6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:15 minutesProblem 6.1.45
Textbook Question
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 = -1 and z = 1 (or within 1 standard deviation of the mean).
Verified step by step guidance1
Step 1: Understand the problem. The question asks us to find the area under the standard normal distribution curve between z = -1 and z = 1. This corresponds to the proportion of data within 1 standard deviation of the mean in a standard normal distribution.
Step 2: Recall the properties of the standard normal distribution. The standard normal distribution is symmetric about the mean (z = 0), and the total area under the curve is 1, representing 100% of the data.
Step 3: Use the z-scores provided (z = -1 and z = 1) to find the cumulative probabilities. The cumulative probability for a z-score represents the area under the curve to the left of that z-score. To find the area between z = -1 and z = 1, calculate the cumulative probability for z = 1 and subtract the cumulative probability for z = -1.
Step 4: Use a standard normal distribution table or a statistical software tool to find the cumulative probabilities. For z = 1, the cumulative probability is approximately 0.8413, and for z = -1, the cumulative probability is approximately 0.1587.
Step 5: Subtract the cumulative probability for z = -1 from the cumulative probability for z = 1 to find the area between z = -1 and z = 1. Multiply the result by 100 to convert it to a percentage. This percentage represents the proportion of data within 1 standard deviation 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 application of the empirical rule and other statistical analyses.
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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, about 95% within two standard deviations, and about 99.7% within three standard deviations. This rule helps in understanding the spread of data and is foundational for making predictions based on normal distributions.
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Range Rule of Thumb
The range rule of thumb is a guideline that suggests the range of a data set 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 the context of the empirical rule, as it helps to understand how much of the data lies within certain standard deviations from the mean.
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