# tsum.test: Summarized t-test

Description Usage Arguments Details Value Null Hypothesis Test Assumptions Confidence Intervals Author(s) References See Also Examples

### Description

Performs a one-sample, two-sample, or a Welch modified two-sample t-test based on user supplied summary information. Output is identical to that produced with `t.test`.

### Usage

 ```1 2 3``` ```tsum.test(mean.x, s.x = NULL, n.x = NULL, mean.y = NULL, s.y = NULL, n.y = NULL, alternative = "two.sided", mu = 0, var.equal = FALSE, conf.level = 0.95) ```

### Arguments

 `mean.x` a single number representing the sample mean of `x` `s.x` a single number representing the sample 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` `s.y` a single number representing the sample 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 just 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`. 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`. For the one-sample 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` is a single number representing the value of the mean or difference in means specified by the null hypothesis. `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 tests. `conf.level` is the confidence level for the returned confidence interval; it must lie between zero and one.

### Details

If `y` is `NULL`, a one-sample t-test is carried out with `x`. If y is not `NULL`, either a standard or Welch modified two-sample t-test is performed, depending on whether `var.equal` is `TRUE` or `FALSE`.

### Value

A list of class `htest`, containing the following components:

 `statistic` the t-statistic, with names attribute `"t"` `parameters` is 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 ` 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 t-test, the null hypothesis is that the mean of the population from which `x` is drawn is `mu`. For the standard and Welch modified two-sample t-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"`, or `"two.sided"`).

### Test Assumptions

The assumption of equal population variances is central to the standard two-sample t-test. This test can be misleading when population variances are not equal, as the null distribution of the test statistic is no longer a t-distribution. If the assumption of equal variances is doubtful with respect to a particular dataset, the Welch modification of the t-test should be used.

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 non-robust with respect to power, and the confidence interval is non-robust with respect to average length, toward these same types of distributions.

### 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`.

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.

`z.test`, `zsum.test`

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```tsum.test(mean.x=5.6, s.x=2.1, n.x=16, mu=4.9, alternative="greater") # Problem 6.31 on page 324 of BSDA states: The chamber of commerce # of a particular city claims that the mean carbon dioxide # level of air polution is no greater than 4.9 ppm. A random # sample of 16 readings resulted in a sample mean of 5.6 ppm, # and s=2.1 ppm. One-sided one-sample t-test. The null # hypothesis is that the population mean for 'x' is 4.9. # The alternative hypothesis states that it is greater than 4.9. x <- rnorm(12) tsum.test(mean(x), sd(x), n.x=12) # 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. Note: above returns same answer as: t.test(x) 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) tsum.test(mean(x), s.x=sd(x), n.x=11 ,mean(y), s.y=sd(y), n.y=8, 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. # Note: above returns same answer as: t.test(x, y) tsum.test(mean(x), s.x=sd(x), n.x=11, mean(y), s.y=sd(y), n.y=8, conf.level=0.90) # Two-sided standard two-sample t-test. 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: above returns same answer as: t.test(x, y, conf.level=0.90) ```

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