Use the following information to construct the confidence intervals specified to estimate the Population mean ( upper and lower limit )

95% confidence for = 85, Population standard deviation = 6.3 and n = 35

Question

Use the following information to construct the confidence intervals specified to estimate the Population mean ( upper and lower limit )

95% confidence for = 85, Population standard deviation  = 6.3 and n = 35

BlackTom AI · Accepted Answer

Clarifying the task: we want a 95% confidence interval for the population mean when the population standard deviation is known (sigma = 6.3) and the sample size is n = 35, with the sample mean given as 85.

Step 1 — compute the standard error: the standard error of the mean is sigma / sqrt(n). Here, sqrt(35) ≈ 5.9161, so SE ≈ 6.3 / 5.9161 ≈ 1.064.

Step 2 — determine the critical value: for a 95% confidence interval with a known sigma, we use the standard normal z value of 1.96 (two-tailed).

Step 3 — compute the margin of error: ME = z * SE ≈ 1.96 * 1.064 ≈ 2.085.

Step 4 — form the confidence interval: lower limit = mean − ME ≈ 85 − 2.085 ≈ 82.915, and upper limit = mean + ME ≈ 85 + 2.085 ≈ 87.085.

Step 5 — rounding considerations: if you round to two decimals, the interval becomes approximately [82.92, 87.09]. If you truncate or round differently, you might obtain [82.91, 87.09] as another common presentation. The provided numbers 82.91 and 87.09 correspond to taking the margins with the computed ME and then applying a specific rounding choice.

Step 6 — interpretation and cross-check: this interval reflects the range in which we are 95% confident the true population mean lies, given the known sigma and the sample information. The use of the z-distribution is appropriate here because sigma is known; if sigma were unknown and the sample standard deviation were used, a t-distribution with df = n − 1 would be appropriate instead.

Summary of calculation path: 85 ± 1.96 × (6.3 / √35) leading to approximately 82.915 to 87.085, which may be reported as about 82.91 to 87.09 depending on rounding conventions.

BSTA011 Week 7 Review

Use the following information to construct the confidence intervals specified to estimate the Population mean ( upper and lower limit ) 95% confidence for = 85, Population standard deviation = 6.3 and n = 35

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