ssrs export to pdf barcode font Sampling Distribution of Ratios of Variances in Software

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Sampling Distribution of Ratios of Variances
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On page 157, we indicated how sampling distributions of differences, in particular differences of means, can be obtained. Using the same idea, we could arrive at the sampling distribution of differences of variances, say, S2 S2. It turns out, however, that this sampling distribution is rather complicated. Instead, we may consider 1 2 the statistic S2 >S2, since a large or small ratio would indicate a large difference while a ratio nearly equal to 1 1 2 would indicate a small difference. Theorem 5-8 Let two independent random samples of sizes m and n, respectively, be drawn from two normal populations with variances s2, s2, respectively. Then if the variances of the random sam1 2 ples are given by S2, S2, respectively, the statistic 1 2 mS2 >(m 1 nS2 >(n 2 1, n 1)s2 1 1)s2 2 S2 >s2 1 1 S2 >s2 2 2
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Many other statistics besides the mean and variance or standard deviation can be defined for samples. Examples are the median, mode, moments, skewness, and kurtosis. Their definitions are analogous to those given for populations in 3. Sampling distributions for these statistics, or at least their means and standard deviations (standard errors), can often be found. Some of these, together with ones already given, are shown in Table 5-1.
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CHAPTER 5 Sampling Theory
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Table 5-1 Standard Errors for Some Sample Statistics
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Standard Error
Remarks This is true for large or small samples where the population is infinite or sampling is with replacement. The sampling distribution of means is very nearly normal (asymptotically normal) for n 30 even when the population is nonnormal. mX m, the population mean in all cases.
Means
s !n
Proportions
p(1 A n
pq An
The remarks made for means apply here as well. mP p in all cases
Medians
smed
p s 2n A 1.2533s !n
For n 30, the sampling distribution of the medians is very nearly normal. The given result holds only if the population is normal or approximately normal. mmed m
(1) sS Standard deviations (2) sS
s 22n m4 s4 A 4ns2
For n 100, the sampling distribution of S is very nearly normal. sS is given by (1) only if the population is normal (or approximately normal). If the population is nonnormal, (2) can be used. Note that (2) reduces to (1) when m4 3s4, which is true for normal populations. For n 100, mS s very nearly. The remarks made for standard deviations apply here as well. Note that (2) yields (1) in case the population is normal.
(1) sS2 Variances (2) sS2
2 s2 n A m4 A n s2
1)s2 >n 30).
which is very nearly s2 for large n (n
Frequency Distributions
If a sample (or even a population) is large, it is difficult to observe the various characteristics or to compute statistics such as mean or standard deviation. For this reason it is useful to organize or group the raw data. As an illustration, suppose that a sample consists of the heights of 100 male students at XYZ University. We arrange the data into classes or categories and determine the number of individuals belonging to each class, called the class frequency. The resulting arrangement, Table 5-2, is called a frequency distribution or frequency table. The first class or category, for example, consists of heights from 60 to 62 inches, indicated by 60 62, which is called a class interval. Since 5 students have heights belonging to this class, the corresponding class frequency is 5. Since a height that is recorded as 60 inches is actually between 59.5 and 60.5 inches while one recorded as 62 inches is actually between 61.5 and 62.5 inches, we could just as well have recorded the class interval as 59.5 62.5. The next class interval would then be 62.5 65.5, etc. In the class interval 59.5 62.5 the numbers 59.5 and 62.5 are often called class boundaries. The width of the jth class interval, denoted by cj, which is usually the same for all classes (in which case it is denoted by c), is the difference between the upper and lower class boundaries. In this case c 62.5 59.5 3. The midpoint of the class interval, which can be taken as representative of the class, is called the class mark. In the above table the class mark corresponding to the class interval 60 62 is 61. A graph for the frequency distribution can be supplied by a histogram, as shown shaded in Fig. 5-1, or by a polygon graph (often called a frequency polygon) connecting the midpoints of the tops in the histogram. It is of
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