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,
sigma.x = NULL,
y = NULL,
sigma.y = NULL,
sigma.d = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0,
paired = FALSE,
conf.level = 0.95,
...
)
a (non-empty) numeric vector of data values
a single number representing the population standard deviation for x
an optional (non-empty) numeric vector of data values
a single number representing the population standard deviation for y
a single number representing the population standard deviation for the paired differences
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
.
a single number representing the value of the mean or difference in means specified by the null hypothesis
a logical indicating whether you want a paired z-test
confidence level for the returned confidence interval, restricted to lie between zero and one
Other arguments passed onto z.test()
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
the value of the mean or difference of 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
If y
is NULL
, a one-sample z-test is carried out with x
provided sigma.x
is not NULL
. If y is not NULL
, a standard two-sample z-test is performed provided both sigma.x
and sigma.y
are finite. If paired = TRUE
, a paired z-test where the differences are defined as x - y
is performed when the user enters a finite value for sigma.d
(the population standard deviation for the differences).
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-test, the null hypothesis is that the population mean for x
less that for y
is mu
. For the paired z-test, the null hypothesis is that the mean difference between x
and 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"
, or "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
.
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.
with(data = GROCERY, z.test(x = amount, sigma.x = 30, conf.level = 0.97)$conf)
#> [1] 108.7473 132.5194
#> attr(,"conf.level")
#> [1] 0.97
# Example 8.3 from PASWR.
x <- rnorm(12)
z.test(x, sigma.x = 1)
#>
#> One Sample z-test
#>
#> data: x
#> z = 0.30947, p-value = 0.757
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> -0.4764578 0.6551279
#> sample estimates:
#> mean of x
#> 0.08933503
#>
# 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)