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Understanding Normal Distributions: Characteristics, Applications, and Problem Solving

This chapter focuses on recognizing normally distributed data and utilizing the characteristics of the normal distribution to solve various problems. It explains how standardized tests, such as the SAT, adhere to a normal distribution with specific mean and standard deviation values. The chapter covers both discrete and continuous probability distributions, emphasizing the significance of the normal curve, including properties like symmetry and area representing probabilities. Key examples illustrate calculations related to SAT scores, providing practical applications of the concepts learned.

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Understanding Normal Distributions: Characteristics, Applications, and Problem Solving

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  1. Chapter 8 Extension Normal Distributions

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

  3. Normal Distribution • Standardized test results, like those used for college admission, 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. Normal Distribution • The binomial distribution 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. Normal Distribution • In a continuous probability distribution, the outcome can be any real number – for example, the time it takes to complete at 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. Normal Distributions • The probability assigned to a real-number interval is the area under the normal curve in that interval. Because the area under the curve represents probability, the total area under the curve is 1 • The maximum value of a normal curve occurs at the mean • The normal curve is symmetric about a vertical line through the mean • The normal curve has a horizontal asymptote at y = 0

  7. Normal Distributions The figure shows the percent of data in a normal distribution that falls within a number of standard deviations from the mean Addition shows the following: • About 68% of the data lies within 1 standard deviation of the mean • About 95% of the data lie within 2 standard deviations of the mean • Close to 99.8% of the data lie within 3 standard deviations of the mean

  8. Example • 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?

  9. Example (Continued) • What is the probability that an SAT score is below 700?

  10. Example (Continued) • What is the probability that an SAT score is less than 400 or greater than 600? • What is the probability that an SAT score is above 300?

  11. Homework • Page 595 #1-8

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