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Statistics

 ShapiroWilkWTest
 apply Shapiro and Wilk's W-test for normality of a sample

 Calling Sequence ShapiroWilkWTest(X, options)

Parameters

 X - options - (optional) equation(s) of the form option=value where option is one of level, output, or summarize; specify options for the ShapiroWilkWTest function

Description

 • The ShapiroWilkWTest function computes Shapiro and Wilk's W-test applied to a data set X.  This test attempts to determine how closely a given sample matches a normal distribution.
 • The first parameter X is the data sample to use in the analysis. It should contain between $3$ and $2000$ data points.

Options

 The options argument can contain one or more of the options shown below.
 • level=float
 This option is used to specify the level of the analysis (minimum criteria for a data set to be considered roughly normal).  By default this value is 0.05.
 • output='report', 'statistic', 'pvalue', 'hypothesis', or list('statistic', 'pvalue', 'hypothesis')
 This option is used to specify the desired format of the output from the function.  If 'report' is specified then a module containing all output from this test is returned.  If a single parameter name is specified other than 'report' then that quantity alone is returned.  If a list of parameter names is specified then a list containing those quantities in the specified order will be returned.
 • summarize= 'true', 'false', 'embed'
 This option controls the display of a printed or embedded summary for the hypothesis test. Unlike the output option, the displayed summary is not assignable output.

Notes

 • This test generates a complete report of all calculations in the form of a userinfo message.  In order to access this report, specify infolevel[Statistics] := 1 or use the summarize option.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Specify the data sample.

 > $S≔\mathrm{Sample}\left(\mathrm{Normal}\left(5,2\right),10\right):$
 > $T≔\mathrm{Sample}\left(\mathrm{Uniform}\left(4,6\right),10\right):$

Calculate Shapiro and Wilk's W-test on the normally distributed sample.

 > $\mathrm{ShapiroWilkWTest}\left(S,\mathrm{level}=0.05,\mathrm{summarize}=\mathrm{embed}\right):$

Null Hypothesis:

Sample drawn from a population that follows a normal distribution

Alternative Hypothesis:

Sample drawn from population that does not follow a normal distribution

 Sample Size Computed Statistic Computed p-value ${10.}$ ${0.967479}$ ${0.856736}$

Result:

Accepted: This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

Calculate Shapiro and Wilk's W-test on the uniformly distributed sample.

 > $\mathrm{ShapiroWilkWTest}\left(T,\mathrm{level}=0.05,\mathrm{summarize}=\mathrm{true}\right):$
 Shapiro and Wilk's W-Test for Normality --------------------------------------- Null Hypothesis: Sample drawn from a population that follows a normal distribution Alt. Hypothesis: Sample drawn from population that does not follow a normal distribution   Sample Size:             10 Computed Statistic:      .832591474899495 Computed p-value:        .0351513590317937   Result: [Rejected] This statistical test provides evidence that the null hypothesis is false.
 > 

References

 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.

Compatibility

 • The Statistics[ShapiroWilkWTest] command was updated in Maple 2016.
 • The summarize option was introduced in Maple 2016.