STATISTICS-AND-PROBABILITY-WEEK-9-10.pptx

Stat and prob week 9-10

1. STATISTICS AND PROBABILITY ENGR. JERIC M. MIRANDILLA

2. TOPIC 1 Identifying Independent and Dependent Variables

3. Regression analysis Regression analysis is a statistical treatment of data which involves identifying the relationship between a dependent variable and one or more independent variables. 3

4. Regression analysis Regression analysis is used to: 1. determine the strength of the predictors, that is, identifying the strength of the effect that the independent variable(s) have on a dependent variable. 2. forecast effects or impact of changes, that is, understanding how much the dependent variable changes with a change in one or more independent variables; and 3. predict trends and future values, that is, getting a point estimates 4

5. Linear regression Linear regression estimates are used to explain the relationship between one dependent variable and one or more independent variables. The simplest form of linear regression is called simple linear regression. It is a linear regression model with two-dimensional sample points, one dependent variable and one independent variable. 5

6. VARIABLES An independent variable is a variable that is hypothesized to have an impact on the dependent variable, can be manipulated or changed, and usually denoted by X. The dependent variable is a variable that is being tested, its value relies or depends on the value of the independent variable, and usually denoted by Y. 6

7. VARIABLES A teacher wants to know the effect of attendance on the academic performance of the students. the independent variable is the attendance of the student. The teacher can manipulate the length of time and the students that will participate in the experiment; and the dependent variable is the academic performance. The students’ academic performance can be affected by their attendance. 7

8. VARIABLES 2. A scientist conducts an experiment to test that vitamin C could improve a person’s immune system. the independent variable is the in-take of vitamin C. The scientist can control the timing and the dosage; and the dependent variable is the improved immune system. The person’s immune system can be affected by their in-take of vitamin C. 8

9. TOPIC 2 Slope and Y-intercept of the Regression Line

10. RECAP Slope-Intercept Form of a Line The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. Slope shows the steepness of a straight line, while y-intercept is a point where the line touches the y-axis. 10

11. SIMPLE LINEAR REGRESSION Take note that correlation analysis should be done first, then test the significance of r, before attempting to fit a linear model to observed data. If it happens to be no association between the independent and dependent variables, then fitting a linear regression model to the data will not provide a useful model. 11

12. SIMPLE LINEAR REGRESSION A simple linear regression line has an equation of the form: = bX + a , where X is the independent variable and is the dependent variable. The slope of the regression line is b, and the y-intercept is a, y-intercept is the value of y when x is 0. Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data.   12

13. SIMPLE LINEAR REGRESSION To calculate the value of a and b, we need to find the values of the summations indicated in the formula. 13

14. SIMPLE LINEAR REGRESSION To interpret the slope and y-intercept of the regression line: the slope tells how much Y changes as X changes. It’s a ratio of change in Y per change in X. y-intercept is a point where the regression line crosses the y-axis at x = 0. 14

15. EXAMPLES Five randomly selected students were surveyed about their Statistics 1st quarter test score and their 1st quarter grade in Statistics. Assuming that there is a significant relationship between the two variables, determine the slope and y-intercept of the regression line. Then, interpret the result. 15 This Photo by Unknown Author is licensed under CC BY-SA

16. EXAMPLES Step 1. Identify the dependent and independent variable. The dependent variable is the 1st quarter grade in Statistics and the independent variable is the 1st quarter test score in Statistics. 16 This Photo by Unknown Author is licensed under CC BY-SA

17. EXAMPLES 17 This Photo by Unknown Author is licensed under CC BY-SA

18. EXAMPLES Step 3. Calculate the value of a and b in the formula, substitute the summations found in step 2 and the sample size n given in the problem, which is 5 students, thus, n=5. 18 This Photo by Unknown Author is licensed under CC BY-SA

19. EXAMPLES Step 4. Interpret the result. The slope of the regression line is 0.82, which indicates that for every grade of 0.82, there corresponds a score of 1 in Statistics. The y-intercept of the regression line is 53.47, which indicates that for a test score of 0, there will be an average grade of 53.47 in Statistics. 19 This Photo by Unknown Author is licensed under CC BY-SA

20. TOPIC 3 Regression Line Equation

21. LINEAR REGRESSION Linear regression quantifies the relationship between one or more predictor variables and one outcome variable. It can be used to quantify the relative impacts of age, gender, and diet (the predictor variables) on height (the outcome variable). Y is the outcome or dependent variable whereas X is the predictor or independent variable. 21

22. LINEAR REGRESSION If the average Y distances of the points from this line is the least, then we call this line the regression line or the line that “best fit” in the scatterplot. The regression line is the same as the trend line. The regression line is the same as the point-slope form equation of a line in algebra. The regression line is = bX + a where b is the slope of the line and a is the y-intercept.   22

