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Let's say you have a sample mean, you may wish to know what confidence intervals you can place on that mean. Colloquially: "I want an interval that I can be P% sure contains the true mean". (On a technical point, note that the interval either contains the true mean or it does not: the meaning of the confidence level is subtly different from this colloquialism. More background information can be found on the NIST site).
The formula for the interval can be expressed as:
Where, Y_{s} is the sample mean, s is the sample standard deviation, N is the sample size, /α/ is the desired significance level and t_{(α/2,N-1)} is the upper critical value of the Students-t distribution with N-1 degrees of freedom.
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The quantity α is the maximum acceptable risk of falsely rejecting the null-hypothesis. The smaller the value of α the greater the strength of the test. The confidence level of the test is defined as 1 - α, and often expressed as a percentage. So for example a significance level of 0.05, is equivalent to a 95% confidence level. Refer to "What are confidence intervals?" in NIST/SEMATECH e-Handbook of Statistical Methods. for more information. |
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The usual assumptions of independent and identically distributed (i.i.d.) variables and normal distribution of course apply here, as they do in other examples. |
From the formula, it should be clear that:
The following example code is taken from the example program students_t_single_sample.cpp.
We'll begin by defining a procedure to calculate intervals for various confidence levels; the procedure will print these out as a table:
// Needed includes: #include <boost/math/distributions/students_t.hpp> #include <iostream> #include <iomanip> // Bring everything into global namespace for ease of use: using namespace boost::math; using namespace std; void confidence_limits_on_mean( double Sm, // Sm = Sample Mean. double Sd, // Sd = Sample Standard Deviation. unsigned Sn) // Sn = Sample Size. { using namespace std; using namespace boost::math; // Print out general info: cout << "__________________________________\n" "2-Sided Confidence Limits For Mean\n" "__________________________________\n\n"; cout << setprecision(7); cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n"; cout << setw(40) << left << "Mean" << "= " << Sm << "\n"; cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
We'll define a table of significance/risk levels for which we'll compute intervals:
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
Note that these are the complements of the confidence/probability levels: 0.5, 0.75, 0.9 .. 0.99999).
Next we'll declare the distribution object we'll need, note that the degrees of freedom parameter is the sample size less one:
students_t dist(Sn - 1);
Most of what follows in the program is pretty printing, so let's focus on the calculation of the interval. First we need the t-statistic, computed using the quantile function and our significance level. Note that since the significance levels are the complement of the probability, we have to wrap the arguments in a call to complement(...):
double T = quantile(complement(dist, alpha[i] / 2));
Note that alpha was divided by two, since we'll be calculating both the upper and lower bounds: had we been interested in a single sided interval then we would have omitted this step.
Now to complete the picture, we'll get the (one-sided) width of the interval from the t-statistic by multiplying by the standard deviation, and dividing by the square root of the sample size:
double w = T * Sd / sqrt(double(Sn));
The two-sided interval is then the sample mean plus and minus this width.
And apart from some more pretty-printing that completes the procedure.
Let's take a look at some sample output, first using the Heat flow data from the NIST site. The data set was collected by Bob Zarr of NIST in January, 1990 from a heat flow meter calibration and stability analysis. The corresponding dataplot output for this test can be found in section 3.5.2 of the NIST/SEMATECH e-Handbook of Statistical Methods..
__________________________________ 2-Sided Confidence Limits For Mean __________________________________ Number of Observations = 195 Mean = 9.26146 Standard Deviation = 0.02278881 ___________________________________________________________________ Confidence T Interval Lower Upper Value (%) Value Width Limit Limit ___________________________________________________________________ 50.000 0.676 1.103e-003 9.26036 9.26256 75.000 1.154 1.883e-003 9.25958 9.26334 90.000 1.653 2.697e-003 9.25876 9.26416 95.000 1.972 3.219e-003 9.25824 9.26468 99.000 2.601 4.245e-003 9.25721 9.26571 99.900 3.341 5.453e-003 9.25601 9.26691 99.990 3.973 6.484e-003 9.25498 9.26794 99.999 4.537 7.404e-003 9.25406 9.26886
As you can see the large sample size (195) and small standard deviation (0.023) have combined to give very small intervals, indeed we can be very confident that the true mean is 9.2.
For comparison the next example data output is taken from P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. The values result from the determination of mercury by cold-vapour atomic absorption.
__________________________________ 2-Sided Confidence Limits For Mean __________________________________ Number of Observations = 3 Mean = 37.8000000 Standard Deviation = 0.9643650 ___________________________________________________________________ Confidence T Interval Lower Upper Value (%) Value Width Limit Limit ___________________________________________________________________ 50.000 0.816 0.455 37.34539 38.25461 75.000 1.604 0.893 36.90717 38.69283 90.000 2.920 1.626 36.17422 39.42578 95.000 4.303 2.396 35.40438 40.19562 99.000 9.925 5.526 32.27408 43.32592 99.900 31.599 17.594 20.20639 55.39361 99.990 99.992 55.673 -17.87346 93.47346 99.999 316.225 176.067 -138.26683 213.86683
This time the fact that there are only three measurements leads to much wider intervals, indeed such large intervals that it's hard to be very confident in the location of the mean.