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

```
z.test(
x,
y = NULL,
alternative = "two.sided",
mu = 0,
sigma.x = NULL,
sigma.y = NULL,
conf.level = 0.95
)
```

- x
numeric vector;

`NA`

s and`Inf`

s are allowed but will be removed.- y
numeric vector;

`NA`

s and`Inf`

s are allowed but will be removed.- 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

- sigma.x
a single number representing the population standard deviation for

`x`

- sigma.y
a single number representing the population standard deviation for

`y`

- conf.level
confidence level for the returned confidence interval, restricted to lie between zero and one

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
is the value of the mean or difference in 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`

If `y`

is `NULL`

, a one-sample z-test is carried out with
`x`

. If y is not `NULL`

, a standard two-sample z-test is
performed.

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-tests, the null hypothesis is that the
population mean for `x`

less that for `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"`

, `"two.sided"`

).

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.

```
x <- rnorm(12)
z.test(x,sigma.x=1)
#>
#> One-sample z-Test
#>
#> data: x
#> z = -0.17855, p-value = 0.8583
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> -0.6173354 0.5142503
#> sample estimates:
#> mean of x
#> -0.05154256
#>
# 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-sample z-Test
#>
#> data: x and y
#> z = -1.0516, p-value = 0.293
#> alternative hypothesis: true difference in means is not equal to 2
#> 95 percent confidence interval:
#> 1.300323 2.211040
#> sample estimates:
#> mean of x mean of y
#> 7.018182 5.262500
#>
# 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-sample z-Test
#>
#> data: x and y
#> z = 7.5568, p-value = 4.13e-14
#> alternative hypothesis: true difference in means is not equal to 0
#> 90 percent confidence interval:
#> 1.373533 2.137831
#> sample estimates:
#> mean of x mean of y
#> 7.018182 5.262500
#>
# 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)
```