6.28.2005

Q&A: A researcher wishes to estimate the mean reading speed of...

...high-school graduates and compute the 95% confidence interval from a random sample of 6 graduates: 200 words per minute, 240, 300, 410, 450, and 600.

CppQuant Answer

When constructing confidence intervals, the t-distribution is used when the population standard deviation is unknown.

  • In Excel, m = AVERAGE(200, 240, 300, 410, 450, 600) = 366.6667
  • sample standard deviation = variance^0.5, where variance = sum( (x(0)-m)^2 + ... + (x(n)-m)^2 )/n 60.9736
    • In Excel, s = STDEV(200, 240, 300, 410, 450, 600) = 149.3541652
  • Standard error of the mean = s / (N^0.5) = 149.3541652 / (6^0.5) = 60.97
  • df = 6-1 = 5
  • t = 2.571 (from the t-table)
  • Confidence interval = Point Estimate +/- Margin of Error = Point Estimate +/- Reliability Factor * Standard Error
    • Lower Limit = 366.67 - 2.571*60.87 = 209.904
    • Upper limit = 366.67 + 2.571*60.87 = 523.430
  • the researcher can be 95% sure that the mean reading speed of high-school graduates is between 209.904 and 523.430.

Bonus Points

  • When use t-distribution to construct confidence intervals, one assumes
    • normal distribution
    • the population standard deviation is unknown.
    • Scores are sampled randomly and are independent
  • T-distributions are spread out MORE than a normal distribution with u = 0, s = 1.
  • T-distributions are robust against nonnormality of the population except in the case of outliners or strong skewness.
    • sample size < 15: can use T-distribution only if the data are close to normal.
    • sample size >= 15: can use T-distribution except in the presence of outliners or strong skewness.
    • sample size >= 40: can always use T-distribution.
  • To shorten a given confidence interval, one should increase the sample size.

Category: Quantitative Analysis

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