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CHAPTER 5 Sampling Theory
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Table 5-2 Heights of 100 Male Students at XYZ University
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Height (inches) Number of Students
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60 62 63 65 66 68 69 71 72 74 TOTAL
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5 18 42 27 8 100
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Fig. 5-1
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interest that the shape of the graph seems to indicate that the sample is drawn from a population of heights that is normally distributed.
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Relative Frequency Distributions
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If in Table 5-2 we recorded the relative frequency or percentage rather than the number of students in each class, the result would be a relative, or percentage, frequency distribution. For example, the relative or percentage frequency corresponding to the class 63 65 is 18>100, or 18%. The corresponding histogram is then similar to that in Fig. 5-1 except that the vertical axis is relative frequency instead of frequency. The sum of the rectangular areas is then 1, or 100%. We can consider a relative frequency distribution as a probability distribution in which probabilities are replaced by relative frequencies. Since relative frequencies can be thought of as empirical probabilities (see page 5), relative frequency distributions are known as empirical probability distributions.
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Computation of Mean, Variance, and Moments for Grouped Data
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We can represent a frequency distribution as in Table 5-3 by giving each class mark and the corresponding class frequency. The total frequency is n, i.e., c
Table 5-3
Class Mark x1 x2 ( xk TOTAL Class Frequency f1 f2 ( fk n
Since there are f1 numbers equal to x1, f2 numbers equal to x2, . . . , fk numbers equal to xk, the mean is given by x # f1x1 f2x2 n c fk xk a fx n (24)
CHAPTER 5 Sampling Theory
Similarly the variance is given by s2 f1(x1 x)2 # f2(x2 x)2 # n c fk(xk x)2 # a f (x n x)2 # (25)
Note the analogy of (24) and (25) with the results (2), page 75, and (13), page 77, if fj >n correspond to empirical probabilities. In the case where class intervals all have equal size c, there are available short methods for computing the mean and variance. These are called coding methods and make use of the transformation from the class mark x to a corresponding integer u given by x a cu (26)
0. The coding formulas for the mean and vari-
where a is an arbitrarily chosen class mark corresponding to u ance are then given by x # s2 a c n a fu a fu2 n a cu #
Similar formulas are available for higher moments. The r th moments about the mean and the origin, respectively, are given by f1(x1 x)r c fk(xk x)r # # # a f (x x)r mr (29) n n mr r f1xr 1 c n fk xr k a fxr n (30)
c2 B
a fu n R
(27) c2(u2 # u2) # (28)
The two kinds of moments are related by m1 m2 m3 m4 etc. If we write Mr a f (u n u)r # Mr r a f ur n 0 mr 2 mr 3 mr 4
mr2 1 3mr mr 1 2 4mr mr 1 3
2mr3 1 6mr2mr 1 2
(31) 3mr14
where u is given by (26), then the relations (31) also hold between the M s. But mr a f (x n x)r # a f [(a cu) n (a cu)]r # a fcr(u n u)r # crMr
so that we obtain from (31) the coding formulas m1 m2 m3 m4 0 c2(Mr 2 c3(Mr 3 c4(Mr 4
Mr2) 1 3Mr Mr 1 2 4Mr Mr 1 3
2Mr3) 1 6Mr2Mr 1 2
(32) 3Mr4) 1
etc. The second equation of (32) is, of course, the same as (28). In a similar manner other statistics, such as skewness and kurtosis, can be found for grouped samples.
CHAPTER 5 Sampling Theory
SOLVED PROBLEMS
Sampling distribution of means 5.1. A population consists of the five numbers 2, 3, 6, 8, 11. Consider all possible samples of size two which can be drawn with replacement from this population. Find (a) the mean of the population, (b) the standard deviation of the population, (c) the mean of the sampling distribution of means, (d) the standard deviation of the sampling distribution of means, i.e., the standard error of means.
(a) (b) s2 and s (2 3.29. 6)2 (3 m 6)2 2 (6 5 3 6)2 6 5 (8 8 11 6)2 30 5 (11 6.0 6)2 16 9 0 5 4 25 10.8
(c) There are 5(5) 25 samples of size two which can be drawn with replacement (since any one of five numbers on the first draw can be associated with any one of the five numbers on the second draw). These are (2, 2) (3, 2) (6, 2) (8, 2) (11, 2) (2, 3) (3, 3) (6, 3) (8, 3) (11, 3) (2, 6) (3, 6) (6, 6) (8, 6) (11, 6) (2, 8) (3, 8) (6, 8) (8, 8) (11, 8) (2, 11) (3, 11) (6, 11) (8, 11) (11, 11)
The corresponding sample means are 2.0 2.5 (1) 4.0 5.0 6.5 2.5 3.0 4.5 5.5 7.0 4.0 4.5 6.0 7.0 8.5 5.0 5.5 7.0 8.0 9.5 6.5 7.0 8.5 9.5 11.0
and the mean of the sampling distribution of means is mX illustrating the fact that mX sum of all sample means in (1) above 25 150 25 6.0
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