1. Unit 2: Modeling Distributions of Data of Data

2. Homework Assignment For the A: 1, 3, 5, 9 - 23 Odd, 25 – 30, 33, 35, 39 – 59 Odd and 54, 63, 65 – 67, 69 – 74, R1 – R11 For the C: 1, 3, 5, 9, 11, 15, 19, 21, 23, 33, 35, 41, 45, 49, 53, 54, 57, 59, 63, 65, 67, R1 – R 11 For the D- : 1, 9, 11, 19, 21, 33, 35, 45, 49, 54, 57, 63, 67, R1 – R 11 All problems must be complete, including explanations with complete sentences and or work to show if the question asks for it. All Multiple Choice problems will be graded for correctness.

3. Density Curves Histograms cannot properly show the distribution of a continuous variable Density curves can! Area = probability!!

4. …continued Properties –Total area under the curve is 1 –Curve is above the x-axis Note: since a vertical line has no area, a density curve cannot tell us the probability of obtaining a single value

5. Some Examples

6. Density Curves Recap… Can be created by smoothing histograms ALWAYS on or above the horizontal axis Has an area of exactly one underneath it Describes the proportion of observations that fall within a range of values Is often a description of the overall distribution Uses  &  to represent the mean & standard deviation

7. Example 01 Is this a valid density curve?

8. Example 02 Is this a valid density curve?

9. Example 03 Is this a valid density curve?

10. Example 04 Is this a valid density curve?

11. The Normal Distribution Most important distribution of all Symmetric Two Parameters: –Mean (center) –Standard Deviation There is a trick for finding the standard deviation!

12. Self Check #8

13. Assignment #4

14. Most important distribution of all Symmetric Two Parameters: –Mean (center) –Standard Deviation There is a trick for finding the standard deviation! The Normal Distribution

15. The Points of Inflection As you trace your finger along the graph, something changes at the indicated points

16. …continued These are called Points of Inflection They are the spots where a graph changes from rising/falling steeply to rising/falling gently The distance from the center to the x- coordinate of a POI is the standard deviation More on this in Calculus!

17. Empirical Rule Approximately 68% of the observations are within 1  of  Approximately 95% of the observations are within 2  of  Approximately 99.7% of the observations are within 3  of  Can ONLY be used with normal curves!

18. Normal Curve recap… Bell-shaped, symmetrical curve Transition points between cupping upward & downward occur at  +  and  –  As the standard deviation increases, the curve flattens & spreads As the standard deviation decreases, the curve gets taller & thinner

19. Self Check #9

20. Multiple Choice Test #3

21. The Standard Normal μ=0; σ=1 Any normal distribution can be transformed into a Standard Normal by standardizing Get out your Z – Tables !!!

22. Using the Z-Table Units and tenths down the side Hundredths along the top Left area at the intersection

23. Questions What if you want area to the right? What if you want the area between two values of z? What if your z doesn’t appear on the chart? What if you are given the area, but want to find the value of the variable? Do you know how to use the calculator to bypass the table entirely?

24. What do these z scores mean? -2.3 1.8 6.1 -4.3 2.3  below the mean 1.8  above the mean 6.1  above the mean 4.3  below the mean

25. Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with  = 45 and  = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations below the mean. Jonathan scores 35 on this test. Does he get the job? No, he scored 2.78 SD below the mean

26. Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of 2.8. Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best? She did better on test A.

27. Example 05 Find the probability that the variable takes a value greater than 1.

28. Example 06 Find the probability that the variable takes a value between 1.4 and 2.

29. Self Check #10

30. z score recap… Standardized score Creates the standard normal density curve Has  = 0 &  = 1

31. Calculator Activity

32. The height of male students at BHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. a) What percent of the male students are shorter than 66 inches? b) Taller than 73.5 inches? c) Between 66 & 73.5 inches? About 2.5% About 16% About 81.5%

33. Example 07 In 2001, the reading scores on the ACT were approximately normally distributed with mean 21.3 and standard deviation 6. What proportion of reading scores were less than 9.3? What proportion of reading scores were between 15.3 and 27.3?

34. Example 08 Final grades in a college class are (historically) normally distributed with mean 72 and standard deviation 12.5. What proportion of final grades are at least 97? What proportion of final grades are between 34.5 and 59.5?

35. Example 09 In 2002, salaries for human resource clerks were approximately normally distributed with mean $27,937 and standard deviation $1700. What percentage of salaries were greater than $30,000? What percentage of salaries were between $20,000 and $25,000?

36. Self Check #11

37. Finding a value given a proportion We may instead want to find the observed value with a given proportion of the observations above or below it. To do this, use Standard Normal (Z) Table backward. Find the given proportion in the body of the table, read the corresponding z from the left column and top row, then “un-standardize” to get the observed value.

38. Example 10 The amount of soda in particular size bottle varies normally with mean 32 oz and standard deviation 0.55 oz. What percentage of bottles contain less than 31 oz? The lowest 5% of volumes are less than _____. The greatest 1% of volumes are more than ____.

39. Calculator Activity

40. Interpreting Center Activity

41. Self Check #12

42. Assignment #5

43. Assessing Normality We need to develop methods for checking if the population is normal. One method we already know: a.create a histogram or stemplots. b.Look for non-normal features such as outliers, skewness, gaps, or clusters

44. Normal Probability Plot A normal probability plot provides a good assessment of the adequacy of the normal model for a set of data. Any normal distribution produces a straight line on the plot because standardizing is a transformation that can change the slope and intercept of the line in our plot but cannot change a line into a curved pattern.

45. Data Sampled From a Normal Distribution Notice that the normal probability plot (NPP) is basically straight. That's the idea: Normal data = straight NPP. So, when the NPP is straight you have evidence that the data is sampled from a normal distribution.

46. Data Sampled From a Right Skewed Distribution For right skewed data, the normal probability plot is generally not straight. In general this sort of curvature in the NPP implies right skew.

47. Data Sampled From a Left Skewed Distribution For left skewed data, the normal probability plot is generally not straight. In general this sort of curvature in the NPP implies left skew.

48. Normal Probability Plot Recap As you progress in the course normal probability plots will become clearer. At this point you should be able to recognize whether the sample data appears to be from a population that is normal based on the normal probability plot.

49. Self Check #13

50. Multiple Choice Test #4