1. Week 2 Normal Distributions, Scatter Plots, Regression and Random

2. The Normal Model Density Curves and Normal Distributions

3. A Density Curve: Is always on or above the x axis Has an area of exactly 1 between the curve and the x axis Describes the overall pattern of a distribution The area under the curve above any range of values is the proportion of all the observations that fall in that range.

4. Mean vs Median The median of a density curve is the equal area point that divides the area under the curve in half The mean of a density function is the center of mass, the point where curve would balance if it were made of solid material

5. Normal Curves Bell shaped, Symmetric,Single-peaked Mean = µ Standard deviation = Notation N(µ, ) One standard deviation on either side of µ is the inflection points of the curve

6. 68-95-99.7 Rule 68% of the data in a normal curve at least is within one standard deviation of the mean 95% of the data in a normal curve at least is within two standard deviations of the mean 99.7% of the data in a normal curve at least is within three standard deviations of the mean

7. Why are Normal Distributions Important? Good descriptions for many distributions of real data Good approximation to the results of many chance outcomes Many statistical inference procedures are based on normal distributions work well for other roughly symmetric distributions

8. Standardizing

9. Standardizing (z-score) We standardize to compare items from more than one distribution. (Apples and Oranges) A z score is the number of standard deviations off center (mean) a data item is.

10. To Find a z score Subtract the mean from the item. Divide the result by the standard deviation

11. Standard Normal Curve

12. Standard Normal Distribution A normal distribution with µ = 0 and  = 1, N(0,1) is called a Standard Normal distribution Z-scores are standard normal where z=(x-µ)/

13. Standard Normal Tables Table B (pg 552) in your book gives the percent of the data to the left of the z value. Or in your Standard Normal table Find the 1st 2 digits of the z value in the left column and move over to the column of the third digit and read off the area. To find the cut-off point given the area, find the closest value to the area ‘inside’ the chart. The row gives the first 2 digits and the column give the last digit

14. Solving a Normal Proportion State the problem in terms of a variable (say x) in the context of the problem Draw a picture and locate the required area Standardize the variable using z =(x-µ)/ Use the calculator/table and the fact that the total area under the curve = 1 to find the desired area. Answer the question.

15. Finding a Cutoff Given the Area State the problem in terms of a variable (say x) and area Draw a picture and shade the area Use the table to find the z value with the desired area Go z standard deviations from the mean in the correct direction. Answer the question.

16. Assessing Normality In order to use the previous techniques the population must be normal To assessing normality :  Construct a stem plot or histogram and see if the curve is unimodal and roughly symmetric around the mean

17. Normal Probability Calculator http://www.math.hope.edu/swanson/statla bs/stat_applets.html

18. Scatterplots, Association and Relationships How are two quantitative variables related?

19. Variables Response Variable: measures the outcome of a study (y variable, dependent variable, the “result”) Explanatory or Predictor Variable attempts to explain the observed outcomes (x variable, independent variable, the “cause”)

20. Scatterplot Shows the relationship between two variables measured on the same individual. Graph the explanatory variable on the horizontal axis. (x list) Graph the response variable on the vertical axis (y list)

21. Interpreting Scatterplots Overall pattern  Direction (increasing or decreasing ?)  Form (linear, exponential?)  Strength of relationship  The width of the hallway Outliers and Influencial Points  What does not fit the pattern  Falls outside the usual values of either variable

22. Direction (Think Slope) Two variables are positively associated when the above average values of one are associated with the above average values of the other. Two variables are negatively associated when the above average values of one are associated with the below average values of the other

23. Form Form is the general shape of the dots in the scatterplot Linear, exponential, logarithmic,... Correlation is ONLY relevant with linear data. “Curved” data must be “straightened” before we can use correlation

24. Strength How much is the data lined up. The closer to a straight line the stronger the relationship Correlation is a measurement of the strength of the relationship between predictor and response variables

25. Correlation How strong is the relationship

26. Correlation The average of the product of standardized x’s and y’s NO! We will never use this formula!!!!! Thank you, Technology!!!

27. Correlation Facts It makes no difference which variable is the x and which is the y Positive r indicates a positive association between the variables and negative r indicates a negative association -1 ≤ r ≤ 1 Values near zero indicate a weak association. Values near 1 or -1 indicate strong association

28. Correlation facts 2 There are no units on r so it is immune to changes when units change. Correlation measures the strength of ONLY linear relationships. Do not use correlation describe curved relationships Correlation is greatly affected by extreme values

29. Correlation Facts 3 Outliers can make a strong relationship look weak Outliers can make a weak relationship look strong Report the correlation both with and without the outlier(s)

30. Conditions for Correlation The quantitative condition: Both variables must be quantitative. The linear condition: The form of the relationship must be linear. The outlier condition: Outliers greatly affect correlation. Report the r value both with and without the outliers factored in

31. Linear Regression Can we predict a result from our data?

32. Modeling quantitative variable relationships We want to be able to predict one quantitative variable if we are given the other. We will use a line as our model This line is called the line of best fit or the Least Squares Regression Line(LSRL)

33. Theory of LRSL If we draw a line through our data not all of the data points lie on the line. So, there is some error in our prediction model. It follows that we want the best line and the best line is the one with the least error

34. Theory page 2 It follows then that we need a measure of the error. We will define our error to be the observed value (point) minus the predicted value (line) and call this error the residual Residual = observed - predicted

35. Theory page 3 The “best” line would be the one with the smallest total of the residuals Problem: The residuals can be both positive (line too low) or negative (line too high) so the best line would have a total of zero no matter how good the model was. Solution: Square the residuals before totaling

36. Theory page 4 So, if we try a graphic representation we are placing square between each data point and the prediction line. Big Finish So, the line that is the best fit to the data is the line with the least total area of the squares. Hence the name Least Squares Regression Line

37. Finding the LSRL The LSRL always goes through the point (x-bar,y-bar), the average of the x’s and the average of the y’s) Move one standard deviation (Sx) in the x direction moves us r standard deviations (Sy) in the y direction.

38. LSRL To write the equation of a line we need the slope and a point. Now use your Algebra I skills to create the line

39. Interpreting the LSRL Example: Price = -500(age) + 2000 Slope = -500 tells us that if age increases by one year the price decreases by 500 dollars 2000 tells us that if the age was zero (new) the price was 2000

40. Interpreting the LSRL Example2: Verbal =.9(Math) + 200 Slope =.9 tells us that if the math score increases by one point the verbal score increases by.9 of a point 200 tells us that if the math score was zero the verbal score would be 2000

41. R squared r 2 is the Coefficient of Determination It is the percent of the variation in the y values that can be explained by the difference in the x values in the prediction line An r 2 =.7 means that 70% of the change in the y value can be attributed to the change in the x value in the LSRL

42. Assumptions In order for a LSRL to be valid there are three conditions:  Both Quantitative variables  The scatterplot is “straight enough”  Outliers can have a large effect

43. Regression tool http://science.kennesaw.edu/~plaval/tools/ regression.html

44. Understanding Randomness

45. Random An event is random if: it is impossible to predict what the next result of the event will be. Results are independent. Events that are random are ones where all sequences of results of the event are equally likely to occur. Random does NOT mean haphazard.

46. Randomness is Important The use of randomness is vital to the modern study of statistics. Without which we can not do many of the techniques of modern statistical analysis.

47. Sources of Random Numbers Random number tables. Table A Page 550 in the back of our book. Computer Software Calculator: RandInt (low, high, HowMany)