Statistics and probability pptx lesson 303
- 1. Lesson 3.3 The Normal Curve
- 2. Learning Competency At the end of the lesson, the learners should be able to construct a normal curve.
- 3. Objectives At the end of this lesson, the learners should be able to do the following: ● Construct a normal curve. ● Find the area of a region under the normal curve. ● Solve real-life problems involving the normal curve.
- 4. Learn about It! The normal curve or bell curve is a graph that represents the probability density function of the normal probability distribution. The normal curve is also called the Gaussian curve, named after the mathematician Carl Friedrich Gauss. The Normal Curve
- 5. Learn about It! The graph below represents the normal curve where 𝜇 is the mean and 𝜎 is the standard deviation. The Normal Curve
- 6. Learn about It! This rule states that the area of the region between one standard deviation away from the mean is 0.6826, two standard deviations away from the mean is 0.9544, and three standard deviations away from the mean is 0.9974. Empirical Rule
- 7. Learn about It! Area of the Region between One Standard Deviation Away from the Mean
- 8. Learn about It! Area of the Region between Two Standard Deviations Away from the Mean
- 9. Learn about It! Area of the Region between Three Standard Deviations Away from the Mean
- 10. Try it! Let’s Practice Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 11. Solution to Let’s Practice Solution: 1. Shade the region in the normal curve that is more than one standard deviation above the mean. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 12. Solution to Let’s Practice Solution: 2. Use the empirical rule to find the area of the given region. Find the area of the region one standard deviation away from the mean. By empirical rule, the area of this region is 0.6826. The shaded area is shown on the next slide. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 13. Solution to Let’s Practice Solution: 2. Use the empirical rule to find the area of the given region. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 14. Solution to Let’s Practice Solution: Since the normal curve is symmetric, we can obtain the area of the region between the mean and one standard deviation above the mean by dividing 0.6826 by 2. To get the area of this region, divide 0.6826 by 2. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 15. Solution to Let’s Practice Solution: 0.6826 ÷ 2 = 0.3413 Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 16. Solution to Let’s Practice Solution: To get the area of the region more than one standard deviation above the mean, subtract 0.3413 from 0.5 since half of the area of a normal curve is 0.5. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 17. Solution to Let’s Practice Solution: 0.5 − 0.3413 = 0.1587 Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 18. Solution to Let’s Practice Solution: Thus, the area of the region more than one standard deviation above the mean is 𝟎. 𝟏𝟓𝟖𝟕. Example 1: What is the area of the scores in a normally distributed data that is more than one standard deviation above the mean?
- 19. Try it! Let’s Practice Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 20. Solution to Let’s Practice Solution: 1. Solve for the scores one, two, and three standard deviations away from the mean. 𝜇 = 15.8, 𝜎 = 2.2 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 21. Solution to Let’s Practice Solution: a. One standard deviation away from the mean 𝜇 + 𝜎 = 15.8 + 2.2 = 18 𝜇 − 𝜎 = 15.8 − 2.2 = 13.6 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 22. Solution to Let’s Practice Solution: b. Two standard deviations away from the mean 𝜇 + 2𝜎 = 15.8 + 2(2.2) = 15.8 + 4.4 = 20.2 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 23. Solution to Let’s Practice Solution: b. Two standard deviations away from the mean 𝜇 − 2𝜎 = 15.8 − 2(2.2) = 15.8 − 4.4 = 11.4 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 24. Solution to Let’s Practice Solution: c. Three standard deviations away from the mean 𝜇 + 3𝜎 = 15.8 + 3(2.2) = 15.8 + 6.6 = 22.4 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 25. Solution to Let’s Practice Solution: c. Three standard deviations away from the mean 𝜇 − 3𝜎 = 15.8 − 3(2.2) = 15.8 − 6.6 = 9.2 Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 26. Solution to Let’s Practice Solution: 2. Construct the normal curve using the set of scores obtained from the previous step. Using the mean as the center and the set of scores, we can construct the normal curve as follows: Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 27. Solution to Let’s Practice Solution: Example 2: The mean and standard deviation of a normally distributed scores are 15.8 and 2.2, respectively. Construct the normal curve of the data.
- 28. Try It! Individual Practice: 1. In a normally distributed data, find the area of the scores less than two standard deviations below the mean. 2. Construct the normal curve of a given data with 𝜇 = 37.6 and 𝜎 = 3.9.
- 29. Try It! Group Practice: To be done in groups of five. Mr. Esguerra gave a 50-item test to his class and got a mean score of 34.8 and a standard deviation of 1.5. Given that the scores are normally distributed, what percent of the students got 33.3 and below?
- 30. Key Points ● A normal curve or a bell curve is a graph that represents the probability density function of a normal probability distribution. It is also called a Gaussian curve—named after the mathematician Carl Friedrich Gauss.
- 31. Key Points ● The empirical rule states that the area of the region between one standard deviation away from the mean is 0.6826, two standard deviations away from the mean is 0.9544, and three standard deviations away from the mean is 0.9974.