The OneSampleZTest function computes the one sample z-test upon 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.

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.

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.

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.

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.

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

Standard Z-Test on One Sample
-----------------------------
Null Hypothesis:
Sample drawn from population with mean 5 and known standard deviation 5
Alt. Hypothesis:
Sample drawn from population with mean not equal to 5 and known standard deviation 5

Sample size:             10
Sample mean:             7.6
Distribution:            Normal(0,1)
Computed statistic:      1.64438
Computed pvalue:         0.100097
Confidence interval:     4.50102483864317 .. 10.6989751613568
                         (population mean)

Result: [Accepted]
There is no statistical evidence against the null hypothesis

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:      1.64438
Computed pvalue:         0.949952
Confidence interval:     -infinity .. 10.2007419392404
                         (population mean)

Result: [Accepted]
There is no statistical evidence against the null hypothesis

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:      1.64438
Computed pvalue:         0.0500484
Confidence interval:     4.99925806075965 .. infinity
                         (population mean)

Result: [Accepted]
There is no statistical evidence against the null hypothesis

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.