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Last data update: 2014.03.03

R: Summarized z-test

zsum.test

R Documentation

Summarized z-test

Description

This function is based on the standard normal distribution and creates confidence intervals and tests hypotheses for both one and two sample problems based on summarized information the user passes to the function. Output is identical to that produced with z.test.

Usage

zsum.test(mean.x, sigma.x = NULL, n.x = NULL, mean.y = NULL,
sigma.y = NULL, n.y = NULL, alternative = "two.sided", mu = 0,
conf.level = 0.95)

Arguments

`mean.x`	a single number representing the sample mean of `x`
`sigma.x`	a single number representing the population standard deviation for `x`
`n.x`	a single number representing the sample size for `x`
`mean.y`	a single number representing the sample mean of `y`
`sigma.y`	a single number representing the population standard deviation for `y`
`n.y`	a single number representing the sample size for `y`
`alternative`	is a 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
`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:

`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 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 names `x` and `y` for the two summarized samples

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 of x and y) from mu (i.e., "greater" , "less", "two.sided" ).

Test Assumptions

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.

Confidence Intervals

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.

Author(s)

Alan T. Arnholt

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.

Examples

zsum.test(mean.x=56/30,sigma.x=2, n.x=30, alternative="greater", mu=1.8)
        # Example 9.7 part a. from PASWR.
x <- rnorm(12)
zsum.test(mean(x),sigma.x=1,n.x=12)
        # 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.
        # Note: returns same answer as:
z.test(x,sigma.x=1)
        #
x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7.0, 6.4, 7.1, 6.7, 7.6, 6.8)
y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5.0, 4.1, 5.5)
zsum.test(mean(x), sigma.x=0.5, n.x=11 ,mean(y), sigma.y=0.5, n.y=8, 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.
        # Note: returns same answer as:
z.test(x, sigma.x=0.5, y, sigma.y=0.5)
        #
zsum.test(mean(x), sigma.x=0.5, n.x=11, mean(y), sigma.y=0.5, n.y=8,
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.  Note: returns same answer as:
z.test(x, sigma.x=0.5, y, sigma.y=0.5, conf.level=0.90)
rm(x, y)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(BSDA)
Loading required package: e1071
Loading required package: lattice

Attaching package: 'BSDA'

The following object is masked from 'package:datasets':

    Orange

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/BSDA/zsum.test.Rd_%03d_medium.png", width=480, height=480)
> ### Name: zsum.test
> ### Title: Summarized z-test
> ### Aliases: zsum.test
> ### Keywords: htest
> 
> ### ** Examples
> 
> zsum.test(mean.x=56/30,sigma.x=2, n.x=30, alternative="greater", mu=1.8)

	One-sample z-Test

data:  Summarized x
z = 0.18257, p-value = 0.4276
alternative hypothesis: true mean is greater than 1.8
95 percent confidence interval:
 1.266051       NA
sample estimates:
mean of x 
 1.866667 

>         # Example 9.7 part a. from PASWR.
> x <- rnorm(12)
> zsum.test(mean(x),sigma.x=1,n.x=12)

	One-sample z-Test

data:  Summarized x
z = 0.86663, p-value = 0.3861
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.3156185  0.8159672
sample estimates:
mean of x 
0.2501744 

>         # 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.
>         # Note: returns same answer as:
> z.test(x,sigma.x=1)

	One-sample z-Test

data:  x
z = 0.86663, p-value = 0.3861
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.3156185  0.8159672
sample estimates:
mean of x 
0.2501744 

>         #
> x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7.0, 6.4, 7.1, 6.7, 7.6, 6.8)
> y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5.0, 4.1, 5.5)
> zsum.test(mean(x), sigma.x=0.5, n.x=11 ,mean(y), sigma.y=0.5, n.y=8, mu=2)

	Two-sample z-Test

data:  Summarized 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.
>         # Note: returns same answer as:
> z.test(x, sigma.x=0.5, y, sigma.y=0.5)

	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
95 percent confidence interval:
 1.300323 2.211040
sample estimates:
mean of x mean of y 
 7.018182  5.262500 

>         #
> zsum.test(mean(x), sigma.x=0.5, n.x=11, mean(y), sigma.y=0.5, n.y=8,
+ conf.level=0.90)

	Two-sample z-Test

data:  Summarized 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.  Note: returns same answer as:
> 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 

> rm(x, y)
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>