Mathematics
Empirical Rule
The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical principle that applies to normal distributions. It states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and almost all (99.7%) within three standard deviations. This rule provides a quick way to estimate the spread of data in a normal distribution.
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eBook - PDFFinite Mathematics
An Applied Approach
- Michael Sullivan(Author)
- 2011(Publication Date)
- Wiley(Publisher)
(b) Since $50.68 is two standard deviations below the mean and $112.00 is two standard deviations above the mean, the Empirical Rule states that approximately 95% of the bills are between $50.68 and $112.00. (c) Using Table 6 (p. 520), we find that 69 (97%) of the customers surveyed had electric bills between $50.68 and $112.00. This is close to the percentage predicted by the Empirical Rule. ■ NOW WORK PROBLEM 27. 9.5 Measures of Dispersion 549 * Named after the nineteenth-century Russian mathematician P. L. Chebychev. ▲ 4 Use Chebychev’s Theorem Suppose we are analyzing an experiment with numerical outcomes and that the experiment has population mean and population standard deviation We wish to estimate the probability that a randomly chosen outcome lies within k units of the mean. Chebychev’s Theorem* For any distribution of numbers with population mean and population standard deviation the probability that a randomly chosen outcome lies between and is at least EXAMPLE 8 Using Chebychev’s Theorem Suppose that an experiment with numerical outcomes has population mean 4 and population standard deviation 1. Use Chebychev’s theorem to estimate the probability that an outcome lies between 2 and 6. SOLUTION Here, Since we wish to estimate the probability that an outcome lies between 2 and 6, the value of k is (or ). Then by Chebychev’s theorem, the desired probability is at least That is, we expect at least 75% of the outcomes of this experiment to lie between 2 and 6. ■ NOW WORK PROBLEM 31. EXAMPLE 9 Using Chebychev’s Theorem An office supply company sells boxes containing 100 paper clips. Because of the packaging procedure, not every box contains exactly 100 clips. From previous data it is known that the average number of clips in a box is indeed 100 and the standard deviation is 2.8. If the company ships 10,000 boxes, estimate the number of boxes having between 94 and 106 clips, inclusive. SOLUTION Our experiment involves counting the number of clips in the box.
- Michael W. Trosset(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
10 Notice that, although the true probabilities are P ( { x i } ) = 1 / 6, the empirical probabilities range from 0 . 05 to 0 . 35. The fact that ˆ P 20 differs from P is an example of sampling variation. Statistical inference is concerned with determining what the empirical distribution (the sample) tells us about the true distribution (the population). The empirical distribution, ˆ P n , is an intuitively appealing approximation of the actual probability distribution, P , from which the sample was drawn. Notice that the empirical probability of any event A is just ˆ P n ( A ) = # { x i ∈ A } · 1 n , the observed frequency with which A occurs in the sample. Because the empirical distribution is an authentic probability distribution, all of the methods that we developed for studying (discrete) distributions are available for studying samples. For example, 156 CHAPTER 7. DATA Definition 7.2 The empirical cdf, usually denoted ˆ F n , is the cdf associated with ˆ P n , i.e. ˆ F n ( y ) = ˆ P n ( X ≤ y ) = # { x i ≤ y } n . The empirical cdf of sample (7.1) is graphed in Figure 7.1. y F(y) -2 -1 0 1 2 3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 1.0 Figure 7.1: An empirical cdf. In R , one can graph the empirical cdf of a sample x with the following command: > plot.ecdf(x) 7.2 Plug-In Estimates of Mean and Variance Population quantities defined by expected values are easily estimated by the plug-in principle. For example, suppose that X 1 , . . . , X n ∼ P and that we observe a sample x = { x 1 , . . . , x n } . Let μ = EX i denote the population mean. Then 7.2. PLUG-IN ESTIMATES OF MEAN AND VARIANCE 157 Definition 7.3 The plug-in estimate of μ , denoted ˆ μ n , is the mean of the empirical distibution: ˆ μ n = n i =1 x i · 1 n = 1 n n i =1 x i = ¯ x n . This quantity is called the sample mean . Example 7.2 (continued) The population mean is μ = EX i = 1 · 1 6 +2 · 1 6 +3 · 1 6 +4 · 1 6 +5 · 1 6 +6 · 1 6 = 1 + 2 + 3 + 4 + 5 + 6 6 = 3 .
eBook - PDFStatistics for the Social Sciences
A General Linear Model Approach
- Russell T. Warne(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
After 100,000 trials (Figure 6.4d), the empirical and theoretical probability distributions are indistinguishable to the naked eye, and all the probabilities are equal (to the second decimal place) to their corresponding values in the theoretical probability distribution. The convergence shown in Figures 6.4a–6.4d between theoretical and empirical probabil- ities and probability distributions continues as long as the number of trials in the empirical 0.000 0.050 0.100 0.150 0.200 1 2 3 4 5 6 7 8 9 10 11 12 13 Theoretical probability Outcome (b) Figure 6.3b A theoretical probability distribution for the outcomes of two 6-sided dice. 136 Probability and Central Limit Theorem probability distribution increases. If there were an infinite number of dice rolls, the empirical probability distribution and theoretical probability distribution would be exactly equal to one another. Because of this, most researchers just use theoretical probabilities and theoretical probabil- ity distributions for two reasons. First, they are easier to calculate. After all, it is much easier to use Formula 6.1 than to roll a pair of dice thousands of times. Second, theoretical probabilities and probability distributions are accurate “in the long run. ” In other words, even if a set of trials does not produce results that resemble the theoretical probabilities, additional trials will eventually converge to the theoretical probabilities with enough trials. Readers may feel there is something familiar about the shape of Figure 6.4c. It appears to resemble the normal distribution, which was discussed in Chapters 3 and 5. This is because, 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 1 2 3 4 5 6 7 8 9 10 11 12 13 Empirical probability Outcome (b) Figure 6.4b An empirical probability distribution for the outcomes of 1,000 trials of tossing two 6-sided dice.
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