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Understanding Normal Distributions for Data Analysis

Learn how to recognize and solve problems with normally distributed data and probability distributions based on discrete or continuous data. Discover the characteristics of the normal distribution and standardized test results. Practice finding normal probabilities with examples.

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Understanding Normal Distributions for Data Analysis

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  1. Normal Distributions 11-Ext Lesson Presentation Holt Algebra 2

  2. Objectives Recognize normally distributed data. Use the characteristics of the normal distribution to solve problems.

  3. Standardized test results, like those used for college admissions, follow a normal distribution. Probability distributions can be based on either discrete or continuous data. Usually discrete data result from counting and continuous data result from measurement.

  4. The binomial distributions that you studied in Lesson 11-6 were discrete probability distributions because there was a finite number of possible outcomes. The graph shows the probability distribution of the number of questions answered correctly when guessing on a true-false test.

  5. In a continuous probability distribution, the outcome can be any real number—for example, the time it takes to complete a task. You may be familiar with the bell-shaped curve called the normal curve. A normal distribution is a function of the mean and standard deviation of a data set that assigns probabilities to intervals of real numbers associated with continuous random variables.

  6. The figure shows the percent of data in a normal distribution that falls within a number of standard deviations from the mean.

  7. Addition shows the following: • • About 68% lie within 1 standard deviation of the mean. • • About 95% lie within 2 standard deviations of the mean. • • Close to 99.8% lie within 3 standard deviations of the mean.

  8. Example 1A: Finding Normal Probabilities The SAT is designed so that scores are normally distributed with a mean of 500 and a standard deviation of 100. What percent of SAT scores are between 300 and 500? 300 is 2 standard deviations from the mean, 500. Use the percents from the previous slide. 13.6% + 34.1% = 47.7%

  9. Example 1B: Finding Normal Probabilities What is the probability that an SAT score is below 700? Because the graph is symmetric, the left side of the graph shows 50% of the data. 50% + 47.7% = 97.7% The probability that an SAT score is below 700 is 0.977, or 97.7%.

  10. Example 1C: Finding Normal Probabilities What is the probability that an SAT score is less than 400 or greater than 600? 50% – 34.1% = 15.9 Percent of data > 600 Because the curve is symmetric, the probability that an SAT score is less than 400 or greater than 600 is about 2(15.9%), or 31.8%.

  11. Check It Out! Example 1 Use the information in Example 1 to answer the following. What is the probability that an SAT score is above 300? Percent of scores below 300. 50% – (34.1% + 13.6%) = 2.3% The probability of a score above 300 is 1 – 2.3% or 97.7%.

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