Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.

About __ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).

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Step 1: Understand the problem. The goal is to find the area under the standard normal distribution curve between z = -3.5 and z = 3.5. This area represents the proportion of data within 3.5 standard deviations of the mean.

Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0). The total area under the curve is 1, which corresponds to 100%.

Step 3: Use a standard normal distribution table (z-table) or statistical software to find the cumulative area to the left of z = 3.5. Similarly, find the cumulative area to the left of z = -3.5.

Step 4: Subtract the cumulative area to the left of z = -3.5 from the cumulative area to the left of z = 3.5. This difference gives the area between z = -3.5 and z = 3.5.

Step 5: Convert the area obtained in Step 4 into a percentage by multiplying it by 100. This percentage represents the proportion of the data within 3.5 standard deviations of the mean.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Normal Distribution

The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is used to determine probabilities and areas under the curve for any normal distribution by converting raw scores (z-scores) into standard scores. This allows for the comparison of different datasets and the application of statistical rules, such as the Empirical Rule.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in understanding the spread of data and is foundational for making predictions about probabilities in a normal distribution.

Range Rule of Thumb

The Range Rule of Thumb is a guideline that suggests the range of a dataset can be estimated as four times the standard deviation. This rule provides a quick way to assess the variability of data and is particularly useful in applied statistics for making rough estimates about the spread of data points in relation to the mean.

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