# z.test: z-Test In PASWR2: Probability and Statistics with R, Second Edition

## Description

This function is based on the standard normal distribution and creates confidence intervals and tests hypotheses for both one and two sample problems.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```z.test( x, sigma.x = NULL, y = NULL, sigma.y = NULL, sigma.d = NULL, alternative = c("two.sided", "less", "greater"), mu = 0, paired = FALSE, conf.level = 0.95, ... ) ```

## Arguments

 `x` a (non-empty) numeric vector of data values `sigma.x` a single number representing the population standard deviation for `x` `y` an optional (non-empty) numeric vector of data values `sigma.y` a single number representing the population standard deviation for `y` `sigma.d` a single number representing the population standard deviation for the paired differences `alternative` character string, one of `"greater"`, `"less"`, or `"two.sided"`, or the initial letter of each, indicating the specification of the alternative hypothesis. For one-sample tests, `alternative` refers to the true mean of the parent population in relation to the hypothesized value `mu`. For the standard two-sample tests, `alternative` refers to the difference between the true population mean for `x` and that for `y`, in relation to `mu`. `mu` a single number representing the value of the mean or difference in means specified by the null hypothesis `paired` a logical indicating whether you want a paired z-test `conf.level` confidence level for the returned confidence interval, restricted to lie between zero and one `...` Other arguments passed onto `z.test()`

## Details

If `y` is `NULL`, a one-sample z-test is carried out with `x` provided `sigma.x` is not `NULL`. If y is not `NULL`, a standard two-sample z-test is performed provided both `sigma.x` and `sigma.y` are finite. If `paired = TRUE`, a paired z-test where the differences are defined as `x - y` is performed when the user enters a finite value for `sigma.d` (the population standard deviation for the differences).

## Value

A list of class `htest`, containing the following components:

 `statistic` the z-statistic, with names attribute `z` `p.value` the p-value for the test `conf.int` is a confidence interval (vector of length 2) for the true mean or difference in means. The confidence level is recorded in the attribute `conf.level`. When alternative is not `"two.sided,"` the confidence interval will be half-infinite, to reflect the interpretation of a confidence interval as the set of all values `k` for which one would not reject the null hypothesis that the true mean or difference in means is `k` . Here, infinity will be represented by `Inf`. `estimate` vector of length 1 or 2, giving the sample mean(s) or mean of differences; these estimate the corresponding population parameters. Component `estimate` has a names attribute describing its elements. `null.value` the value of the mean or difference of means specified by the null hypothesis. This equals the input argument `mu`. Component `null.value` has a names attribute describing its elements. `alternative` records the value of the input argument alternative: `"greater"`, `"less"`, or `"two.sided"`. `data.name` a character string (vector of length 1) containing the actual names of the input vectors `x` and `y`

## Null Hypothesis

For the one-sample z-test, the null hypothesis is that the mean of the population from which `x` is drawn is `mu`. For the standard two-sample z-test, the null hypothesis is that the population mean for `x` less that for `y` is `mu`. For the paired z-test, the null hypothesis is that the mean difference between `x` and `y` is `mu`.

The alternative hypothesis in each case indicates the direction of divergence of the population mean for `x` (or difference of means for `x` and `y`) from `mu` (i.e., `"greater"`, `"less"`, or `"two.sided"`).

## Test Assumptions

The assumption of normality for the underlying distribution or a sufficiently large sample size is required along with the population standard deviation to use Z procedures.

## Confidence Intervals

For each of the above tests, an expression for the related confidence interval (returned component `conf.int`) can be obtained in the usual way by inverting the expression for the test statistic. Note that, as explained under the description of `conf.int`, the confidence interval will be half-infinite when alternative is not `"two.sided"` ; infinity will be represented by `Inf`.

## Author(s)

Alan T. Arnholt <arnholtat@appstate.edu>

## References

• Kitchens, L.J. 2003. Basic Statistics and Data Analysis. Duxbury.

• Hogg, R. V. and Craig, A. T. 1970. Introduction to Mathematical Statistics, 3rd ed. Toronto, Canada: Macmillan.

• Mood, A. M., Graybill, F. A. and Boes, D. C. 1974. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill.

• Snedecor, G. W. and Cochran, W. G. 1980. Statistical Methods, 7th ed. Ames, Iowa: Iowa State University Press.

`zsum.test`, `tsum.test`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```with(data = GROCERY, z.test(x = amount, sigma.x = 30, conf.level = 0.97)\$conf) # Example 8.3 from PASWR. x <- rnorm(12) z.test(x, sigma.x = 1) # Two-sided one-sample z-test where the assumed value for # sigma.x is one. The null hypothesis is that the population # mean for 'x' is zero. The alternative hypothesis states # that it is either greater or less than zero. A confidence # interval for the population mean will be computed. x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7., 6.4, 7.1, 6.7, 7.6, 6.8) y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5., 4.1, 5.5) z.test(x, sigma.x=0.5, y, sigma.y=0.5, mu=2) # Two-sided standard two-sample z-test where both sigma.x # and sigma.y are both assumed to equal 0.5. The null hypothesis # is that the population mean for 'x' less that for 'y' is 2. # The alternative hypothesis is that this difference is not 2. # A confidence interval for the true difference will be computed. z.test(x, sigma.x = 0.5, y, sigma.y = 0.5, conf.level = 0.90) # Two-sided standard two-sample z-test where both sigma.x and # sigma.y are both assumed to equal 0.5. The null hypothesis # is that the population mean for 'x' less that for 'y' is zero. # The alternative hypothesis is that this difference is not # zero. A 90\% confidence interval for the true difference will # be computed. rm(x, y) ```