Confidence Intervals
mctad.js’ implementation of confidence intervals draws heavily from Chapter 5 of William Navidi’s Statistics for Engineers and Scientists (McGraw-Hill, 2008). The data used here to illustrate their use (and used in the unit tests) comes from journal artices cited in Navidi’s text.
Confidence Intervals on the Population Mean
mctad.confidenceIntervalOnTheMean(X̄, s, n, α, type)
Consider Z. Yang, Y. Chen, and Y. Yang, “Study on the Life Distribution of Microdrills”, Journal of Engineering Manufacture, 2002:301-305. This journal found that in a sample of 50 low-carbon alloy steel microdrills, the average lifetime (number of holes drilled before failure) was 12.68 with a standard deviation of 6.83.
Because the sample size is large (n > 30), the two-tailed 95% confidence interval can be calculated using a Z Score as X̄ ± zα/2σX̄
> mctad.confidenceIntervalOnTheMean(12.68, 6.83, 50, 0.05);
[10.787739226564065, 14.572260773435934]
>
A one-tailed confidence interval will be calculated if passed either ‘l’ or ‘u’ for the optional type argument. The lower-tailed 99% confidence, then, is
> mctad.confidenceIntervalOnTheMean(12.68, 6.83, 50, 0.01, 'l');
10.435143205815796
>
Now consider K. Tan, K. Tong, and C. Tang, “Direct Strut-and-Tie Model for Prestressed Deep Beams”, Journal of Structural Engineering, 2001:1076-1084. This article presents measurements of the nominal shear strength for a sample of 15 prestressed concrete beams. The average shear strength was 668.27 kN with a standard deviation of 192.089.
Because the sample size is small (n ≤ 30), it is inappropriate to use the Standard Normal Distribution and a Z Score. An examination of the raw sample data reveals no strong asymmetry or outliers, so the two-tailed 99% confidence interval can be calculated using a t Score as X̄ ± tn-1, α/2σX̄
> mctad.confidenceIntervalOnTheMean(668.27, 192.089, 15, 0.01);
[520.6192352323235, 815.9207647676765]
>
mctad.js will switch between using Z Scores and t Scores based on the sample size, but makes no attempt to assess whether an assumption of normality in the sample data is warranted.