9-Locating-Percentiles-under-the-Curve.pptx

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1. LOCATING PERCENTILES UNDER THE NORMAL CURVE DOMINIC DALTON L. CALING Statistics and Probability | Grade 11

2. LESSON OBJECTIVES At the end of this lesson, you are expected to: find z-scores when probabilities are given; and locate percentiles under the normal curve.

3. Check your readiness for this lesson by doing the following activities. Pre-Assessment

4. Lesson Introduction Which of the following expressions are familiar to you? ‘First honor’ ‘Top five’ ‘a score of 98%’ T hese are expressions of order. They indicate relative standing. In real life, many people want to belong to a high level in terms of relative standing.

5. Discussion Points Percentile For any set of measurements (arranged in ascending or descending order), a percentile (or a centile) is a point in the distribution such that a given number of cases is below it. A percentile is a measure of relative standing. It is a descriptive measure of the relationship of a measurement to the rest of the data.

6. Discussion Points Percentile and z -scores A probability value corresponds to an area under the normal curve. In the Table of Areas Under the Normal Curve, the numbers in the extreme left and across the top are z-scores, which are the distances along the horizontal scale. The numbers in the body of the table are areas or probabilities. The z-scores to the left of the mean are negative values.

7. Discussion Points Illustrative Example Find the 95 th percentile of a normal curve. Analysis P 95 means locating an area before (or below) the point. We want to know what the z -value is at this point.

8. Discussion Points STEPS SOLUTION Draw the appropriate normal curve. Express the given percentage as probability. 95% is the same as 0.9500 Split 0.9500 into 0.5000 and 0.4500. 0.9500 = 0.5000 + 0.4500 Shade .5000 of the sketch of the normal curve in Step 1. The proportion of the area above is .8413. Refer to the Table of Areas Under the Normal Curve. Locate the area 0.4500 in the body of the table. This area is not found in the table. It is between the values of 0.4495 and 0.4505. Find the z-score that corresponds to 0.4500 on the leftmost column. Find z by interpolation, as follows.

9. Discussion Points STEPS SOLUTION Find the z-value that corresponds to 0.4505 0.4505 ↔ z = 1.65 Find the z-value that corresponds to 0.4495. 0.4495 ↔ z = 1.64 Find the average of the two z-values. z = 1.645 Locate z = 1.645 under the curve in Step 1 and make a statement. The 95th percentile is z = 1.645. Draw a line through under the curve in Step 1. Do this under the sketch of the curve in Step 1. Shade the region to the left of z = 1.645. Do this under the sketch of the curve in Step 1. Describe the shaded region. The shaded region is 95% of the distribution.

10. Example: Find the upper 10% of the normal curve. STEPS SOLUTION Draw the appropriate normal curve. Express the given percentage as probability. 10% is the same as 0.1000 With respect to the mean, locate the upper 10%. To the right of the mean Using the upper side of the mean, find the remaining area. 0.5000 – 0.1000 = 0.4000 Refer to the Table of Areas Under the Normal Curve. Locate the area 0.4 00 in the body of the table. This area is not found in the body of the table, so we take 0.3997, which is the closest value. Find the z-score that corresponds to 0. 3997 on the leftmost column. z = 1.28.

11. STEPS SOLUTION Locate z=1.28 under the sketch of the curve in Step 1 and make a statement. That is, the upper 10% is above z = 1.28. Draw a line through under the curve in Step 1. Do this under the sketch of the curve in Step 1. Shade the region to the left of z = 1.645. Do this under the sketch of the curve in Step 1. Describe the shaded region. The shaded region is the upper 10% of the normal curve. Solution to Example (continuation)

12. EXERCISES Find each of the following percentile points under the normal curve. 1. P 99 2. P 90 3. P 68 4. P 40 5. P 32

13. Summary A percentile is a measure of relative standing. It is a descriptive measure of the relationship of a measurement to the rest of the data. In the Table of Areas Under the Normal Curve, the numbers in the extreme left and across the top are z-scores, which are the distances along the horizontal scale. The numbers in the body of the table are areas or probabilities . The z-scores to the left of the mean are negative values.

14. THANK YOU!