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Chapter 4: Problem 43

A sample of concrete specimens of a certain type is selected, and the compressive strength of each specimen is determined. The mean and standard deviation are calculated as \(\bar{x}=3000\) and \(s=500\), and the sample histogram is found to be well approximated by a normal curve. a. Approximately what percentage of the sample observations are between 2500 and 3500 ? b. Approximately what percentage of sample observations are outside the interval from 2000 to 4000 ? c. What can be said about the approximate percentage of observations between 2000 and 2500 ? d. Why would you not use Chebyshev's Rule to answer the questions posed in Parts (a)-(c)?

Short Answer

Expert verified
a) Approximately 68%. b) Approximately 5%. c) Approximately 13.5%. d) Because the data is normally distributed, it's more accurate to use the empirical rule rather than Chebyshev's rule which provides wider intervals.

Step by step solution

01

Apply the Empirical Rule to answer question (a)

Approximately 68% of the sample observations fall within one standard deviation of the mean. So, it's about 68% of the sample observations fall between 2500 and 3500.
02

Apply the Empirical Rule to answer question (b)

Approximately 95% of the sample observations fall within two standard deviations of the mean. So the percentage of observations outside this range is 100% - 95% = 5%.
03

Apply the Empirical Rule to answer question (c)

The percentage of observations between 2000 and 2500 is roughly equivalent to the observations between one standard deviation and two standard deviations below the mean, which subtracts the percentage of observations within one standard deviation from the percentage within two standard deviations. Therefore, it's gotten by getting 95% - 68% = 27%, and divide by 2, because only one side (below the mean) is considered, so the result is about 13.5%.
04

Analyze the role of Chebyshev's Rule for question (d)

In this case, Chebyshev鈥檚 rule is not applied because that rule doesn't require the distribution to be normal, which is the case in this exercise. Moreover, Chebyshev鈥檚 rule provides wider, less precise intervals when compared with the empirical rule. Hence, it's more accurate to use the empirical rule for normally distributed data.

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