1## Copyright (C) 1995-2017 Kurt Hornik 2## 3## This program is free software: you can redistribute it and/or 4## modify it under the terms of the GNU General Public License as 5## published by the Free Software Foundation, either version 3 of the 6## License, or (at your option) any later version. 7## 8## This program is distributed in the hope that it will be useful, but 9## WITHOUT ANY WARRANTY; without even the implied warranty of 10## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 11## General Public License for more details. 12## 13## You should have received a copy of the GNU General Public License 14## along with this program; see the file COPYING. If not, see 15## <http://www.gnu.org/licenses/>. 16 17## -*- texinfo -*- 18## @deftypefn {} {[@var{pval}, @var{z}] =} z_test (@var{x}, @var{m}, @var{v}, @var{alt}) 19## Perform a Z-test of the null hypothesis @code{mean (@var{x}) == @var{m}} 20## for a sample @var{x} from a normal distribution with unknown mean and known 21## variance @var{v}. 22## 23## Under the null, the test statistic @var{z} follows a standard normal 24## distribution. 25## 26## With the optional argument string @var{alt}, the alternative of interest 27## can be selected. If @var{alt} is @qcode{"!="} or @qcode{"<>"}, the null 28## is tested against the two-sided alternative 29## @code{mean (@var{x}) != @var{m}}. If @var{alt} is @qcode{">"}, the 30## one-sided alternative @code{mean (@var{x}) > @var{m}} is considered. 31## Similarly for @qcode{"<"}, the one-sided alternative 32## @code{mean (@var{x}) < @var{m}} is considered. The default is the two-sided 33## case. 34## 35## The p-value of the test is returned in @var{pval}. 36## 37## If no output argument is given, the p-value of the test is displayed along 38## with some information. 39## @end deftypefn 40 41## Author: KH <Kurt.Hornik@wu-wien.ac.at> 42## Description: Test for mean of a normal sample with known variance 43 44function [pval, z] = z_test (x, m, v, alt) 45 46 if (nargin < 3 || nargin > 4) 47 print_usage (); 48 endif 49 50 if (! isvector (x)) 51 error ("z_test: X must be a vector"); 52 endif 53 if (! isscalar (m)) 54 error ("z_test: M must be a scalar"); 55 endif 56 if (! (isscalar (v) && (v > 0))) 57 error ("z_test: V must be a positive scalar"); 58 endif 59 60 n = length (x); 61 z = sqrt (n/v) * (sum (x) / n - m); 62 cdf = stdnormal_cdf (z); 63 64 if (nargin == 3) 65 alt = "!="; 66 endif 67 68 if (! ischar (alt)) 69 error ("z_test: ALT must be a string"); 70 elseif (strcmp (alt, "!=") || strcmp (alt, "<>")) 71 pval = 2 * min (cdf, 1 - cdf); 72 elseif (strcmp (alt, ">")) 73 pval = 1 - cdf; 74 elseif (strcmp (alt, "<")) 75 pval = cdf; 76 else 77 error ("z_test: option %s not recognized", alt); 78 endif 79 80 if (nargout == 0) 81 s = ["Z-test of mean(x) == %g against mean(x) %s %g,\n", ... 82 "with known var(x) == %g:\n", ... 83 " pval = %g\n"]; 84 printf (s, m, alt, m, v, pval); 85 endif 86 87endfunction 88 89 90%!test 91%! ## Two-sided (also the default option) 92%! x = rand (10,1); n = length (x); 93%! u0 = 0.5; v = 1/12; # true mean, var 94%! pval = z_test (x, u0, v, "!="); 95%! if (mean (x) >= u0) 96%! zval = abs (norminv (0.5*pval)); 97%! else 98%! zval = -abs (norminv (0.5*pval)); 99%! endif 100%! unew = zval * sqrt (v/n) + u0; 101%! assert (mean (x), unew, 100*eps); 102 103%!test 104%! x = rand (10,1); n = length (x); 105%! u0 = 0.5; v = 1/12; 106%! pval = z_test (x, u0, v, ">"); 107%! zval = norminv (1-pval); 108%! unew = zval * sqrt (v/n) + u0; 109%! assert (mean (x), unew, 100*eps); 110 111%!test 112%! x = rand (10,1); n = length (x); 113%! u0 = 0.5; v = 1/12; 114%! pval = z_test (x, u0, v, "<"); 115%! zval = norminv (pval); 116%! unew = zval * sqrt (v/n) + u0; 117%! assert (mean (x), unew, 100*eps); 118