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Statistics

 OneSampleZTest
 apply the one sample z-test for the population mean of a sample

 Calling Sequence OneSampleZTest(X, mu0, sigma, test_options) OneSampleZTest[SampleSize](width, sigma, samplesize_options)

Parameters

 X - mu0 - realcons; the test value for the mean sigma - realcons; the standard deviation of the sample X was drawn from 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 OneSampleZTest function width - realcons; the desired width of the confidence interval sigma - realcons; the known value of the standard deviation for the population samplesize_options - (optional) an equation of the form confidence=value; specify options for the OneSampleZTest[SampleSize] utility function

Description

 • The OneSampleZTest function computes the one sample z-test on a dataset X.  This calculation is used to determine the significance of the difference between the sample mean and an assumed population mean when the standard deviation of the population is known.
 • The first parameter X is the data sample to use in the analysis.
 • The second parameter mu0 is the assumed population mean, specified as a real constant.
 • The third parameter sigma is the known population standard deviation, specified as a positive real constant.
 • The OneSampleZTest[SampleSize] utility computes the number of samples required in a data set in order to get a confidence interval with the specified width using this test.
 • The first parameter of the utility, width, specifies the desired width of the confidence interval (difference between the upper bound and the lower bound).  This value must be strictly greater than 0.
 • The second parameter of the utility, sigma, is the known population standard deviation, specified as a positive real constant.

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 data 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 OneSampleZTest 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.

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.
 • A weaker version of the z-test, the t-test is available if the standard deviation of the sample is not known.

Examples

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

Specify the data sample.

 > $X≔\mathrm{Array}\left(\left[9,10,8,4,8,3,0,10,15,9\right]\right):$
 > $\mathrm{Mean}\left(X\right)$
 ${7.60000000000000}$ (1)

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

 > $\mathrm{OneSampleZTest}\left(X,5,5,\mathrm{confidence}=0.95,\mathrm{summarize}=\mathrm{embed}\right):$

Null Hypothesis:

Sample drawn from population with mean 5 and known standard deviation 5

Alternative Hypothesis:

Sample drawn from population with mean not equal to 5 and known standard deviation 5

 Sample Size Sample Mean Distribution Computed Statistic Computed p-value Confidence Interval ${10.}$ ${7.60000}$ ${\mathrm{Normal}}{}\left({0}{,}{1}\right)$ ${1.64438}$ ${0.100097}$ ${4.50102}{..}{10.6990}$

Result:

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

Calculate the lower tail z-test.

 > $\mathrm{OneSampleZTest}\left(X,5,5,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{lowertail}',\mathrm{summarize}=\mathrm{true}\right)$
 Standard Z-Test on One Sample ----------------------------- Null Hypothesis: Sample drawn from population with mean greater than 5 and known standard deviation 5 Alt. Hypothesis: Sample drawn from population with mean less than 5 and known standard deviation 5   Sample Size:             10 Sample Mean:             7.6 Distribution:            Normal(0,1) Computed Statistic:      1.64438438337511 Computed p-value:        .949951585583421 Confidence Interval:     -infinity .. 10.2007419392404                          (population mean)   Result: [Accepted] This statistical test does not provide enough evidence to conclude that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{true}}{,}{\mathrm{confidenceinterval}}{=}{-}{\mathrm{∞}}{..}{10.2007419392404}{,}{\mathrm{distribution}}{=}{\mathrm{Normal}}{}\left({0}{,}{1}\right){,}{\mathrm{pvalue}}{=}{0.949951585583421}{,}{\mathrm{statistic}}{=}{1.64438438337511}$ (2)

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

 > ${\mathrm{infolevel}}_{\mathrm{Statistics}}≔1:$

Calculate the upper tail z-test.

 > $\mathrm{OneSampleZTest}\left(X,5,5,\mathrm{confidence}=0.95,\mathrm{alternative}='\mathrm{uppertail}'\right)$
 Standard Z-Test on One Sample ----------------------------- Null Hypothesis: Sample drawn from population with mean less than 5 and known standard deviation 5 Alt. Hypothesis: Sample drawn from population with mean greater than 5 and known standard deviation 5   Sample Size:             10 Sample Mean:             7.6 Distribution:            Normal(0,1) Computed Statistic:      1.64438438337511 Computed p-value:        .0500484144165788 Confidence Interval:     4.99925806075965 .. infinity                          (population mean)   Result: [Accepted] This statistical test does not provide enough evidence to conclude that the null hypothesis is false.
 ${\mathrm{hypothesis}}{=}{\mathrm{true}}{,}{\mathrm{confidenceinterval}}{=}{4.99925806075965}{..}{\mathrm{∞}}{,}{\mathrm{distribution}}{=}{\mathrm{Normal}}{}\left({0}{,}{1}\right){,}{\mathrm{pvalue}}{=}{0.0500484144165788}{,}{\mathrm{statistic}}{=}{1.64438438337511}$ (3)

Calculate the number of samples required to compute a confidence interval of size 3.

 > $\mathrm{OneSampleZTest}[\mathrm{SampleSize}]\left(3,5\right)$
 ${43}$ (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[OneSampleZTest] command was updated in Maple 2016.
 • The summarize option was introduced in Maple 2016.