R: Z Test for Known Population Standard Deviation

ZTest {DescTools} | R Documentation

## Z Test for Known Population Standard Deviation

### Description

Compute the test of hypothesis and compute confidence interval on the mean of a population when the standard deviation of the population is known.

### Usage

    ZTest(x, ...)

    ## Default S3 method:
    ZTest(x, y = NULL, alternative = c("two.sided", "less", "greater"),
          paired = FALSE, mu = 0, sd_pop, conf.level = 0.95, ... )

    ## S3 method for class 'formula'
    ZTest(formula, data, subset, na.action, ...)

### Arguments

`x` | numeric vector of data values. Non-finite (e.g. infinite or missing) values will be omitted.
`y` | an optional numeric vector of data values: as with x non-finite values will be omitted.
`mu` | a number specifying the hypothesized mean of the population.
`sd_pop` | a number specifying the known standard deviation of the population.
`alternative` | a character string specifying the alternative hypothesis, must be one of `"two.sided"` (default), `"greater"` or `"less"`. You can specify just the initial letter. For one-sample tests, `alternative` refers to the true mean of the parent population in relation to the hypothesized value of the mean.
`paired` | a logical indicating whether you want a paired z-test.
`conf.level` | confidence level for the interval computation.
`formula` | a formula of the form `lhs ~ rhs` where `lhs` gives the data values and `rhs` a factor with two levels giving the corresponding groups.
`data` | an optional matrix or data frame (or similar: see `model.frame`) containing the variables in the formula `formula`. By default the variables are taken from `environment(formula)`.
`subset` | an optional vector specifying a subset of observations to be used.
`na.action` | a function which indicates what should happen when the data contain `NA`s. Defaults to `getOption("na.action")`.
`...` | further arguments to be passed to or from methods.

### Details

Most introductory statistical texts introduce inference by using the z-test and z-based confidence intervals based on knowing the population standard deviation. However statistical packages often do not include functions to do z-tests since the t-test is usually more appropriate for real world situations. This function is meant to be used during that short period of learning when the student is learning about inference using z-procedures, but has not learned the t-based procedures yet. Once the student has learned about the t-distribution the `t.test()` function should be used instead of this one (but the syntax is very similar, so this function should be an appropriate introductory step to learning `t.test()`).

The formula interface is only applicable for the 2-sample tests.

### Value

A list with class "`htest`" containing the following components:

`statistic` | the value of the z-statistic.
`p.value` | the p-value for the test
`conf.int` | a confidence interval for the mean appropriate to the specified alternative hypothesis.
`estimate` | the estimated mean or difference in means depending on whether it was a one-sample test or a two-sample test.
`null.value` | the specified hypothesized value of the mean or mean difference depending on whether it was a one-sample test or a two-sample test.
`alternative` | a character string describing the alternative hypothesis.
`method` | a character string indicating what type of test was performed.
`data.name` | a character string giving the name(s) of the data.

### Author(s)

Andri Signorell <andri@signorell.net>, based on R-Core code of `t.test`,
documentation partly from Greg Snow <greg.snow@imail.org>

### References

Stahel, W. (2002) Statistische Datenanalyse, 4th ed, vieweg

### See Also

`t.test`, `print.htest`

### Examples

    x <- rnorm(25, 100, 5)
    ZTest(x, mu=99, sd_pop=5)

    # the classic interface
    with(sleep, ZTest(extra[group == 1], extra[group == 2], sd_pop=2))

    # the formula interface
    ZTest(extra ~ group, data = sleep, sd_pop=2)

    # Stahel (2002), pp. 186, 196

    d.tyres <- data.frame(A=c(44.5,55,52.5,50.2,45.3,46.1,52.1,50.5,50.6,49.2),
                          B=c(44.9,54.8,55.6,55.2,55.6,47.7,53,49.1,52.3,50.7))
    with(d.tyres, ZTest(A, B, sd_pop=3, paired=TRUE))

    d.oxen <- data.frame(ext=c(2.7,2.7,1.1,3.0,1.9,3.0,3.8,3.8,0.3,1.9,1.9),
                         int=c(6.5,5.4,8.1,3.5,0.5,3.8,6.8,4.9,9.5,6.2,4.1))
    with(d.oxen, ZTest(int, ext, sd_pop=1.8, paired=FALSE))

[Package DescTools version 0.99.28 Index]