Confidence Intervals For Estimating The Population Mean In

Confidence Intervals for estimating the Population Mean In each problem assume a normal distribution

Confidence Intervals for estimating the Population Mean µ In each problem assume a normal distribution

Review and solve the two problems involving confidence intervals for estimating a population mean (µ) provided below. For each, determine the critical value (z or t), compute the margin of error, and construct the confidence interval. Assume normal distribution in both cases and use the given data.

Paper For Above instruction

Problem 1

Given data:

  • Sample size (n) = 64
  • Sample mean (x̄) = 23,228
  • Population standard deviation (σ) = 8,779
  • Confidence level = 92%

Solution:

First, identify the appropriate critical value (z) since the population standard deviation (σ) is known and the sample size is sufficiently large (n > 30). For a 92% confidence level, the remaining alpha (α) is 8%, thus the area in the two tails combined is 0.08, with each tail having 0.04. Therefore, the critical z-value (z) corresponds to the 96th percentile of the standard normal distribution.

From standard normal distribution tables, the z* for 96% confidence interval (i.e., 1 - (1 - 0.92)/2 = 0.96) is approximately 1.75.

Calculating the Margin of Error (ME):

  • ME = z (σ/√n) = 1.75 (8779 / √64) = 1.75 (8779 / 8) = 1.75 1097.375 ≈ 1922.53

Constructing the confidence interval:

  • Lower limit = x̄ - ME = 23,228 - 1,922.53 ≈ 21,305.47
  • Upper limit = x̄ + ME = 23,228 + 1,922.53 ≈ 25,150.53

Final confidence interval: (21,305.47, 25,150.53)

Problem 2

Given data:

  • Sample size (n) = 31
  • Sample mean (x̄) = 134.5
  • Sample standard deviation (s) = 3.48
  • Confidence level = 95%

Solution:

Since the population standard deviation is unknown, and the sample size n=31 is less than 30 but generally acceptable with the t-distribution, we use the t-distribution with degrees of freedom (df) = n - 1 = 30.

The critical value (t) corresponds to a 95% confidence level with df=30. Looking at t-distribution tables or using statistical software, t ≈ 2.042.

Calculating the Margin of Error (ME):

  • ME = t (s/√n) = 2.042 (3.48 / √31) ≈ 2.042 (3.48 / 5.567) ≈ 2.042 * 0.625 ≈ 1.277

Constructing the confidence interval:

  • Lower limit = x̄ - ME = 134.5 - 1.277 ≈ 133.223
  • Upper limit = x̄ + ME = 134.5 + 1.277 ≈ 135.777

Final confidence interval: (133.223, 135.777)

Summary

In these problems, the appropriate z- or t-values were used based on the known population standard deviation and sample size. The margin of error was calculated accordingly, and the confidence intervals give a range where the true population mean is likely to fall with specified confidence levels. Such calculations are fundamental in inferential statistics to estimate population parameters based on sample data.

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