Z-Test for Testing Means With Known Population Standard Deviations

A parametric test for the mean of a normal distribution when the population variance is known, or for comparing two means with known variances, based on the standard normal distribution.

Usage

zTest(x, ...)

# S3 method for class 'formula'
zTest(formula, data, subset, na.action = na.pass, ...)

# Default S3 method
zTest(
  x,
  y = NULL,
  alternative = c("two.sided", "less", "greater"),
  paired = FALSE,
  mu = 0,
  sd_pop,
  conf.level = 0.95,
  ...
)

Arguments

x: numeric vector of data values. Non-finite (e.g. infinite or missing) values will be omitted.
...: further arguments to be passed to or from methods.
formula: a formula of the form lhs ~ rhs where lhs gives the data values and rhs a factor with two levels giving the corresponding groups.
data: an optional matrix or data frame (or similar: see model.frame()) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").
y: an optional numeric vector of data values: as with x non-finite values will be omitted.
alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
For one-sample tests, alternative refers to the true mean of the parent population in relation to the hypothesized value of the mean.
paired: a logical indicating whether you want a paired z-test.
mu: a number specifying the hypothesized mean of the population.
sd_pop: a number specifying the known standard deviation of the population. For the two-sample test, this single value is assumed to be the common known standard deviation of both populations.
conf.level: confidence level for the interval computation.

Value

A list with class "htest" containing the following components:

statistic: the value of the z-statistic.
p.value: the p-value for the test
conf.int: a confidence interval for the mean appropriate to the specified alternative hypothesis.
estimate: the estimated mean or difference in means depending on whether it was a one-sample test or a two-sample test.
null.value: the specified hypothesized value of the mean or mean difference depending on whether it was a one-sample test or a two-sample test.
alternative: a character string describing the alternative hypothesis.
method: a character string indicating what type of test was performed.
data.name: a character string giving the name(s) of the data.

Details

Compute the test of hypothesis and compute confidence interval on the mean of a population when the standard deviation of the population is known.

Most introductory statistical texts introduce inference by using the z-test and z-based confidence intervals based on knowing the population standard deviation. However statistical packages often do not include functions to do z-tests since the t-test is usually more appropriate for real world situations. This function is meant to be used during that short period of learning when the student is learning about inference using z-procedures, but has not learned the t-based procedures yet. Once the student has learned about the t-distribution the t.test() function should be used instead of this one (but the syntax is very similar, so this function should be an appropriate introductory step to learning t.test()).

The formula interface is only applicable for the 2-sample tests.

References

Stahel, W. (2002) Statistische Datenanalyse, 4th ed, vieweg

Examples


x <- rnorm(25, 100, 5)
zTest(x, mu=99, sd_pop=5)
#> 
#> 	One Sample z-test
#> 
#> data:  x
#> z = -0.51593, Std. Dev. Population = 5, p-value = 0.6059
#> alternative hypothesis: true mean is not equal to 99
#> 95 percent confidence interval:
#>   96.5241 100.4440
#> sample estimates:
#> mean of x 
#>  98.48407 
#> 

# the classic interface
with(sleep, zTest(extra[group==1], extra[group==2], sd_pop=2))
#> 
#> 	Two Sample z-test
#> 
#> data:  extra[group == 1] and extra[group == 2]
#> z = -1.7665, Std. Dev. Population = 2, p-value = 0.07731
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -3.3330451  0.1730451
#> sample estimates:
#> mean of x mean of y 
#>      0.75      2.33 
#> 

# the formula interface
zTest(extra ~ group, data=sleep, sd_pop=2)
#> 
#> 	Two Sample z-test
#> 
#> data:  
#> z = -1.7665, Std. Dev. Population = 2, p-value = 0.07731
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -3.3330451  0.1730451
#> sample estimates:
#> mean of x mean of y 
#>      0.75      2.33 
#> 


# Stahel (2002), pp. 186, 196

Tyres <- data.frame(A=c(44.5,55,52.5,50.2,45.3,46.1,52.1,50.5,50.6,49.2),
                      B=c(44.9,54.8,55.6,55.2,55.6,47.7,53,49.1,52.3,50.7))
with(Tyres, zTest(A, B, sd_pop=3, paired=TRUE))
#> 
#> 	Paired z-test
#> 
#> data:  A and B
#> z = -2.4139, Std. Dev. Population = 3, p-value = 0.01578
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -4.1493851 -0.4306149
#> sample estimates:
#> mean of the differences 
#>                   -2.29 
#> 


Oxen <- data.frame(ext=c(2.7,2.7,1.1,3.0,1.9,3.0,3.8,3.8,0.3,1.9,1.9),
                   int=c(6.5,5.4,8.1,3.5,0.5,3.8,6.8,4.9,9.5,6.2,4.1))
with(Oxen, zTest(int, ext, sd_pop=1.8, paired=FALSE))
#> 
#> 	Two Sample z-test
#> 
#> data:  int and ext
#> z = 3.9324, Std. Dev. Population = 1.8, p-value = 8.411e-05
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  1.513865 4.522498
#> sample estimates:
#> mean of x mean of y 
#>  5.390909  2.372727 
#>