PSUnit_II_Lesson_3_Identifying_Regions_of_Areas_Under_the_Normal_Curve.pptx

dentifying Regions_of_Areas_Under_the_Normal_Curve

1. Identifying Regions of Areas Under the Normal Curve

2. Lesson Objectives At the end of this lesson, you are expected to: identify the regions of the areas under the normal curve; express the areas under the normal curve as probabilities or percentages; and determine the areas under the normal curve given z-values.

3. Lesson Introduction T he area under the curve is 1. So, we can make the correspondence between area and probability .

4. Pre-Assessment

5. Discussion Points Identifying Regions Under the Normal Curve z-table provides the proportion of the area (or probability or percentage) between any two specific values under the curve, regions under the curve can be described in terms of area. For example, the area of the region between z = 0 and z = 1 is given in the z-table to be .3413

6. Discussion Points T o find the area of the region between z = 1 and z = 2, we subtract .3413 from .4772 resulting in .1359. It is graphically shown below.

7. The area of the region between and is given in the z-table to be . So, the area of the region between and is given by  

8. Discussion Points T he regions under the normal curve in terms of percent, the graph of the distribution would look like this:

9. Discussion Points Understanding Proportions of Areas Under the Normal Curve

10. The total area between and is The total area between and The total area between and is or . The area of is and the area of is the total area between and is the sum of or  

11. The area of is and the area of is ; the total area between is the difference of or The area of is and the area of is ; the total area between and is the difference of or  

12. Discussion Points Using the z-Table in Determining Areas Under the Normal Curve when z is Given Step 1. Write the given z-value into a three-digit form. Step 2. Find the first two digits in r ow . Step 3. Locate the third digit in Column Step 4. Take the area value at the intersection of Row and Column.

13. Example 1 Find the area that corresponds to z =1.96.

14. Example 2 Find the area that corresponds to z =-1.15.

15. Exercise 1 Use the z -table to find the area that corresponds to each of the following: Note that area are between zero and the given z-score . 1. z=0.56 2. z=1.32 3. z = –1.05 4. z = –2.18 5. z = –2.58

16. Performance Task 3 Find the total area of the following: Between and Between Between Between Between Between Between Between  

17. Summary Properties of the Normal Probability Distribution The distribution curve is bell-shaped. The curve is symmetrical about its center. The mean, the median, and the mode coincide at the center. The width of the curve is determined by the standard deviation of the distribution. The tails of the curve flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve is asymptotic to the base line. The area under the curve is 1. Thus, it represents the probability or proportion or the percentage associated with specific sets of measurement values.

18. Summary Using the z-Table in Determining Areas Under the Normal Curve when z is Given Step 1. Write the given z-value into a three-digit form. Step 2. Find the first two digits in r ow . Step 3. Locate the third digit in Column Step 4. Take the area value at the intersection of Row and Column.