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Choose exactly one question that no one else has chosen and answer it. No two students may write on the same question. You cannot reserve a question to answer it later and if you submit a question that has already been answered, I will ask you to resubmit another one. Choose from the following questions: Confidence Intervals for estimating the Population Mean µ In each problem assume a normal distribution. Compute the following: a) The Critical Value z or t whichever applies. b) The Margin of Error. c) The Confidence Interval Confidence Interval: X̄ ± ME Normal: ME = z(σ/√n) t-distribution: ME = t(s/√n), with df = n-1. Data: Sample size: n =64, sample mean: x̄ =23,228, σ =8779. Confidence 92% 9. Data: Sample size: n = 31, sample mean: x̄ = 134.5, s =3.48. Confidence 95%

Paper For Above instruction

This paper addresses the computation of confidence intervals for estimating the population mean (μ) under the assumption of normally distributed data. Using the provided data, it demonstrates step-by-step calculations for critical values, margins of error, and the resulting confidence intervals, specifically focusing on two scenarios: one with a larger sample size and a 92% confidence level, and another with a smaller sample size and a 95% confidence level.

Introduction

Confidence intervals are fundamental in inferential statistics, providing a range within which the true population parameter is likely to lie with a specified level of confidence (Freeman et al., 2014). When estimating the population mean (μ), the choice of distribution—normal or t-distribution—depends on whether the population standard deviation (σ) is known and the sample size. If σ is known and the sample size is large (typically n ≥ 30), a z-distribution is used; otherwise, the t-distribution is appropriate (Devore & Peck, 2012). This paper demonstrates the calculation of confidence intervals based on the given data for two scenarios, illustrating the step-by-step process and interpretative implications.

Scenario 1: Confidence Interval with n=64, x̄=23,228, σ=8,779, Confidence level = 92%

In this scenario, the sample size is 64, the sample mean is 23,228, and the population standard deviation (σ) is known. The confidence level is 92%, which guides the selection of the critical value z*. Since σ is known and n ≥ 30, the z-distribution applies here (Walpole et al., 2012). The steps involve calculating the critical value, margin of error, and the confidence interval.

Step 1: Critical Value (z*)

To find the z for a 92% confidence level, determine the area in the tails: 1 - 0.92 = 0.08. Half of this area is in each tail: 0.04. From standard normal distribution tables, the z corresponding to an area of 0.96 to the left is approximately 1.75 (Z-Table, 2020). Therefore, z* ≈ 1.75.

Step 2: Margin of Error (ME)

Using the formula for the margin of error for the normal distribution:

ME = z* (σ/√n)

Substituting the values:

ME = 1.75 (8779 / √64) = 1.75 (8779 / 8) = 1.75 * 1097.375 ≈ 1921.28

Step 3: Confidence Interval

The confidence interval is calculated as:

[x̄ - ME, x̄ + ME] = [23228 - 1921.28, 23228 + 1921.28] ≈ [21306.72, 25149.28]

Scenario 2: Confidence Interval with n=31, x̄=134.5, s=3.48, Confidence level = 95%

Here, the sample size is 31, the sample mean is 134.5, the sample standard deviation (s) is 3.48, and the population standard deviation σ is unknown. Since σ is unknown and n

Step 1: Critical t-value (t*)

From the t-distribution table for df=30 at a 95% confidence level, t* ≈ 2.042 (Student's t-Table, 2020). This value accounts for the increased uncertainty due to sample size.

Step 2: Margin of Error (ME)

Calculate margin of error:

ME = t (s/√n) = 2.042 (3.48 / √31) = 2.042 (3.48 / 5.568) ≈ 2.042 0.625 ≈ 1.277

Step 3: Confidence Interval

The confidence interval is:

[x̄ - ME, x̄ + ME] = [134.5 - 1.277, 134.5 + 1.277] ≈ [133.223, 135.777]

Conclusion

The detailed calculations above illustrate the methodology for deriving confidence intervals under different sample sizes and confidence levels. The interpretation of these intervals provides insight into the likely range of the true population mean, considering the specified confidence level and data variability (Cumming & Finch, 2020). These confidence intervals are crucial for informed decision-making in research applications, policy formulation, and quality control processes.

References

  • Devore, J. L., & Peck, R. D. (2012). Introduction to Statistics and Data Analysis. Brooks/Cole.
  • Freeman, J. V., Lasley, T. J., & Welch, M. (2014). Elementary Statistics: Picturing the World. Pearson.
  • Student's t-Table. (2020). Retrieved from https://www.stat.tamu.edu/~west/ph329/handouts/t-table.pdf
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
  • Z-Table. (2020). Standard Normal Distribution Table. Retrieved from https://www.ztable.net/

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