NAG Toolbox: nag_univar_ttest_2normal (g07ca)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{tail}$ – string (length ≥ 1)

Indicates which tail probability is to be calculated, and thus which alternative hypothesis is to be used.

$\mathbf{tail}=\text{'T'}$

The two tail probability, i.e., $H_1:{\mu }_x\ne {\mu }_y$.

$\mathbf{tail}=\text{'U'}$

The upper tail probability, i.e., $H_1:{\mu }_x>{\mu }_y$.

$\mathbf{tail}=\text{'L'}$

The lower tail probability, i.e., $H_1:{\mu }_x<{\mu }_y$.

Constraint: $\mathbf{tail}=\text{'T'}$, $\text{'U'}$ or $\text{'L'}$.

2:     $\mathrm{equal}$ – string (length ≥ 1)

Indicates whether the population variances are assumed to be equal or not.

$\mathbf{equal}=\text{'E'}$

The population variances are assumed to be equal, that is ${\sigma }_x^2={\sigma }_y^2$.

$\mathbf{equal}=\text{'U'}$

The population variances are not assumed to be equal.

Constraint: $\mathbf{equal}=\text{'E'}$ or $\text{'U'}$.

3:     $\mathrm{nx}$ – int64int32nag_int scalar

$n_x$, the size of the $X$ sample.

Constraint: $\mathbf{nx}\ge 2$.

4:     $\mathrm{ny}$ – int64int32nag_int scalar

$n_y$, the size of the $Y$ sample.

Constraint: $\mathbf{ny}\ge 2$.

5:     $\mathrm{xmean}$ – double scalar

$\stackrel{-}{x}$, the mean of the $X$ sample.

6:     $\mathrm{ymean}$ – double scalar

$\stackrel{-}{y}$, the mean of the $Y$ sample.

7:     $\mathrm{xstd}$ – double scalar

$s_x$, the standard deviation of the $X$ sample.

Constraint: $\mathbf{xstd}>0.0$.

8:     $\mathrm{ystd}$ – double scalar

$s_y$, the standard deviation of the $Y$ sample.

Constraint: $\mathbf{ystd}>0.0$.

9:     $\mathrm{clevel}$ – double scalar

The confidence level, $1-\alpha$, for the specified tail. For example $\mathbf{clevel}=0.95$ will give a $95\%$ confidence interval.

Constraint: $0.0<\mathbf{clevel}<1.0$.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{t}$ – double scalar

Contains the test statistic, $t_{\mathrm{obs}}$ or $t_{\mathrm{obs}}^{\prime }$.

2:     $\mathrm{df}$ – double scalar

Contains the degrees of freedom for the test statistic.

3:     $\mathrm{prob}$ – double scalar

Contains the significance level, that is the tail probability, $p$, as defined by tail.

4:     $\mathrm{dl}$ – double scalar

Contains the lower confidence limit for ${\mu }_x-{\mu }_y$.

5:     $\mathrm{du}$ – double scalar

Contains the upper confidence limit for ${\mu }_x-{\mu }_y$.

6:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{tail}\ne \text{'T'}$, $\text{'U'}$ or $\text{'L'}$,
or	$\mathbf{equal}\ne \text{'E'}$ or $\text{'U'}$,
or	$\mathbf{nx}<2$,
or	$\mathbf{ny}<2$,
or	$\mathbf{xstd}\le 0.0$,
or	$\mathbf{ystd}\le 0.0$,
or	$\mathbf{clevel}\le 0.0$,
or	$\mathbf{clevel}\ge 1.0$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g07ca_example

function g07ca_example


fprintf('g07ca example results\n\n');

nx = int64(4);
ny = int64(8);
xmean  = 25;
ymean  = 21;
xstd   = 0.8185;
ystd   = 4.2083;

% Display data
fprintf('Sample X\n');
fprintf('  Sample size        = %5d\n', nx);
fprintf('  Mean               = %10.4e\n', xmean);
fprintf('  Standard deviation = %10.4e\n\n', xstd);
fprintf('Sample Y\n');
fprintf('  Sample size        = %5d\n', ny);
fprintf('  Mean               = %10.4e\n', ymean);
fprintf('  Standard deviation = %10.4e\n', ystd);

% Loop over different assumptions and requests
clevel = [0.95,    0.95];
tail   = {'Two';   'Two'};
equal  = {'Equal'; 'Unequal'};

for j = 1:numel(clevel)
  % Calculate statistic
  [t, df, prob, dl, du, ifail] = ...
  g07ca( ...
         tail{j}, equal{j}, nx, ny, xmean, ymean, xstd, ystd, clevel(j));

  if equal{1}=='Equal'
    fprintf('\nAssuming population variances are equal.\n\n');
  else
    fprintf('\nNo assumptions about population variances.\n\n');
  end
  fprintf('t test statistic         = %11.4f\n', t);
  fprintf('Degrees of freedom       = %8.1f\n', df);
  fprintf('Significance level       = %11.4f\n\n', prob);
  fprintf('Difference in means\n');
  fprintf('  Value                  = %10.4f\n', xmean - ymean);
  fprintf('  Lower confidence limit = %10.4f\n', dl);
  fprintf('  Upper confidence limit = %10.4f\n', du);
  fprintf('  Confidence level       = %10.4f\n', clevel(j));
end

g07ca example results

Sample X
  Sample size        =     4
  Mean               = 2.5000e+01
  Standard deviation = 8.1850e-01

Sample Y
  Sample size        =     8
  Mean               = 2.1000e+01
  Standard deviation = 4.2083e+00

Assuming population variances are equal.

t test statistic         =      1.8403
Degrees of freedom       =     10.0
Significance level       =      0.0955

Difference in means
  Value                  =     4.0000
  Lower confidence limit =    -0.8429
  Upper confidence limit =     8.8429
  Confidence level       =     0.9500

Assuming population variances are equal.

t test statistic         =      2.5922
Degrees of freedom       =      8.0
Significance level       =      0.0320

Difference in means
  Value                  =     4.0000
  Lower confidence limit =     0.4410
  Upper confidence limit =     7.5590
  Confidence level       =     0.9500