t.test(x, y=NULL, alternative="two.sided", mu=0, paired=F,
       var.equal=T, conf.level=.95)

x:

numeric vector. NAs and Infs are allowed but will be removed.

y:

numeric vector. NAs and Infs are allowed but will be removed. If paired=TRUE, then x and y must have the same length, and observation pairs (x[i], y[i]) with at least one NA or Inf will be removed.

alternative:

character string, one of "greater", "less" or "two.sided", or just the initial letter of each, indicating the specification of the alternative hypothesis. For the one-sample and paired t-tests, alternative refers to the true mean of the parent population in relation to the hypothesized value mu. For the standard and Welch modified two-sample t-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.

paired:

logical flag: if TRUE, x and y are considered as paired vectors.

var.equal:

logical flag: if TRUE, the variances of the parent populations of x and y are assumed equal. Argument var.equal should be supplied only for the two-sample (i.e., unpaired) tests.

conf.level:

confidence level for the returned confidence interval, restricted to lie between zero and one.

statistic:

the t-statistic, with names attribute "t".

parameters:

the degrees of freedom of the t-distribution associated with statistic. Component parameters has names attribute "df".

p.value:

the p-value for the test.

conf.int:

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 NA.

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".

method:

character string giving the name of the test used.

data.name:

a character string (vector of length 1) containing the actual names of the input vectors x and y.

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").

The t-test and the associated confidence interval are quite robust with respect to level toward heavy-tailed non-Gaussian distributions (e.g., data with outliers). However, the t-test is quite non-robust with respect to power, and the confidence interval is quite non-robust with respect to average length, toward these same types of distributions.

The arguments y, paired and var.equal determine the type of test. If y is NULL, a one-sample t-test is carried out with x. Here statistic is given by: t <- (mean(x) - mu) / ( sqrt(var(x)) / sqrt(length(x)) ) If x was drawn from a normal population, t has a t-distribution with length(x) - 1 degrees of freedom under the null hypothesis.

(b) Paired t-Test.

If y is not NULL and paired=TRUE, a paired t-test is performed; here statistic is defined through t <- (mean(d) - mu) / ( sqrt(var(d)) / sqrt(length(d)) ) where d is the vector of differences x - y. Under the null hypothesis, t follows a t-distribution with length(d) - 1 degrees of freedom, assuming normality of the differences d.

(c) Standard Two-Sample t-Test.

If y is not NULL and paired=FALSE, either a standard or Welch modified two-sample t-test is performed, depending on whether var.equal is TRUE or FALSE. For the standard t-test, statistic is t <- (mean(x) - mean(y) - mu) / s1, with s1 <- sp * sqrt(1/nx + 1/ny), sp <- sqrt( ( (nx-1)*var(x) + (ny-1)*var(y) ) / (nx + ny - 2) ), nx <- length(x), ny <- length(y). Assuming that x and y come from normal populations with equal variances, t has a t-distribution with nx + ny - 2 degrees of freedom under the null hypothesis.

(d) Welch Modified Two-Sample t-Test.

If y is not NULL, paired=FALSE and var.equal=FALSE, the Welch modified two-sample t-test is performed. In this case statistic is t <- (mean(x) - mean(y) - mu) / s2 with s2 <- sqrt( var(x)/nx + var(y)/ny ), nx <- length(x), ny <- length(y). If x and y come from normal populations, the distribution of t under the null hypothesis can be approximated by a t-distribution with (non-integral) degrees of freedom 1 / ( (c^2)/(nx-1) + ((1-c)^2)/(ny-1) ) where c <- var(x) / (nx * s2^2).

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.

t.test(x)
        # Two-sided one-sample t-test. 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.
t.test(data.after, data.before, alternative="less", paired=T)
        # One-sided paired t-test. The null hypothesis is that
        # the population mean "before" and the one "after" are
        # the same, or equivalently that the mean change ("after"
        # minus "before") is zero. The alternative hypothesis is
        # that the mean "after" is less than the one "before",
        # or equivalently that the mean change is negative. A
        # confidence interval for the mean change will be
        # computed.
t.test(x, y, mu=2)
        # Two-sided standard two-sample t-test. 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.
t.test(x, y, var.equal=F, conf.level=0.90)
        # Two-sided Welch modified two-sample t-test. The null
        # hypothesis is that the population means for 'x' and 'y'
        # are the same. The alternative hypothesis is that they
        # are not. The confidence interval for the difference in
        # true means ('x' minus 'y') will have a confidence level
        # of 0.90.