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Chapter 5: Problem 41

Empirical Rule for Normal Distributions Pick any positive values for the mean and the standard deviation of a normal distribution. Use your selection of a normal distribution to answer the questions below. The results of parts (a) to (c) form what is often called the Empirical Rule for the standard deviation in a normal distribution. (a) Verify that about \(95 \%\) of the values fall within two standard deviations of the mean. (b) What proportion of values fall within one standard deviation of the mean? (c) What proportion of values fall within three standard deviations of the mean? (d) Will the answers to (b) and (c) be the same for any normal distribution? Explain why or why not.

Short Answer

Expert verified
According to the empirical rule, about \(95 \%\) of the values fall within two standard deviations of the mean, about \(68 \%\) of the values fall within one standard deviation of the mean, and about \(99.7 \%\) fall within three standard deviations of the mean. These percentages hold true for all normal distributions.

Step by step solution

01

Select Mean and Standard Deviation

Choose any positive values for the mean and the standard deviation of a normal distribution. For instance, let's take the mean (\( \mu \)) as 100 and the standard deviation (\( \sigma \)) as 15.
02

Apply Empirical Rule for two standard deviations

The empirical rule states that 95% of the data falls within two standard deviations of the mean. Therefore, this means that 95% of the values will fall between \( \mu - 2\sigma \) and \( \mu + 2\sigma \). In our case, this is between (100 - 2*15) and (100 + 2*15), which is between 70 and 130. So, \(95 \%\) of the values fall within two standard deviations of the mean.
03

Empirical Rule for one standard deviation

According to empirical rule, 68% of the data falls within one standard deviation of the mean. That means 68% of the values will fall between \( \mu - \sigma \) and \( \mu + \sigma \). With our values, this is between (100 - 15) and (100 + 15), which is between 85 and 115.
04

Empirical Rule for three standard deviations

For three standard deviations from the mean, the empirical rule states that 99.7% of the values in a normal distribution fall within this range. Thus, 99.7% of the data will be between \( \mu - 3\sigma \) and \( \mu + 3\sigma \). In our case, this is between (100 - 3*15) and (100 + 3*15) which is between 55 and 145.
05

Explanation

The empirical rule applies to all normal distributions, so the answers to (b) and (c) will be the same regardless of the chosen mean and standard deviation. But for distributions that are not normal, the empirical rule may not hold.

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