| z.test {PASWR} | R Documentation |
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; 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 |
y is NULL, a one-sample
z-test is carried out with x.NULL, a standard
two-sample z-test is performed.
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 |
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").
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.
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.
Alan T. Arnholt
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.
attach(Grocery)
z.test(x=groceries,sigma.x=30,conf.level=.97)$conf
detach(Grocery)
# 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.