A Random Sample From A Population With Mean And Std Dev Prod
A Random Sample Frm A Population With Mean And Std Dev Produced The Fo
A random sample from a population with a known mean and standard deviation produced the following sample information: sample size \( n=110 \), sample mean \( \bar{x}=699 \), sample standard deviation \( s=20.4 \). The problem asks to (1) find the 95% confidence interval for the population mean \( \mu \) and interpret the resulting interval, and (2) find the 99% confidence interval for \( \mu \) and interpret that interval.
Paper For Above instruction
The calculation of confidence intervals for a population mean is a fundamental aspect of inferential statistics, especially when the population standard deviation is known, and the sample size is sufficiently large, as in this case with \( n=110 \). Utilizing the sample data provided, we can estimate the population mean \( \mu \) with a specified level of confidence.
Understanding the Confidence Interval Concept
A confidence interval (CI) provides a range within which the true population parameter—in this case, the population mean \( \mu \)—is expected to fall with a certain probability. It is constructed based on sample data and the sampling distribution of the mean.
When the population standard deviation \( \sigma \) is known, the appropriate distribution for constructing the confidence interval is the standard normal distribution \( Z \). The formula for the confidence interval is:
\[
\text{CI} = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]
where \( Z_{\alpha/2} \) is the critical value from the standard normal distribution corresponding to the desired confidence level, \( \bar{x} \) is the sample mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
Calculating the 95% Confidence Interval
Given data:
- \( n=110 \)
- \( \bar{x}=699 \)
- \( s=20.4 \), which is an estimate of \( \sigma \) since the population standard deviation is assumed known, but typically should be replaced by the known \( \sigma \). Assuming the sample standard deviation approximates the population standard deviation, the value used for calculations will be \( \sigma=20.4 \).
The critical value \( Z_{\alpha/2} \) for a 95% confidence level is approximately 1.96.
Calculating the standard error:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{20.4}{\sqrt{110}} \approx \frac{20.4}{10.488} \approx 1.944
\]
Constructing the 95% confidence interval:
\[
699 \pm 1.96 \times 1.944 \approx 699 \pm 3.811
\]
Thus, the 95% confidence interval is approximately:
\[
(695.189, 702.811)
\]
Interpreting the 95% Confidence Interval
This interval suggests that we are 95% confident that the true population mean \( \mu \) lies between approximately 695.19 and 702.81. It implies that, based on the data and the assumptions of the normal distribution, repeated sampling would produce similar intervals that contain the true mean 95% of the time.
Calculating the 99% Confidence Interval
For a 99% confidence level, the critical value \( Z_{\alpha/2} \) is approximately 2.576.
Using the same standard error:
\[
SE \approx 1.944
\]
Construct the interval:
\[
699 \pm 2.576 \times 1.944 \approx 699 \pm 5.017
\]
The 99% confidence interval becomes:
\[
(693.983, 704.017)
\]
Interpreting the 99% Confidence Interval
The wider interval indicates that with a higher confidence level, the estimate of the population mean \( \mu \) is more uncertain and thus covers a broader range. We can be 99% confident that the true population mean \( \mu \) falls between roughly 693.98 and 704.02, reflecting increased confidence at the expense of precision.
Conclusion
Constructing confidence intervals at different confidence levels helps balance the precision of the estimate against the certainty we require. In this case, both intervals suggest that the population mean is around 700, with the 95% interval being narrower and more precise, while the 99% interval offers higher confidence but at the cost of a wider range. These estimates are vital for decision-making, quality control, and further statistical analysis, especially in fields like manufacturing, medicine, and social sciences where sample data inform critical insights.
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