Today s Goals

1. Today�s Goals Calculate quantities using the normal distribution Find percentiles of normal distributions Homework #8 (due Wednesday April 1) : CH3:79, 110, 122, CH4: 5,13, 40 plus three web problems. Midterm Wednesday April 8 (esp lectures 10 � 23) Office hours this week: Today until 3:30. Tuesday 2:30 � 4:30

2. Newspaper articles Correlation versus causation �Win-win situation on immigration� In states with a higher percentage of HB-1 visas, both immigrants and non-immigrants have more patent applications. What is the implication of the headline? What is causing what?

3. Newspaper articles Correlation versus causation �Win-win situation on immigration� In states with a higher percentage of HB-1 visas, both immigrants and non-immigrants have more patent applications. What is the implication of the headline? What is causing what? Headline implies that having more immigrant scientists and engineers is good for science: immigrants are causing better science But, another possibility is that areas with active science communities are doing better economically and have more need to hire immigrants: better science leads to more immigrants.

4. Normal Probability Calculations Probability density function: Theoretical range of X is -8 to +8. The parameters of the Normal Distribution are the mean m and the standard deviation s.

5. Standard Normal The normal distribution with parameters 0 and 1 is called a standard normal distribution. A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by Z. The pdf of Z is:

7. Normal Probability Calculations: Standardization Then, we can use the standard tables

10. Standard Normal Tables (z-Tables)

12. Standard Normal Tables (z-Tables)

13. Reading the Table A.3 on page 740 The values of z are listed down the rows (up to first decimal digit) and across the top of the columns (second decimal digit). The probability that Z is less than or equal to z is listed within the appropriate row.

14. Standard Normal Tables (z-Tables)

15. Example Resistors made by a certain manufacturer have resistances that are normally distributed with a mean of 9.9 ohms and SD of 0.1ohms. If the specification limits are 10 � 0.2 ohms, what fraction of the resistors conform to the specification limits? X: resistance of a randomly selected resistor.

16. Solution Fraction conforming to specification limits: Thus 84% of the resistors conform to the specification limits and 16% do not.

17. Percentiles The 100pth percentile is identified by the row and column in which the entry p appears.

18. Percentile To find the (100p)th percentile, find the value z that has probability of p. Find the 30th percentile of the standard normal.

19. Percentile To find the (100p)th percentile, find the value z that has probability of p. Find the 30th percentile of the standard normal.

20. Percentile To find the (100p)th percentile, find the value z that has probability of p. Find the 30th percentile of the standard normal.

21. Percentiles The 100pth percentile is identified by the row and column in which the entry p appears. What is the 99th percentile of X~N(50,20)? We want the x such that p(X<x) = .99

22. Percentiles The 100pth percentile is identified by the row and column in which the entry p appears. What is the 99th percentile of X~N(50,202)? We want the x such that p(X<x) = .99 Translate to: Find 99th percentile of Z: 2.33 (x-50)/20 = 2.33 x = 2.33 * 20 + 50 = 96.6

23. Percentiles The 100pth percentile is identified by the row and column in which the entry p appears. What is the 99th percentile of X~N(50,202)? We want the x such that p(X<x) = .99 Translate to: Find 99th percentile of Z: 2.33 (x-50)/20 = 2.33 x = 2.33 * 20 + 50 = 96.6 In general: x(percentile) = z(percentile) * s + m

24. Percentiles The 99th percentile of Z is 2.33 p(Z < 2.33) = .99 What is the 1st percentile of Z?

25. Percentiles The 99th percentile of Z is 2.33 p(Z < 2.33) = .99 What is the 1st percentile of Z? What is the 1st percentile of X ~ N(50,202)?

26. Percentiles The 99th percentile of Z is 2.33 p(Z < 2.33) = .99 What is the 1st percentile of Z? -2.33 What is the 1st percentile of X ~ N(50,202)? x = -2.33*20+50 = 3.4 There is only a 1% chance that X is less than 3.4.

27. Example Test scores X ? N(? =500, ? =100). What does your score have to be to assure that you are among the top 10%? And to be among the top 5%? How well have you done in relation to the others if your score is 750?

28. Solution Let's calculate the corresponding percentiles of the distribution: Thus, you need at least a score of 628 to be among the top 10%.

29. Example Test scores X ? N(? =500, ? =100). What does your score have to be to assure that you are among the top 10%? And to be among the top 5%? How well have you done in relation to the others if your score is 750?

30. Solution Let's calculate the corresponding percentiles of the distribution: Thus, you need at least a score of 628 to be among the top 10% and at least 664.5 to be on the top 5%.

31. Example Test scores X ? N(? =500, ? =100). What does your score have to be to assure that you are among the top 10%? And to be among the top 5%? How well have you done in relation to the others if your score is 750?

32. Example Now, your score of 750 corresponds to a standard value 2.5 standard deviations above the mean. Then, P(X < 2.5)=0.938 ? you are among the 6.2% top students.