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