1.2 the normal curve
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Before mass, a church bell is usually rung. The document discusses the normal distribution and standard normal distribution. It provides examples of calculating areas under the normal curve using z-scores and standard normal tables. Regions under the normal curve correspond to probabilities and standard normal variables.
1.2 the normal curve
- 2. Before the mass will start, what do you usually hear from the church?
- 4. Learning Competencies The learner will be able to: 1. illustrate a normal random variable and its properties; 2. construct a normal curve; and 3. identify regions under the normal curve corresponding to different standard normal variables.
- 5. The most important of all continuous probability distribution is the normal distribution. Its graph is called the normal curve, is a bell-shaped curve. It lies entirely above the horizontal axis. It is symmetrical, unimodal, and asymototic to the horizontal axis.
- 6. The area between the curve and the horizontal axis is exactly equal to 1. Half of the area is above the mean and the remaining half is below the mean.
- 7. The normal distribution is determined by two parameters: the mean and the standard deviation. If the mean is 0 and the standard deviation is 1 then the normal distribution is a standard normal distribution.
- 9. However, the mean is not always equal to 0 and the standard deviation is not always equal to 1. In the normal curve below, mean=40 and sd=12.
- 10. In the normal curve below, mean=75 and sd=10
- 11. Suppose two curves are sketched above the same horizontal axis and these normal curves have the same standard deviations but different means.
- 12. Suppose the normal curves have the same means but different standard deviations.
- 13. Suppose the normal curves have different means and different standard deviations.
- 14. Areas under the Normal Curve
- 15. Areas under the standard normal curve can be found using the Areas under the Standard Normal Curve table (Z Table). These areas are regions under the normal curve.
- 16. Example 1 Find the area between z=0 and z=1.54. Step 1. Sketch the normal curve. Since 1.54 is positive. It is somewhere to right of 0.
- 17. Step 2. Locate the area for z=1.54 from the Areas under the normal Curve Table (Z table). Proceed down the column marked z until you reach 1.5. Then proceed to the right along this row until you reach the column marked .04. The intersection of the row that contains 1.5 and the column marked .04 is the area. The area is 0.4382.
- 19. Example 2 Find the area between z=1.52 and z=2.5
- 20. Example 2 Find the area between z=1.52 and z=2.5 Step 1. Sketch the normal curve
- 21. Step 2. Let A=area between z=1.52 and z=2.5 A1= area between z=0 and z=1.52 A2= area between z=0 and z=2.5 Locate from the Z table, A1= 0.4357 A2= 0.4938 A= A2-A1 = 0.4938-0.4357 = 0.0581 Hence, the area between z=1.52 and z=2.5 is 0.0581.
- 22. Example 3 Find the area to right of z=1.56.
- 23. Example 3 Find the area to right of z=1.56. Step 1. Sketch the normal curve.
- 24. Step 2. Let A= area to right of z=1.56 A1= area between z=0 and z=1.56. Locate from the Z table, A1= 0.4406 A=0.5-A1 since the area of the half the curve is 0.5 =0.5-0.4406 =0.0594 Hence, the area to the right of z=1.56 is 0.0594.
- 25. Example 4 Find the area between z=0 and z=-1.65
- 26. Example 4 Find the area between z=0 and z=-1.65 Step 1. Sketch the normal curve. Since z=-1.65 is negative, the area is located to the left of z=0. The same table will be used to find the area to the left z=0 since the area to the left of z=0 is also positive.
- 27. Step 2. Locate the area of z=-1.65 from the table. Proceed down the column marked z until you reach 1.6. Disregard the negative sign (-). Then proceed right along this row until you reached the column marked .05. The intersection of the row and the column marked .05 is the area. Hence, the area is 0.4505.
- 28. Example 5 Find the area between z=-1.5 and z=-2.5
- 29. Example 5 Find the area between z=-1.5 and z=-2.5 Step 1. Sketch the normal curve.
- 30. Step 2. Let A=area between z= -1.5 and z= -2.5 A1= area between z= -1.5 and z=0 A2= area between z= -2.5 and z= 0 From the table, A1= 0.4332 A2= 0.4938 A=A2- A1 = 0.4938-0.4332 = 0.0606 Hence, the area between z=-1.5 and z=-2.56 is 0.0606
- 31. Example 6 Find the area between z=-1.35 and z=2.95
- 32. Example 6 Find the area between z=-1.35 and z=2.95 Step 1. Sketch the normal curve.
- 33. Step 2. Let A=area between z= -1.35 and z= 2.95 A1= area between z= 0and z=-1.35 A2= area between z= 0and z= 2.95 From the table, A1= 0.4115 A2= 0.4984 A=A1+ A2 = 0.4115+ 0.4984 = 0.9099 Hence, the area between z=-1.35 and z=2.95 is 0.9099
- 34. Example 7 Find the area to left of z=2.32.
- 35. Example 7 Find the area to left of z=2.32. Step 1. Sketch the normal curve.
- 36. Step 2. Let A=area to the left of z= 2.32 A1= area of the half curve A2= area between z= 0 and z= 2.32 From the table, A2= 0.4898 A=A1+ A2 = 0.5+ 0.4898 = 0.9898 Hence, the area to the left of z=2.32 is 0.9898
- 37. Example 8 Find the area to right of z=-1.8
- 38. Example 8 Find the area to right of z=-1.8 Step 1. Sketch the normal curve.
- 39. Step 2. Let A=area to the right of z= -1.8 A1= area between z=-1.8 and z=0 A2= area half of the curve From the table, A1= 0.4641 A=A1+ A2 = 0.4641+ 0.5 = 0.9641 Hence, the area to the right of z= -1.8 is 0.9641
- 40. Example 9 Find the area to left of z=-1.52
- 41. Example 9 Find the area to left of z=-1.52 Step 1. Sketch the normal curve.
- 42. Step 2. Let A=area to the left of z= -1.52 A1= area between z=0 and z= -1.52 A2= area of the half of the curve From the table, A1= 0.4357 A=A2- A1 = 0.5- 0.4357 = 0.0643 Hence, the area to the left of z=-1.52 is 0.0643