Sample Of 8 Observations From A Normal Population
A Sample Of 8 Observations Is Selected From A Normal Population For Wh
A sample of 8 observations is selected from a normal population for which the population standard deviation is known to be 5. The sample mean is 23. (Round your answers to 3 decimal places.) (a) The standard error of the mean is . (b) The 95 percent confidence interval for the population mean is between
Paper For Above instruction
In statistical analysis, confidence intervals provide a range of plausible values for a population parameter, such as the mean, based on sample data. When the population standard deviation is known, the calculation of confidence intervals relies on the standard normal distribution (z-distribution). This paper demonstrates the process of calculating the standard error of the mean and constructing a 95% confidence interval for the population mean, based on a sample of 8 observations from a normal population with a known standard deviation.
Given Data: The sample size (n) is 8, the known population standard deviation (σ) is 5, and the sample mean (\(\bar{x}\)) is 23. The goal is to determine the standard error of the mean (Part a) and the 95% confidence interval (Part b).
Calculating the Standard Error of the Mean
The standard error of the mean (SEM) measures the variability of the sample mean estimate of the population mean. When the population standard deviation is known, SEM is calculated using the formula:
SEM = σ / √n
Substituting the known values:
SEM = 5 / √8 ≈ 5 / 2.828 ≈ 1.768
Thus, the standard error of the mean is approximately 1.768 (rounded to three decimal places).
Constructing the 95% Confidence Interval
The confidence interval (CI) for the population mean when σ is known is given by:
\(\bar{x} \pm Z_{\alpha/2} \times SEM\)
where \(Z_{\alpha/2}\) is the critical value from the standard normal distribution for a 95% confidence level. For 95%, \(Z_{0.025} ≈ 1.960\).
Calculating the margin of error (ME):
ME = 1.960 × 1.768 ≈ 3.464
Therefore, the lower and upper bounds of the confidence interval are:
Lower bound: 23 - 3.464 = 19.536
Upper bound: 23 + 3.464 = 26.464
Hence, the 95% confidence interval for the population mean is approximately between 19.536 and 26.464 (rounded to three decimal places).
Conclusion
The calculation of the standard error and confidence interval demonstrates how sample data, combined with known population parameters, facilitate the estimation of the population mean with a specified level of confidence. In cases where the population standard deviation is known and the sample size is small, the z-distribution provides a reliable basis for such inferential procedures.
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