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Probability, Statistics and Queueing

The standard deviation of a population is known to be 10. A sample of size 100 drawn from it has a sample mean of 50.0. Taking the two-sided 95% value of the standard normal distribution to be 1.96, which of the following is the 95% confidence interval for the population mean?

  1. A30.4 to 69.6
  2. B48.04 to 51.96
  3. C49.804 to 50.196
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Answer: B. 48.04 to 51.96

The standard error of the sample mean is 10 ÷ √100 = 1. The 95% confidence interval is "sample mean ± 1.96 × standard error", so 50.0 ± 1.96, which is 48.04 to 51.96. With a sample four times as large, 400, the half-width of the interval narrows to 1.96 × 10 ÷ 20 = 0.98.

  • A. This is 50.0 ± 1.96 × 10, without dividing the standard deviation by √n. It is a rough range containing about 95% of the individual values.
  • B. Correct. 50.0 ± 1.96 × 10/√100 = 50.0 ± 1.96.
  • C. This is 50.0 ± 1.96 × 10/100, dividing the standard deviation by n instead of √n.
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