!s21 S

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s22 S

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(14)

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assuming the samples are independent. Results similar to (11) can then be obtained.

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The Sample Variance

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If X1, X2, . . . , Xn denote the random variables for a random sample of size n, then the random variable giving the variance of the sample or the sample variance is defined in analogy with (14), page 77, by S2 (X1 # X)2 (X2 # X)2 n c (Xn # X)2 (15)

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CHAPTER 5 Sampling Theory

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# Now in Theorem 5-1 we found that E(X) m, and it would be nice if we could also have E(S2) s2. Whenever the expected value of a statistic is equal to the corresponding population parameter, we call the statistic an unbiased estimator, and the value an unbiased estimate, of this parameter. It turns out, however (see Problem 5.20), that E(S2) which is very nearly

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only for large values of n (say, n S2

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30). The desired unbiased estimator is defined by # X)2 n 1 c (Xn # X)2 (17) (18)

n n 1

# X)2

so that

E(S2)

Because of this, some statisticians choose to define the sample variance by S2 rather than S 2 and they simply replace n by n 1 in the denominator of (15). We shall, however, continue to define the sample variance by (15) because by doing this, many later results are simplified.

EXAMPLE 5.6

Referring to Example 5.5, page 155, the sample variance has the value (4 6)2 (7 6)2 (5 5 6)2 (8 6)2 (6 6)2

while an unbiased estimate is given by 5 2 s 4 (4 6)2 (7 6)2 (5 4 6)2 (8 6)2 (6 6)2

The above results hold if sampling is from an infinite population or with replacement from a finite population. If sampling is without replacement from a finite population of size N, then the mean of the sampling distribution of variances is given by E(S2) As N S `, this reduces to (16). mS2 a N N 1 ba n n 1 bs2 (19)

Sampling Distribution of Variances

By taking all possible random samples of size n drawn from a population and then computing the variance for each sample, we can obtain the sampling distribution of variances. Instead of finding the sampling distribution ^ of S2 or S2 itself, it is convenient to find the sampling distribution of the related random variable nS2 s2 (n 1)S2 s2

# X)2

# X)2 s2

# X)2

(20)

The distribution of this random variable is described in the following theorem. Theorem 5-6 If random samples of size n are taken from a population having a normal distribution, then the sampling variable (20) has a chi-square distribution with n 1 degrees of freedom.

CHAPTER 5 Sampling Theory

Because of Theorem 5-6, the variable in (20) is often denoted by x2. For a proof of this theorem see Problem 5.22.

Case Where Population Variance Is Unknown

In Theorems 5-4 and 5-5 we found that the standardized variable Z # X m s> !n (21)

is normally distributed if the population from which samples of size n are taken is normally distributed, while it is asymptotically normal if the population is not normal provided that n 30. In (21) we have, of course, assumed that the population variance s2 is known. It is natural to ask what would happen if we do not know the population variance. One possibility is to estimate the population variance by using the sample variance and then put the corresponding standard deviation in ^ (21). A better idea is to replace the s in (21) by the random variable S giving the sample standard deviation and to seek the distribution of the corresponding statistic, which we designate by T # X

S > !n

# X m S> !n 1

(22)

We can then show by using Theorem 4-6, page 116, that T has Student s t distribution with n 1 degrees of freedom whenever the population random variable is normally distributed. We state this in the following theorem, which is proved in Problem 5.24. Theorem 5-7 If random samples of size n are taken from a normally distributed population, then the statistic (22) has Student s distribution with n 1 degrees of freedom.