Chebychev\'s Rule. A more precise version of the three-standard-deviations rule
ID: 3354973 • Letter: C
Question
Chebychev's Rule. A more precise version of the three-standard-deviations rule (Key Fact 3.2) can be obtained from Chebychev's rule, which can be stated as follows: CHEBYCHEV'S RULE For any data set and any real number k>1, at least 100(1-1/k2)% of the observations lie within k standard deviations to either side of the mean. Two special cases of Chebychev's rule are applied frequently, namely, when k =2 and k = 3. These state, respectively, that at least 75% of the observations in any data set lie within two standard devia- tions to either side of the mean. at least 89% of the observations in any data set lie within three standard devia- tions to either side of the mean. S3.31) Book Costs. Chebychev's rule also permits us to make pertinent state- ments about a data set when we know only its mean and standard deviation, and frequently that is all we do know. Here is an example of this use of Chebychev's rule. The R. R. Bowker Company of New York compiles information on costs of new books and publishes its findings in Library Journal. A sample of 40 so- ciology books taken a few years ago had a mean cost of $53.75 and a standard deviation of $10.42. Use this information and the two aforementioned special cases of Chebychev's rule to complete the following statements. a. At least 30 of the 40 sociology books cost between _ and b. At leastof the 40 sociology books cost between $22.49 and $85.01. The Empirical Rule. For data sets that have approximately bell-shaped distribu- tions, we can improve on the estimates given by Chebychev's rule by using the empirical rule, which is as follows: EMPIRICAL RULE For any data set having approximately a bell-shaped distribution, we have that roughly 68% of the observations lie within one standard deviation to either side of the mean. roughly 95% of the observations lie within two standard deviations to either side of the mean. . . roughly 99.7% of the observations lie within three standard deviations to either side of the mean.Explanation / Answer
S3.31 It is given as per the cheby slave's rule that 75% of the observations lie within 2 std deviation and here
(i) it has been asked to find the range for 30 out of 40 boos that is nothing but 30/40 = 75% of the books price range and so it wil be between mean-2*sd to mean+2*sd
So., ans = (53.75-2*10.42 , 53.75+2*10.42) = (32.91,74.59)
(ii) Now it is asked to find the 5 of books cost between 22.49 to 85.01 i.e. nothing but 3*sd apart
i.e. (53.75-3*10.42 , 53.75+3*10.42) = (22.49,85.01) and so we can say that 1-1/k^2 = 1-1/3^2 =1-1/9 = 88.88% = approx 89%
So we can say that 89% of the books will fall within 22.49 to 85.01
As per the Chegg policy and time crunch I have answerd the 1st question with all its sub parts. I would like to answer the rest as well but the policy and time is not in favour. If you want the ans for the rest of the que as well kindly post each seperately. Hope you understand.
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