The ChiSquareGoodnessOfFitTest function computes the chi-square test for goodness-of-fit. This tests whether an observed sample follows the distribution specified by the expected sample.

The first parameter Ob is a data sample of categorized observed data; that is, each entry of the sample is a count of observations of a particular category, as in a histogram. This parameter must have the same length as Ex.

The second parameter Ex is a data sample of categorized expected data. Each entry of Ex is an expected count of observations, as in a histogram, just like with Ob. The categories, or bins of the histograms, are expected to be the same for both samples. In particular, the samples must have the same length.

level=float

This option is used to specify the level of the analysis (minimum criteria for the observed data to be considered well-fit to the expected data).  By default, this value is 0.05.

fitparameters=posint

This option is used to specify if this goodness-of-fit test is used to indicate the number of categories used when fitting this data to a distribution. A positive value for this parameter negatively affects the number of degrees of freedom used in the calculation, and so should be no greater than rtable_num_elems(Ex)-1.

To compare observed samples against a distribution rather than a categorized data set, the chi-square suitable model test should be applied instead.

If the option output is not included or is specified to be output=report, then the function will return a report. If output=plot is specified, then the function will return a plot of the sample test. If output=both is specified, then both the report and the plot will be returned.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$\mathrm{Ob1}≔\left[25\,17\,15\,23\,24\,16\right]\:$

$\mathrm{Ex1}≔\left[20\,20\,20\,20\,20\,20\right]\:$

$\mathrm{ChiSquareGoodnessOfFitTest}\left(\mathrm{Ob1}\,\mathrm{Ex1}\,\mathrm{level}=0.05\right)$

Chi-Square Test for Goodness-of-Fit
-----------------------------------
Null Hypothesis:
Observed sample does not differ from expected sample
Alt. Hypothesis:
Observed sample differs from expected sample
 
Categories:              6
Distribution:            ChiSquare(5)
Computed Statistic:      5.000000000
Computed p-value:        .415879723998286
Critical Values:         11.0704974062099
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

$\left[\mathrm{hypothesis}=\mathrm{true}\,\mathrm{criticalvalue}=11.0704974062099\,\mathrm{distribution}=\mathrm{ChiSquare}\left(5\right)\,\mathrm{pvalue}=0.415879723998286\,\mathrm{statistic}=5.000000000\right]$

$\mathrm{Ob2}≔\left[1\,2\,3\,2\,2\,2\,4\,1\,6\right]\:$

$\mathrm{Ex2}≔\left[5\,2\,2\,1\,1\,1\,8\,5\,2\right]\:$

$\mathrm{ChiSquareGoodnessOfFitTest}\left(\mathrm{Ob2}\,\mathrm{Ex2}\right)$

Chi-Square Test for Goodness-of-Fit
-----------------------------------
Null Hypothesis:
Observed sample does not differ from expected sample
Alt. Hypothesis:
Observed sample differs from expected sample
 
Categories:              9
Distribution:            ChiSquare(8)
Computed Statistic:      19.90000000
Computed p-value:        .0107210773034122
Critical Values:         15.5073130558655
 
Result: [Rejected]
This statistical test provides evidence that the null hypothesis is false.

$\left[\mathrm{hypothesis}=\mathrm{false}\,\mathrm{criticalvalue}=15.5073130558655\,\mathrm{distribution}=\mathrm{ChiSquare}\left(8\right)\,\mathrm{pvalue}=0.0107210773034122\,\mathrm{statistic}=19.90000000\right]$

$\mathrm{ChiSquareGoodnessOfFitTest}\left(\mathrm{Ob2}\,\mathrm{Ex2}\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{report},\mathrm{graph}≔\mathrm{ChiSquareGoodnessOfFitTest}\left(\mathrm{Ob2}\,\mathrm{Ex2}\,\mathrm{output}=\mathrm{both}\right)\:$

Chi-Square Test for Goodness-of-Fit
-----------------------------------
Null Hypothesis:
Observed sample does not differ from expected sample
Alt. Hypothesis:
Observed sample differs from expected sample
 
Categories:              9
Distribution:            ChiSquare(8)
Computed Statistic:      19.90000000
Computed p-value:        .0107210773034122
Critical Values:         15.5073130558655
 
Result: [Rejected]
This statistical test provides evidence that the null hypothesis is false.
Histogram Type:  default
Data Range:      1 .. 6
Bin Width:       1/6
Number of Bins:  30
Frequency Scale: relative
Histogram Type:  default
Data Range:      1 .. 8
Bin Width:       7/30
Number of Bins:  30
Frequency Scale: relative

$\mathrm{report}$

$\left[\mathrm{hypothesis}=\mathrm{false}\,\mathrm{criticalvalue}=15.5073130558655\,\mathrm{distribution}=\mathrm{ChiSquare}\left(8\right)\,\mathrm{pvalue}=0.0107210773034122\,\mathrm{statistic}=19.90000000\right]$

$\mathrm{graph}$

Kanji, Gopal K. 100 Statistical Tests. London: SAGE Publications Ltd., 1994.

Sheskin, David J. Handbook of Parametric and Nonparametric Statistical Procedures. London: CRC Press, 1997.

The Student[Statistics][ChiSquareGoodnessOfFitTest] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.