# T-Distribution Stats Assignment Homework Help

8.7   T-Distribution

In Section 8.5 we discussed the utility of the central limit theorem. Its applica­tions revolve around inferences on a population mean or the difference between two population means. Use of the central limit theorem and the normal distribution is certainly helpful in this context. However, it was assumed that the population standard deviation is known. This assumption may not be unreasonable in situa­tions where the engineer is quite familiar with the system or process. However, in many experimental scenarios knowledge of <r is certainly no more reasonable than knowledge of the population mean n- Often, in fact, an estimate of a must be supplied by the same sample information that produced the sample average x. As a result, a natural statistic to consider to deal with inferences on n is ™     X-n

since S is the sample analog to a. If the sample size is. small, the values of S2 fluc­tuate considerably from sample to sample (see Exercise 8.45 on page 265) and the distribution of T deviates appreciably from that of a standard normal distribution. If the sample size is large enough, say n > 30. the distribution of T does not differ considerably from the standard normal. However, for n < 30, it is useful to deal with the exact distribution of T. In developing the sampling distribution of T, we shall assume that our random sample was selected from a normal population. We can then, write where has the standard normal distribution and has a chi-squared distribution with v = n –1 degrees of freedom. In sampling from normal populations, we can show that X and S2 arc independent, and consequently so are Z and V. Tlu: following theorem gives the definition of a T random variable as a function of Z (standard normal) and a \2. For completeness the density function of the t-distributibn is given.

Theorem 8.5;

Let Z be a standard normal random variable and V a chi-squared random variable with v degrees of freedom. If Z and V are independent, then the distribution of the random variable T, where is given by the density function.  This is known as the distribution with v degrees of frq&om. From the foregoing and the theorem above we have the following corollary.

Corollary 8.1;

Let •X^,Xa,…,Xn be independent random variables that are all normal within mean p and standard deviation a. Let The probability distribution of T was first published in 1908 in a paper by W. S. Cosset. At the time, Cosset was employed by an Irish brewery that disallowed publication of research by members of its staff. To circumvent this restrict.., >>e published his work secretly under the name “Student.” Consequently, the uLtri’->•-‘-tion of T is usually called the Student t-distribution, or simply the t-distribution. In deriving the equation of this distribution, Cosset assumed that the samples were selected from a normal population. Although this would seem to be a very restrictive assumption, it can be shown that nonnormal populations possessing nearly bell-shaped distributions will still provide values of T that approximate the t-distribution very closely.

What Does the for-Distribution Look Like?

The distribution of T is similar to the distribution of Z in that they both are symmetric about a mean of zero. Both distributions are bell shaped, but the t-distribution is more variable, owing to the fact that the T-values depend on the fluctuations of two quantities, X and S2, whereas the Z-values depend only on the changes of X from sample to sample. The distribution of T differs from that of Z in fiat the variance of T depends on the sample size n and is always greater than 1. Only when the sample size n —» 60 will the two distributions become the same. In Figure 8.13 we show the relationship between a standard normal distribution (v = oo) and t-distributions with 2 and 5 degrees of freedom. The percentage points of the t-distribution are given in Table A.4.

-2                -1            0             1                2

Figure 8.13: The t-distribution for v              Figure 8.14: Symmetry property of the distribution.

It is customary to let t represent the t-value above which we find an area equal to q. Hence, the t-value with 10 degrees of freedom leaving an area of 0.025 to the right is t = 2.228. Since the t-distribution is symmetric about a mean of zero, we have that is, the t-value leaving an area of 1 — a to the right and therefore an area of a to the left is equal to the negative t-value that leaves an area of a in the right tail of the distribution (see Figure 8.14).

Solution:

From Table A.4 we note that 1.761 corresponds to to.os when v = 14. Therefore,! -to.os = —1.761.  Since k in the original probability statement is’to the left = —1-761, let k = —to,. Then, from Figure 8.15, we have

0.045 = 0.05-a   or   a = 0.005.

Hence, from Table A.4 with t; = 14, fc =-to.oo5 =-2.977   and   P(-2.977 < T <*-1.76l) = 0.045. Exactly 95% of the values of a t-distribution with v = n -1 degrees of freedom lie between — t0.o25 and £0.025- Of course, there are other t-values that contain 95% I of the distribution, such as -(0.02 and to os, but these values do not appear in l Table A.4, and furthermore the shortest possible interval is obtained by choosing] t-values that leave exactly the same area in the two tails of our distribution. Ai t-value that falls below —to.ozs or above to.oas would tend to make us believe that] either a very rare event has taken place or perhaps our assumption about is in error. Should this happen, we shall make the latter decision and claim that our assumed value of p is in error. In feet, a t-value falling below –to oi or above to oi would provide even stronger evidence that our assumed value of m is quite unlikely. I General procedures for testing claims concerning the value of the parameter will be treated in Chapter 10.  A preliminary look into the foundation of these] procedures is illustrated by the following example.

What Is the t-Distribution Used for?

The .t-distribution is used extensively in problems that deal with inference about the population mean (as illustrated in Example 8.14) or in problems that involve comparative samples (i.e., in cases where one is trying to determine if means from two samples are significantly different). The use of the distribution will be extended in Chapters 9.10.11, and 12. The reader should note that use of the t-distribution for the statistic requires that X\, Xy, Xn be normal. The use of the t-distribution and the sample size consideration do not relate to the central limit theorem. The use of the standard normal distribution rather than T for 30 merely implies that S is a sufficiently good estimator of a in this case. In chapters that follow the t-distribution finds extensive usage.

Posted on July 5, 2014 in Statistics
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