2: CONFIDENCE INTERVALS FOR THE MEAN; UNKNOWN VARIANCE

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1 : CONFIDENCE INTERVALS FOR THE MEAN; UNKNOWN VARIANCE Now, we suppose that X,...,X 1 n are iid with unknown mean µ and unknown variance σ. Clearly, we will now have to estimate σ from the available data. The most commonly-used estimator of σ is the sample variance, S Σ n (X i X ). i = 1 = 1 n 1 The reason for using the n 1 in the denominator is that this makes S an unbiased estimator of σ.in other words, E [S ] =σ. We will prove this later. A proof in the normal case follows from Section 4. 8 of Hogg & Craig.

2 -- Note: Hogg and Craig use a denominator of n in their S. We, most tetbooks, and most practitioners, however, use n 1. To minimize confusion, we will try for now to avoid using the symbol S. Question: What would happen if we used S in place of σ in the formula for the CI? Answer: It depends on whether the sample size is " large" or not.

3 -3- Large-Sample Confidence Interval; Population Not Necessarily Normal Theorem: The interval X ± z α/ S is an asymp- n totic level 1 α CI for µ. In other words, when the sample size is large, we can use S in place of the unknown σ, and the CI will still work. Proof: It can be shown that S converges in probab ility to σ. In other words, lim Pr ( S σ >ε) 0 for any ε>0. n As a result, the distribution of

4 -4- X µ S / n converges to the standard normal distribution. Simi- larly to the proof from the previous handout, we get X µ Pr (CI Contains µ) = Pr ( z α/ < S / n < z α/ ) 1 α. Small-Sample Confidence Interval; Normal Population If the sample size is small (the usual guideline is n dity of the CI we will present here, we must assume 30), and σ is unknown, then to assure the valithat the population distribution is normal. This assumption is hard to check in small samples!

5 The CI is X ± t α/ -5- S. (t α/ is defined below.) n The Basics of t Distributions When n is small, the quantity t = X µ does not S / n have a normal distribution, even when the popula- tion is normal. Instead, t has a "Student s t distribution with n 1 degrees of freedom". There is a different t distribution for each value of the degrees of freedom, ν. The quantity t α/ denotes the t value such that the

6 -6- area to its right under the Student s t distribution ( with ν=n 1) is α/. Note that we use ν=n 1, even though the sample size is n. Values of t α are listed in Table, Page 599 of Jobson. Note that the last row of Table is denoted by " ". For practical purposes, any value of ν beyond 9 is usually considered "infinite". (Most tables stop at ν=9. Jobson s table is somewhat better, since he also has entries for ν= 30, 40, 50, 60, and 10.) In this case, the corresponding t distribution is essentially identical to the standard normal distribution. Here, it doesn t matter whether we use the CI X ± t α/ S or X ± z n α/ S since they will be n

7 -7 almost the same. Since t is asymptotically standard normal, the t values given in the " " row of Table α are identical to the z α values defined earlier. On the other hand, if ν 9 the t distribution has " longer tails" (i.e., contains more outliers) than the normal distribution, and it is important to use the t values of Table, assuming that σ is unknown. - Here, the CI based on t α/ will be wider than the (incorrect) one based on z α/. ( Why does this happen, and why does it make sense?) Eg 1: A random sample of 8 "Quarter Pounders" yields a mean weight of =. pounds, with a

8 -8- sample standard deviation of s =.07 pounds. Con struct a 95% CI for the unknown population mean weight for all "Quarter Pounders". Background: Definitions of χ and t distributions As in Section of Jobson, we define the χ dis- ν tribution with ν degrees of freedom to be the distribution of the random variable χ = Z i, where i =1 Z 1,..., Z are iid standard normal. The distribu- ν tion is positive valued and is skewed to the right. The mean and variance are E [χ ] =ν, var [χ ] = ν. I 1 n ν Σ ν ν f X,...,X are iid N (µ,σ ), then it can be shown that (n 1)S /σ has a χ n 1 distribution.

9 n 1-9- Therefore, S σ χ /(n 1), and we find that E [S ] =σ, so that S is unbiased for σ. The random variable Z χ /ν ν is said to have a t distribution with ν degrees of freedom if Z is standard normal and χ ν dent of Z and has a χ is indepenν distribution. Establishing the Small-Sample CI Theorem: If X,...,X are iid N (µ, σ 1 n ), then the interval X ± t S α/ is a level 1 α CI for µ. n

10 -10- Proof: It can be shown that X and S are indepen- dent. (We will prove this later). Define Z = n (X µ)/σ, which is standard normal. n 1 bution. Define n 1 - Define χ =(n 1)S /σ, which has a χ distri t = Z = χ /(n 1) n 1 n (X µ) S X µ =. S / n By its definition, t has a t distribution with n 1 degrees of freedom. Therefore, similarly to the earlier proofs, Pr (CI Contains µ) = Pr ( t α/ < X µ < t α/ ) = 1 α. S / n