z.test
Z-test
This function is based on the standard normal distribution and creates confidence intervals and tests hypotheses for both one and two sample problems.
- Keywords
- htest
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;
NA
s andInf
s are allowed but will be removed.- y
numeric vector;
NA
s andInf
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 valuemu
. For the standard two-sample tests,alternative
refers to the difference between the true population mean forx
and that fory
, in relation tomu
.- 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:
the z-statistic, with names attribute "z"
the p-value for the test
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
.
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.
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.
records the
value of the input argument alternative: "greater"
, "less"
or
"two.sided"
.
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"
).
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
Examples
# NOT RUN {
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)
# }