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; 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 |
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.