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Statistics

 OneSampleChiSquareTest
 apply the one sample chi-square test for the population standard deviation

 Calling Sequence OneSampleChiSquareTest(X, sigma0, test_options) OneSampleChiSquareTest[SampleSize](widthratio, samplesize_options)

Parameters

 X - sigma0 - realcons; the test value for the standard deviation test_options - (optional) equation(s) of the form option=value where option is one of alternative, confidence, ignore, output, summarize or weights; specify options for the OneSampleChiSquareTest function widthratio - realcons; the ratio between the upper bound on the confidence interval and the lower bound on the confidence interval samplesize_options - (optional) equation(s) of the form option=value where option is one of confidence or iterations; specify options for the OneSampleChiSquareTest[SampleSize] utility function

Description

 • The OneSampleChiSquareTest function computes the one sample chi-square test on a dataset X.  This calculation is used to determine the significance of the difference between the sample standard deviation and an assumed population standard deviation.
 • The first parameter X is the data sample to use in the analysis.
 • The second parameter sigma0 is the assumed population standard deviation, specified as a real constant.
 • The OneSampleChiSquareTest[SampleSize] utility computes the number of samples required in a data set in order to get a confidence interval with the specified width ratio using this test.
 • The first parameter of the utility, widthratio, specifies the ratio between the upper bound on the confidence interval and the lower bound on the confidence interval.  This value must be strictly greater than 1.

Test Options

 The test_options argument can contain one or more of the options shown below.
 • alternative='twotailed', 'lowertail', or 'uppertail'
 This option is used to specify the type or interval used in the analysis, or similarly, the alternative hypothesis to consider when performing the analysis.
 • confidence=float
 This option is used to specify the confidence level of the interval and must be a floating-point value between 0 and 1.  By default this is set to 0.95.
 • ignore=truefalse
 This option is used to specify how to handle non-numeric data. If ignore is set to true all non-numeric items in X will be ignored.
 • output='report', 'statistic', 'pvalue', 'confidenceinterval', 'distribution', 'hypothesis', or list('statistic', 'pvalue', 'confidenceinterval', 'distribution', '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.
 • weights=rtable
 Vector of weights (one-dimensional rtable). If weights are given, the OneSampleChiSquareTest function will scale each data point to have given weight. Note that the weights provided must have type realcons and the results are floating- point, even if the problem is specified with exact values. Both the data array and the weights array must have the same number of elements.

Sample Size Options

 The samplesize_options argument can contain one or more of the options shown below.
 • confidence=float
 This option is used to specify the confidence level of the interval and must be a floating-point value between 0 and 1.  By default this is set to 0.95.
 • iterations=posint
 This option specifies the maximum number of iterations to process when attempting to calculate the number of samples required.  By default, this is set to 100.

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.
 • The chi-square test can be extended to consider the ratio of two population variances, which is available as the F-test.

Examples

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

Specify the data sample.

 > $X≔\mathrm{Array}\left(\left[9,10,8,4,8,3,0,10,15,9\right]\right):$
 > $\mathrm{StandardDeviation}\left(X\right)$
 ${4.24787528588640}$ (1)

Calculate the one sample chi-square test on an array of values.

 > $\mathrm{OneSampleChiSquareTest}\left(X,7,\mathrm{confidence}=0.95,\mathrm{summarize}=\mathrm{embed}\right):$

Null Hypothesis:

Sample drawn from population with standard deviation equal to 7

Alternative Hypothesis:

Sample drawn from population with standard deviation not equal to 7

 Sample Size Sample Standard Deviation Distribution Computed Statistic Computed p-value Confidence Interval ${10.}$ ${4.24788}$ ${\mathrm{ChiSquare}}{}\left({9}\right)$ ${3.31429}$ ${0.0989571}$ ${2.92184}{..}{7.75496}$

Result:

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

Calculate the lower tail chi-square test.

 > $\mathrm{OneSampleChiSquareTest}\left(X,7,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{lowertail}',\mathrm{summarize}=\mathrm{true}\right)$
 Chi-Square Test on One Sample
 -----------------------------
 Null Hypothesis: Sample drawn from population with standard deviation greater than 7
 Alt. Hypothesis: Sample drawn from population with standard deviation less than 7
 Sample Size:             10
 Sample Standard Dev.:    4.24788
 Distribution:            ChiSquare(9)
 Computed Statistic:      3.31428571428571
 Computed p-value:        .0494785373485
 Confidence Interval:     0 .. 6.98859392393519
 (population standard deviation)
 Result: [Rejected] This statistical test provides evidence that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{false}}{,}{\mathrm{confidenceinterval}}{=}{0}{..}{6.98859392393519}{,}{\mathrm{distribution}}{=}{\mathrm{ChiSquare}}{}\left({9}\right){,}{\mathrm{pvalue}}{=}{0.0494785373485000}{,}{\mathrm{statistic}}{=}{3.31428571428571}$ (2)

As an alternative to using the summarize option, setting infolevel[Statistics] := 1 also returns the printed summary.

 > $\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔1:$

Calculate the upper tail chi-square test.

 > $\mathrm{OneSampleChiSquareTest}\left(X,7,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{uppertail}'\right)$
 Chi-Square Test on One Sample
 -----------------------------
 Null Hypothesis: Sample drawn from population with standard deviation less than 7
 Alt. Hypothesis: Sample drawn from population with standard deviation greater than 7
 Sample Size:             10
 Sample Standard Dev.:    4.24788
 Distribution:            ChiSquare(9)
 Computed Statistic:      3.31428571428571
 Computed p-value:        .9505214626515
 Confidence Interval:     3.09817508200398 .. infinity
 (population standard deviation)
 Result: [Accepted] This statistical test does not provide enough evidence to conclude that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{true}}{,}{\mathrm{confidenceinterval}}{=}{3.09817508200398}{..}{\mathrm{\infty }}{,}{\mathrm{distribution}}{=}{\mathrm{ChiSquare}}{}\left({9}\right){,}{\mathrm{pvalue}}{=}{0.950521462651500}{,}{\mathrm{statistic}}{=}{3.31428571428571}$ (3)

Determine the number of samples required to compute a confidence interval with width ratio equal to 1.5.

 > $\mathrm{OneSampleChiSquareTest}\left[\mathrm{SampleSize}\right]\left(1.5\right)$
 ${49}$ (4)
 > 

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