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; |
|---|---|
| y | numeric vector; |
| alternative | character string, one of |
| 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 |
| sigma.y | a single number representing the population standard
deviation for |
| conf.level | confidence level for the returned confidence interval, restricted to lie between zero and one |
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
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.
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").
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=groceries,sigma.x=30,conf.level=.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 = -0.36301, p-value = 0.7166 #> alternative hypothesis: true mean is not equal to 0 #> 95 percent confidence interval: #> -0.6705857 0.4610001 #> sample estimates: #> mean of x #> -0.1047928 #># 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)