An Introduction to Statistics - Lesson 9

The Student t Distribution

Lesson Overview

• Hypothesis Testing
• Type I and Type II Errors
• Computing a test statistic
• Making a decision about H₀
• Student t Distribution
• Degrees of Freedom
• Table of t Values
• Homework

Hypothesis Testing

Our two hypotheses have special names: the null hypothesis represented by H₀ and the alternative hypothesis by H_a. Historically, the null (invalid, void, amounting to nothing) hypothesis was what the researcher hoped to reject. In theory, it is now common practice not to associate any special meaning to which hypothesis is which. (In practice, this may be different, so check early with your research advisor. The research hypothesis becomes the alternate hypothesis and the null hypothesis or "straw man" to be knocked down is so determined.) Although simple hypotheses would be easiest to test, it is much more common to have one of each type or perhaps for both to be composite. If the values specified by H_a are all on one side of the value specified by H₀, then we have a one-sided test (one-tailed), whereas if the H_a values lie on both sides of H₀, then we have a two-sided test (two-tailed). A one-tailed test is sometimes called a directional test and a two-tailed test is sometimes called a nondirectional test.

The outcome of our test regarding the population parameter will be that we either reject the null hypothesis or fail to reject the null hypothesis. It is considered poor form to "accept" the null hypothesis, although if we fail to reject it, that is in fact essentially what we are doing. When we reject the null hypothesis we have only shown that it is highly unlikely to be true---we have not proven it in the mathematical sense. The research hypothesis is supported by rejecting the null hypothesis. The null hypothesis locates the sampling distribution, since it is (usually) the simple hypothesis, testing against one specific value of the population parameter. Establishing the null and alternative hypotheses is sometimes considered the first step in hypothesis testing.

Type I and Type II Errors

Reject\Truth	H0 True	Ha True
Reject Ha	no error	False positive, Type II, beta=P(Reject Ha|Ha true)
Reject H0	False negative, Type I, alpha=P(Reject H0|H0 true)	no error

The term false positive for type II errors comes from perhaps a blood test where the test results came back positive, but it is not the case (false) that the person has whatever was being tested for. The term false negative for type I errors then would mean that the person does indeed have whatever was being tested for, but the test didn't find it. When testing for pregnancy, AIDS, or other medical conditions, both types of errors can be a very serious matter. Formally, alpha=P(Accept H_a|H₀ true), meaning the probability that we "accepted" H_a when in fact H₀ was true. Alpha is the term used to express the level of significance we will accept. For 95% confidence, alpha=0.05. For 99% confidence, alpha=0.01. These two alpha values are the ones most frequently used. If our P-value, the high unlikeliness of H₀ being true, is less than alpha, we can reject the null hypothesis. Alpha and beta usually cannot both be minimized—there is a trade-off between the two. Ideally, of course, we would minimize both. Historically, a fixed level of significance was selected (alpha=0.05 for the social sciences and alpha=0.01 or alpha=0.001 for the natural sciences, for instance). This was due to the fact that the null hypothesis was considered the "current theory" and the size of Type I errors was much more important than that of Type II errors. Now both are usually considered together when determining an adequately sized sample. Instead of testing against a fixed level of alpha, now the P-value is often reported. Obviously, the smaller the P-value, the stronger the evidence (higher significance, smaller alpha) provided by the data is against H₀. Establishing threshhold error levels is often considered step two in hypothesis testing.

Example: On July 14, 2005 the AU EDRM611 class took 10 samples of 20 pennies set on edge and the table banged. The resultant mean of heads was 14.5 with a standard deviation of 2.12. Since this is a small sample, and the population variance is unknown, after we calculate a t value and obtain t=6.71=(14.5-10)/(2.12/ (10)), we apply the t-test and find a P-value of either 8.73×10^-5 or 4.36×10^-5 depending on whether we do a one-tailed or two-tailed test. In either case our results are statistically significant at the 0.0001 level.

The P-value of a test is the probability that the test statistic would take a value as extreme or more extreme than that actually observed, assuming H0 is true.

Computing a Test Statistic

Making a decision about H₀

Student t Distribution

Testing a hypothesis at the alpha=0.05 level or establishing a 95% confidence interval are again essentially the same thing. In both cases the critical values and the region of rejection are the same. However, we will more formally develop the confidence intervals elsewhere.

