard deviation of 12, based on a sample of 34 observations. These observations likely encompass different areas within northern New Jersey, reflecting regional variability. The assumption of normality permits the application of parametric statistical methods such as the t-distribution for confidence interval estimation.
Furthermore, a separate sample size of 50 is considered, with an unknown mean but a known standard deviation of 0.289. For this larger sample, the primary goal is to compute the margin of error for a 95% confidence interval, a key component in assessing the precision of the estimate.
Calculating the 90% Confidence Interval for the Mean
To determine the 90% confidence interval (CI) for the mean number of bears based on the 2014 data:
- Sample mean (\(\bar{x}\)) = 124
- Standard deviation (s) = 12
- Sample size (n) = 34
- Confidence level = 90%
Since the population standard deviation is unknown and the sample size is relatively small, the t-distribution is appropriate. The critical value \( t_{\alpha/2} \) for a 90% CI with degrees of freedom (df) = n - 1 = 33 can be obtained from t-tables or statistical software.
Using standard resources, \( t_{0.05,33} \approx 1.692 \).
The standard error (SE) is computed as:
\[
SE = \frac{s}{\sqrt{n}} = \frac{12}{\sqrt{34}} \approx 2.055
\]
The margin of error (ME) is then:
\[
ME = t_{\alpha/2} \times SE = 1.692 \times 2.055 \approx 3.481
\]
Therefore, the 90% confidence interval for the mean number of bears is:
\[
124 \pm 3.481 \Rightarrow (120.519, 127.481)
\]
This interval suggests that, with 90% confidence, the true mean number of bears in northern New Jersey falls between approximately 120.52 and 127.48.
Calculating the Margin of Error for the 95% Confidence Interval
For the second part, with a sample size n = 50 and a known standard deviation (\(\sigma\)) = 0.289, the objective is to compute the margin of error for a 95% confidence interval.
Since \(\sigma\) is known and the sample size is sufficiently large, the Z-distribution applies. The critical Z-value for 95% confidence:
\[
Z_{0.025} \approx 1.960
\]
The standard error:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.289}{\sqrt{50}} \approx 0.0409
\]
The margin of error:
\[
ME = Z_{0.025} \times SE = 1.960 \times 0.0409 \approx 0.0802
\]
Even without knowing the sample mean, this margin of error indicates the level of precision expected in the estimate of the true mean bear population, given the standard deviation and sample size.
Significance and Application
These statistical calculations play a vital role in wildlife management. The confidence interval around the mean provides policymakers and conservationists with an estimate of current bear population levels, aiding in developing appropriate response strategies. The margin of error's magnitude reflects the reliability of these estimates; smaller margins indicate greater precision and confidence in the data.
In practical terms, understanding the variability and spread of bear populations supports efforts to implement targeted interventions, reduce human-wildlife conflicts, and preserve ecological balance. As bear populations potentially increase, ongoing statistical monitoring is essential to adapt management practices accordingly.
Conclusion
Accurate estimation of bear populations using confidence intervals informs effective wildlife management policies. The calculation of a 90% confidence interval based on historical data estimates the true mean bear population to likely be between 120.52 and 127.48 bears. Meanwhile, the margin of error for a 95% confidence interval, given the standard deviation and sample size, is approximately 0.0802. Together, these statistical measures provide a foundation for data-driven decision-making aimed at safeguarding both human interests and bear conservation in northern New Jersey.
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