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`

.

```
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
)
```

- 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

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

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 of
`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.

```
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 = 0.86264, p-value = 0.3883
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> -0.3167716 0.8148142
#> sample estimates:
#> mean of x
#> 0.2490213
#>
# 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 = 0.86264, p-value = 0.3883
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> -0.3167716 0.8148142
#> sample estimates:
#> mean of x
#> 0.2490213
#>
#
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)
```