The OneSampleZTest function computes the one sample z-test upon a data sample X. This tests whether the mean of the population is equal to mu0, under the assumption that the population is normally distributed with standard deviation sigma.

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.

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.

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.

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

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

$X≔\left[9\,10\,8\,4\,8\,3\,0\,10\,15\,9\right]\:$

$\mathrm{Mean}\left(X\right)$

$\frac{38}{5}$

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

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:      1.64438438337511
Computed p-value:        .100096828833158
Confidence Interval:     4.50102483864317 .. 10.6989751613568
                         (population mean)
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

$\left[\mathrm{hypothesis}=\mathrm{true}\,\mathrm{confidenceinterval}=4.50102483864317..10.6989751613568\,\mathrm{distribution}=\mathrm{Normal}\left(0\,1\right)\,\mathrm{pvalue}=0.100096828833158\,\mathrm{statistic}=1.64438438337511\right]$

$Y≔\mathrm{Sample}\left(\mathrm{NormalRandomVariable}\left(10\,15\right)\,1000\right)\:$

$\mathrm{OneSampleZTest}\left(Y\,11\,15\right)$

Standard Z-Test on One Sample
-----------------------------
Null Hypothesis:
Sample drawn from population with mean 11 and known standard deviation 15
Alt. Hypothesis:
Sample drawn from population with mean not equal to 11 and known standard deviation 15
 
Sample Size:             1000
Sample Mean:             10.9169
Distribution:            Normal(0,1)
Computed Statistic:      -.175176935595206
Computed p-value:        .86094060682115
Confidence Interval:     9.98721373509595 .. 11.8465988319101
                         (population mean)
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.

$\left[\mathrm{hypothesis}=\mathrm{true}\,\mathrm{confidenceinterval}=9.98721373509595..11.8465988319101\,\mathrm{distribution}=\mathrm{Normal}\left(0\,1\right)\,\mathrm{pvalue}=0.860940606821150\,\mathrm{statistic}=\mathrm{-0.175176935595206}\right]$

$\mathrm{OneSampleZTest}\left(Y\,11\,15\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{report},\mathrm{graph}≔\mathrm{OneSampleZTest}\left(Y\,11\,15\,\mathrm{output}=\mathrm{both}\right)\:$

Standard Z-Test on One Sample
-----------------------------
Null Hypothesis:
Sample drawn from population with mean 11 and known standard deviation 15
Alt. Hypothesis:
Sample drawn from population with mean not equal to 11 and known standard deviation 15
 
Sample Size:             1000
Sample Mean:             10.9169
Distribution:            Normal(0,1)
Computed Statistic:      -.175176935595206
Computed p-value:        .86094060682115
Confidence Interval:     9.98721373509595 .. 11.8465988319101
                         (population mean)
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.
Histogram Type:  default
Data Range:      -30.7094021886569 .. 54.4808425292522
Bin Width:       2.8396748239303
Number of Bins:  30
Frequency Scale: relative

$\mathrm{report}$

$\left[\mathrm{hypothesis}=\mathrm{true}\,\mathrm{confidenceinterval}=9.98721373509595..11.8465988319101\,\mathrm{distribution}=\mathrm{Normal}\left(0\,1\right)\,\mathrm{pvalue}=0.860940606821150\,\mathrm{statistic}=\mathrm{-0.175176935595206}\right]$

$\mathrm{graph}$

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][OneSampleZTest] command was introduced in Maple 18.

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