Statistics 5102 (Geyer, Spring 2003) t-tests

Contents

• One-Sample Test
• One-Sample Confidence Interval
• Two-Sample Tests
  • Welch's Approximation
  • Assuming Equal Variances
• Two-Sample Confidence Intervals

One-Sample Test

R has a function called t.test that does one-sample and two-sample tests and confidence intervals about mean parameters of samples from normally distributed populations.

In order to use t.test you need actual data, not just sample mean and standard deviation. Hence I have made up some normal-looking data with the same sample size, sample mean, and sample standard deviation as in Example 8.5.1 in DeGroot and Schervish.

Then the following statement does the test

The only things you have to do are

• get the data into R somehow.
• specify the null hypothesis (here mu = 5.2)
• specify the alternative hypothesis (here alternative = "greater"). The choices are
  • alternative = "two.sided", which is the default and so does not need to be specified.
  • alternative = "less"
  • alternative = "greater"

This function is geared toward P-values. It does not mention the critical value, and you don't need to know it. It does report the test statistic t = 1.8329 and the P-value p-value = 0.04408.

If you want to convert the P-value into a decision, you can do that in the usual way. Reject the null hypothesis as level 0.05 if the P-value is less than 0.05. Here p-value = 0.04408 is indeed less than 0.05 so reject the null.

One-Sample Confidence Interval

Suppose we want a confidence interval instead of a test. The same R function t.test does both. The alternative optional argument controls which kind of confidence interval is made.

By far the most common type of confidence interval is two-sided, which is specified by alternative = "two-sided", which is the default and so is what happens when you specify no alternative argument.

Very rarely, one actually wants a one-sided confidence interval, which is produced by one of the other alternatives. But the main way one gets one-sided confidence intervals is as an unwanted byproduct of doing a one-tailed test.

So just get rid of the alternative argument to get an ordinary confidence interval.

Another optional argument is conf.level specifies the confidence level. The default is 0.95, so this argument can be omitted when you want a 95% confidence interval.

The result: the 90% confidence interval for the true mean μ computed from these data is (5.207815, 5.592185) or when rounded to reasonable precision (5.21, 5.59).

Two-Sample Tests

Welch's Approximation

For a test or confidence interval about the difference of the means of two normal distributions from which we have two independent samples, say x and y we use the form

t.test(x, y, alternative = "less")

or something similar.

As in the one-sample case, in order to use t.test you need actual data, not just sample means and standard deviations. Hence I have made up some normal-looking data with the same sample sizes, sample means, and sample standard deviations as in Example 8.6.2 in DeGroot and Schervish.

As the printout says, we get test statistic t = 3.3213 and P-value p-value = 0.005796 (two-tailed test with null hypothesis μ₁ = μ₂).

This is not the old-fashioned two sample t-test based on the assumption of equal variances described at the beginning of Section 8.6 in DeGroot and Schervish and in the following section. Rather it is the newer test without the equal variance assumption described starting at the bottom of p. 501 under the heading Unequal Variances. Note that the test statistic and P-value agree with those given in Example 8.6.3 on p. 503. Also in agreement is the reported non-integer degrees of freedom df = 12.492, although when we use t.test to do the calculations we don't really need to know this.

Assuming Equal Variances

If one wants to do the old-fashioned test, the optional argument var.equal = TRUE to the t.test function makes it do that.

Why would one want to do this? Here are the rationales for the two procedures.

• The old-fashioned test is exact if (big if)
  • the data are exactly normal,
  • the data are exactly independent,
  • the variances of the two populations are exactly equal,
  and if these assumptions are not exactly correct (particularly the last), then it is only approximate. The closer the assumptions are to correct, the closer the test is to correct.
• The test using Welch's approximation is not exact, but it is closer to correct than the old-fashioned test unless the population variances are almost exactly equal.

Unless you know somehow that the variances are equal (a direct pipeline to God perhaps), you should use Welch's approximation.

Now we get test statistic t = 3.4427 and P-value p-value = 0.003345 (two-tailed test with null hypothesis μ₁ = μ₂). This matches what DeGroot and Schervish get in their Example 8.6.2.

Two-Sample Confidence Intervals

Not much to say here. The preceding two examples produced 95% confidence intervals for the parameter μ₁ &minus μ₂.

The example in the Welch's approximation section produces the confidence interval (0.1040533, 0.4959467).

The example in the equal variance section produces the confidence interval (0.1152669, 0.4847331). This latter relies on a generally unverifiable assumption of equal population variances.