The Normal Distribution

Transcription

1 The Normal Distribution Lecture 20 Section Robb T. Koether Hampden-Sydney College Wed, Feb 17, 2009 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

2 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

3 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

4 Homework Review Exercise 6.3, page 376. Consider the following density function for the variable X: 2/9 Density 1/ X Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

5 Homework Review Exercise 6.3, page 376. (a) The median is (select one) (i) equal to 6. (ii) less than 6. (iii) more than 6. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

6 Homework Review Exercise 6.3, page 376. (a) The median is (select one) (i) equal to 6. (ii) less than 6. (iii) more than 6. The median is more than 6. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

7 Homework Review Exercise 6.3, page 376. (a) The median is (select one) (i) equal to 6. (ii) less than 6. (iii) more than 6. The median is more than 6. (b) Explain your answer to part (a) using one well-written sentence. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

8 Homework Review Exercise 6.3, page 376. (a) The median is (select one) (i) equal to 6. (ii) less than 6. (iii) more than 6. The median is more than 6. (b) Explain your answer to part (a) using one well-written sentence. It is clear from the graph that more than half the area is greater than 6, so the median must be greater than 6. (In fact, 6 is the 33rd percentile. That is because the right-hand rectangle has twice the area of the left-hand rectangle.) Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

9 Homework Review Solution (c) We are interested in learning about the proportion of units with values between 9 and 12. Explain in words or show on a graph what corresponds to this proportion; then calculate the proportion. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

10 Homework Review Solution (c) We are interested in learning about the proportion of units with values between 9 and 12. Explain in words or show on a graph what corresponds to this proportion; then calculate the proportion. The area under the graph and between 9 and 12 corresponds to the proportion. 2/9 Density 1/ X Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

11 Homework Review Solution (c) Using the formula A = 1 2bh for the area of a triangle, the right-hand triangle has area 1 2 (6) ( ) 2 = Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

12 Homework Review Solution (c) Using the formula A = 1 2bh for the area of a triangle, the right-hand triangle has area 1 2 (6) ( ) 2 = The triangle from 6 to 9 has area ( ) (3) 9 = 1 6. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

13 Homework Review Solution (c) Using the formula A = 1 2bh for the area of a triangle, the right-hand triangle has area 1 2 (6) ( ) 2 = The triangle from 6 to 9 has area ( ) (3) 9 = 1 6. Therefore, the area between 9 and 12 is = 1 2. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

14 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

15 The Rule The Rule For any normal distribution, Approximately 68% of the values lie within one standard deviation of the mean. Approximately 95% of the values lie within two standard deviations of the mean. Approximately 99.7% of the values lie within three standard deviations of the mean. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

16 The Empirical Rule The Empirical Rule For any mound-shaped distribution, Approximately 68% lie within one standard deviation of the mean. Approximately 95% lie within two standard deviations of the mean. Nearly all lie within three standard deviations of the mean. The well-known Empirical Rule is more general than the Rule. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

17 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

18 The Standard Normal Distribution Definition (Standard normal distribution) The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. It is denoted by the letter Z. That is, Z is N(0, 1). Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

19 The Standard Normal Distribution Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

20 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

21 Areas Under the Standard Normal Curve Easy questions: What is the total area under the curve? What proportion of values of Z will fall below 0? What proportion of values of Z will fall above 0? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

22 Areas Under the Standard Normal Curve What proportion of values will fall below +1? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

23 Areas Under the Standard Normal Curve It turns out that the area to the left of +1 is Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

24 Areas Under the Standard Normal Curve What is the area to the right of +1? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

25 Areas Under the Standard Normal Curve What is the area between 1 and 1? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

26 Areas Under the Standard Normal Curve What is the area between 0 and 1? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

27 Areas Under the Standard Normal Curve There are two methods to finding standard normal areas: The TI-83 function normalcdf. Standard normal table. We will use the TI-83 (unless you want to use the table). Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

28 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

29 TI-83 - Standard Normal Areas TI-83 Standard Normal Areas Press 2nd DISTR. Select normalcdf (Item #2). Enter the lower and upper bounds of the interval. If the interval is infinite to the left, enter E99 as the lower bound. If the interval is infinite to the right, enter E99 as the upper bound. Press ENTER. The area appears in the display. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

30 Standard Normal Areas Practice Use the TI-83 to find the following. The area between 1 and 1. The area to the right of 1. The area to the left of What standard normal percentile is 1.645? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

31 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

32 Other Normal Curves If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve? Use the same procedure as before, except enter the mean and standard deviation as the 3rd and 4th parameters of the normalcdf function. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

33 TI-83 - Area Under Normal Curves Example (TI-83 Normal Areas) Find area between 25 and 38 in the distribution N(30, 5). In the TI-83, enter normalcdf(25,38,30,5). Press ENTER. The answer appears. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

34 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

35 IQ Scores Read the article Understanding and Interpreting IQ. IQ scores are normalized to have a mean of 100 and a standard deviation of 15. Psychologists often assume a normal distribution of IQ scores as well. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

36 IQ Scores Practice What percentage of the population has an IQ above 120? above 140? What percentage of the population has an IQ between 75 and 125? Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

37 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

38 Assignment Homework Read Section 6.3.1, pages Let s Do It! 6.2, 6.3, 6.4, 6.5, 6.6 Exercises 4-9, 11, 12, 15, 16, 18, page 376. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

39 Outline 1 Homework Review 2 The Rule 3 The Standard Normal Distribution Standard Normal Areas TI-83 Standard Normal Areas 4 Areas under Other Normal Curves 5 IQ Scores 6 Assignment 7 Answers to Even-numbered Exercises Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

40 Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, A score at the mean. 6.6 µ = 5 and σ = 1. The Rule says that about 95% lie between µ 2σ and µ + 2σ, which is the interval from 1 to (a) Stats: 0.41; Logic: 0.70; Verbal: (b) Stats. (c) 65.82%. (d) 24.15%. (e) 2.83%. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

41 Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, (a) (b) Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

42 Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, The means are the same, but the standard deviation for men is larger than the standard deviation for women Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33

43 Answers to Even-numbered Exercises Page 376, Exercises 4, 6, 8, 12, 16, (a) (b) minutes. (c) minutes. Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Feb 17, / 33