This function is based on the standard normal distribution and
creates confidence intervals and tests hypotheses for both one
and two sample problems.
Usage
z.test(x, y = NULL, alternative = "two.sided", mu = 0, sigma.x = NULL,
sigma.y = NULL, conf.level = 0.95)
Arguments
x
numeric vector; NAs and Infs are allowed
but will be removed.
y
numeric vector; NAs and Infs 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
Details
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.
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
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
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-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").
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 Interval
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
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.
See Also
zsum.test, tsum.test
Examples
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)
Results
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> library(BSDA)
Loading required package: e1071
Loading required package: lattice
Attaching package: 'BSDA'
The following object is masked from 'package:datasets':
Orange
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/BSDA/z.test.Rd_%03d_medium.png", width=480, height=480)
> ### Name: z.test
> ### Title: Z-test
> ### Aliases: z.test
> ### Keywords: htest
>
> ### ** Examples
>
> x <- rnorm(12)
> z.test(x,sigma.x=1)
One-sample z-Test
data: x
z = -0.085, p-value = 0.9323
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
-0.5903304 0.5412554
sample estimates:
mean of x
-0.0245375
> # 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)
>
>
>
>
>
> dev.off()
null device
1
>