Questions: What is the 68-95-99.7 rule for normal distributions? Explain how it can be used to answer questions about frequencies of data values in a normal distribution. Choose the correct answer below. A. The rule states that about 0,1 , and 2 data points lie in 68%, 95%, and 99.7% of the data points, respectively, in a normal distribution. B. The rule states that about 68%, 95%, and 99.7% of the data points in a normal distribution lie within 0,1 , and 2 standard deviations of the mean, respectively. C. The rule states that about 68%, 95%, and 99.7% of the data points in a normal distribution lie within 1,2 , and 3 standard deviations of the mean, respectively. D. The rule states that about 1,2 , and 3 data points lie in 68%, 95%, and 99.7% of the data points, respectively, in a normal distribution.

Solution Steps

The 68-95-99.7 rule, also known as the empirical rule, describes how data is distributed in a normal distribution. Specifically, it states that:

• Approximately \(68\%\) of the data points lie within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\)).
• Approximately \(95\%\) of the data points lie within \(2\) standard deviations of the mean.
• Approximately \(99.7\%\) of the data points lie within \(3\) standard deviations of the mean.

Using the standard normal distribution, we can calculate the probabilities for each of these ranges:

1. For \(1\) standard deviation: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827 \]

2. For \(2\) standard deviations: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.0) - \Phi(-2.0) = 0.9545 \]

3. For \(3\) standard deviations: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(3.0) - \Phi(-3.0) = 0.9973 \]

The calculated probabilities confirm the empirical rule:

• Probability within \(1\) standard deviation (68% rule): \(0.6827\)
• Probability within \(2\) standard deviations (95% rule): \(0.9545\)
• Probability within \(3\) standard deviations (99.7% rule): \(0.9973\)

Final Answer