First, a little history about this distribution's curious name. William Gosset (1876–1937) was a Guinness Brewery chemist who needed a distribution that could be used with small samples. Since the Irish brewery did not allow publication of research results, he published in 1908 under the pseudonym of Student. We know that large samples approach a normal distribution. What Gosset showed was that small samples taken from an essentially normal population have a distribution characterized by the sample size. The population does not have to be exactly normal, only unimodal and basically symmetric. This is often characterized as heap-shaped or mound shaped.

Following are the important properties of the Student t distribution.

1. The Student t distribution is different for different sample sizes.
2. The Student t distribution is generally bell-shaped, but with smaller sample sizes shows increased variability (flatter). In other words, the distribution is less peaked than a normal distribution and with thicker tails (platykurtic). As the sample size increases, the distribution approaches a normal distribution. For n > 30, the differences are negligible.
3. The mean is zero (much like the standard normal distribution).
4. The distribution is symmetrical about the mean.
5. The variance is greater than one, but approaches one from above as the sample size increases (²=1 for the standard normal distribution).
6. It takes into account the fact that the population standard deviation is unknown.
7. The population is essentially normal (unimodal and basically symmetric)

To use the Student t distribution which is often referred to just as the t distribution, the first step is to calculate a t-score. This is much like finding the z-score. The formula is:

t = ( - µ) ÷ (s/sqrt(n))

Actually, since the population mean is likely also unknown, the sample mean must be used. The critical t-score can be looked up based on the level of confidence desired and the degrees of freedom. Degrees of freedom is a fairly technical term which permeates all of inferential statistics. It is usually abbreviated df. In this case, it is the very common value n-1.

In general, the degrees of freedom is the number of values that can vary after certain restrictions have been imposed on all values.

Where does the term degrees of freedom come from? Suppose, for example, that you have a phone bill from Ameritech that says your household owes $100. Your mother and father state that $70 of it is theirs and that your younger sibling owes only $5. How much does that leave you? Here, n=3 (parents, sibling, you), but once you have the total (or mean) and two more pieces of information, the last data element is constrained. The same is true with the degrees of freedom, you can arbitrarily use any n-1 data points, but the last one will be determined for a given mean. Another example is with 10 tests that averaged 55, if you assign nine people random grades, the last test score is not random, but constrained by the overall mean. Thus for 10 tests and a mean, there are nine degrees of freedom.

If the interval calls for a 90% confidence level, then alpha = 0.10 and alpha/2 = 0.05 (for a two-tailed test). Tables of t values typically have a column for degrees of freedom and then columns of t values corresponding with various tail areas. An abbreviated table is given below. For a complete set of values consult a larger table or your TI-84+ graphing calculator. DISTR 5 gives tcdf. tcdf expects three arguments: lower t value, upper t value, and degrees of freedom. Since no inverse t function is given on the calculator, some guessing may be involved. Note how tcdf(9.9,9E99,2) indicates a t value of about 9.9 for a one tailed area of 0.005 with two degrees of freedom. Please locate the corresponding value of 9.925 in the table.

As with other confidence intervals, we use the t-score to obtain the margin of error term which is added and subtracted from the statistic of interest (in this case, the sample mean) to obtain a confidence interval for the parameter of interest (in this case, the population mean). In this case the margin of error is defined (since you don't have population standard deviation you use the sample's) as:

ME = talpha/2 • (s ÷ sqrt(n))

Your confidence interval should look like: - ME < µ < + ME or possibly (low value, high value).

Table of t Values

Degrees of Freedom\1/2 tails	.005/.01	.01/.02	.025/.05	.05/.10	.10/.20
1	63.66	31.82	12.71	6.314	3.078
2	9.925	6.965	4.303	2.920	1.886
3	5.841	4.541	3.182	2.353	1.638
4	4.604	3.747	2.776	2.131	1.533
5	4.032	3.365	2.571	2.015	1.476
10	3.169	2.764	2.228	1.812	1.372
15	2.947	2.602	2.132	1.753	1.341
20	2.845	2.528	2.086	1.725	1.325
25	2.787	2.485	2.060	1.708	1.316
z	2.576	2.326	1.960	1.645	1.282

Although the t procedure is fairly robust, that is it does not change very much when the assumptions of the procedure are violated, you should always plot the data to check for skewness and outliers before using it on small samples. Here small can be interpreted as n < 15. If your sample is small and the data is clearly nonnormal or outliers are present, do not use the t. If your sample is not small, but n < 40, and there are outliners or strong skewness, do not use the t. Since the assumption that the samples are random is more important that the normality of the population distribution, the t statistic can be safely used even when the sample indicates the population is clearly skewed, if n > 40.

The two sample t tests will be discussed next year.

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