**5.2****
****One Sample Z-Test** (.NET, C#, CSharp, VB, Visual Basic, F#)

Class **OneSampleZTest** determines
whether a sample from a normal distribution with known standard deviation
could have a given mean. For example, suppose we wish to determine whether
the IQs of children from a particular school are above average, given
that Wechsler IQ scores are normally distributed with a mean of 100 and standard deviation of 15.
Sample scores from 9 students are 116 110
111 113 112 113 111 109 121, with a mean of 112.8.

As described Section 5.1,
all hypothesis test classes provide two paths for constructing instances
of that type: a parameter-based method and a data-based method. Thus,
you can construct a **OneSampleZTest**
object by explicitly specifying a sample mean (), sample size (), population mean (), and population standard
deviation (), like
so:

Code Example – C# z-test

double xbar = 112.8; int n = 9; double mu0 = 100; double sigma = 15; var test = new OneSampleZTest( xbar, n, mu0, sigma );

Or by supplying a set of sample data, and the necessary population parameters:

Code Example – C# z-test

var data = new DoubleVector( "[ 116 110 111 113 112 113 111 109 121 ]" ); double mu0 = 100; double sigma = 15; var test = new OneSampleZTest( data, mu0, sigma );

In this case, the sample mean and sample size are calculated from the given data.

In addition to the properties common to all hypothesis
test objects (Section 5.1),
a **OneSampleZTest** object provides
the following read-only properties:

● Xbar gets the sample mean.

● N gets the sample size.

● Mu0 gets the population mean.

● Sigma gets the population standard deviation.

By default, a **OneSampleZTest**
object performs a two-sided hypothesis test () with . In this example, we
wish to test the one-sided form to the right (; that is, we wish to
test whether the children in our sample have a *higher*
than average IQ. Suppose also that we wish to set the alpha level to
0.05. Non-default test parameters can be
specified at the time of construction using constructor overloads, or
after construction using the provided Alpha
and Type properties, like so:

Code Example – C# z-test

test.Type = HypothesisType.Right; test.Alpha = 0.05;

Once you've constructed and configured a **OneSampleZTest** object, you can access
the test results using the provided properties, as described in Section 5.1:

Code Example – C# z-test

Console.WriteLine( "z-statistic = " + test.Statistic ); Console.WriteLine( "p-value = " + test.P ); Console.WriteLine( "reject the null hypothesis? " + test.Reject);

The output is:

z-statistic = 2.56 p-value = 0.00523360816355578 reject the null hypothesis? true

This indicates that we can reject the null hypotheses (). We can conclude that the children have IQs significantly above average.

Finally, remember that the ToString() method returns a formatted string representation of the complete test results:

One Sample Z Test ----------------- Sample mean = 112.8 Sample size = 9 Population mean = 100 Population standard deviation = 15 Computed Z statistic: 2.56 Hypothesis type: one-sided to the right Null hypothesis: sample mean = population mean Alt hypothesis: sample mean > population mean P-value: 0.00523360816355578 REJECT the null hypothesis for alpha = 0.05 0.95 confidence interval: 104.575731865243 Infinity