23. LINEAR REGRESSION If the average Y distances of the points from this line is the least, then we call this line the regression line or the line that “best fit” in the scatterplot. The regression line is the same as the trend line. The regression line is the same as the point-slope form equation of a line in algebra. The regression line is = bX + a where b is the slope of the line and a is the y-intercept.   23

24. EXAMPLES In the regression line, Y’= 4X + 6 predict Y’ if the given value of X=4 Solution: Step 1: Copy the linear equation Y = 4X + 6 Step 2: Substitute the given value of X = 4 in the equation Y = 4(4) + 6 24 This Photo by Unknown Author is licensed under CC BY-SA

25. EXAMPLES Step 3: Solve for Y, evaluate Y= 16+6 Y = 22 Therefore, the predicted value of Y is 22 when X = 4. 25 This Photo by Unknown Author is licensed under CC BY-SA

26. EXAMPLES In the regression line, Y = 2X - 4 predict Y if the given value of X = 3. Solution: Y = 2X - 4 Y = 2(3) – 4 Y = 6 - 4 Y = 2 Therefore, the predicted value of Y is 2 when X = 3. 26 This Photo by Unknown Author is licensed under CC BY-SA

27. TOPIC 4 Problem Solving Involving Regression Analysis

28. TESTING THE SIGNIFICANCE OF r The relationship or correlation must be significant. This means that the actual relationship exists in the population, not just in the sample. The regression analysis is then used to predict the value of one variable in terms of the other variable. Thus, we do correlation analysis first before performing regression analysis. 28

29. TESTING THE SIGNIFICANCE OF r To solve for the correlation coefficient (r) The formula for t: where: df = n – 2 29

30. regression analysis 1. Find the value of the correlation coefficient (r) 2. Test the significance of r. If r is significant, proceed to regression analysis (Proceed to Step 3). If r is not significant , regression analysis cannot be done (Stop) 3. Find the values of a and b. 4. Substitute the values of a and b in the regression line Y = bX + a. 30

31. regression analysis STEPS IN TESTING THE SIGNIFICANCE OF r a. State the null and alternative hypothesis b. Compute for the value of t c. Compare the computed value of t with the critical value of t. 31

32. EXAMPLES A researcher would like to know if IQ scores are related to age. Using 10 high school students, he found out that the computed r is 0.58. At 0.05 level of significance, can he conclude that the relationship really exists in the population? 32 This Photo by Unknown Author is licensed under CC BY-SA

33. EXAMPLES 1. State the null and alternative hypotheses Ho: There is no significant relationship between IQ scores and age (r = 0) Ha : There is a significant correlation between IQ scores and age (r ≠ 0) 33 This Photo by Unknown Author is licensed under CC BY-SA

34. EXAMPLES 2. Compute for the value of t: 34 This Photo by Unknown Author is licensed under CC BY-SA

35. EXAMPLES 3. Compare the computed value of t with the critical value of t: Using df = n – 2 =10 – 2 = 8, a = 0.05, two-tailed test, we get from the table of t-values that the critical value of t is 2.306. 35 This Photo by Unknown Author is licensed under CC BY-SA

36. EXAMPLES 4. Make a decision. Since the computed value of t = 2.01 is less than the critical value of t which is 2.306, we accept the null hypothesis. So, we say that there is no significant relationship between IQ scores and age. 36 This Photo by Unknown Author is licensed under CC BY-SA

37. EXAMPLES 5. Summarize the results. We conclude that the relationship between IQ scores and age does not really exist in the population. Thus, regression analysis should not be performed since the test of significance of r yields no significant result. 37 This Photo by Unknown Author is licensed under CC BY-SA

38. EXAMPLES The following data pertains to the heights of fathers and their eldest sons in inches. Is there a significant relationship between the two variables, predict the height of the son if the height of his father is 78 inches. 38 This Photo by Unknown Author is licensed under CC BY-SA

39. EXAMPLES 1. Compute the correlation coefficient (r) using the formula 39 This Photo by Unknown Author is licensed under CC BY-SA

40. EXAMPLES 1. Compute the correlation coefficient (r) using the formula 40 This Photo by Unknown Author is licensed under CC BY-SA

41. EXAMPLES 2. State the null and alternative hypotheses Ho: There is no significant relationship between the number of height of the father and height of the son. (r = 0) Ha: There is a significant relationship between the two variables. (r ≠ 0) 41 This Photo by Unknown Author is licensed under CC BY-SA

42. EXAMPLES 3. Test the significance of r using the formula: 42 This Photo by Unknown Author is licensed under CC BY-SA

43. EXAMPLES 4. Compare the computed t-value to the critical t-value Solution: Using df = n – 2 =10 – 2 = 8, a=0.05, two-tailed test, we find from the table that the critical value of t is 2.306 43 This Photo by Unknown Author is licensed under CC BY-SA

44. EXAMPLES 5. Make a decision Solution: Since the computed t = 8.61 is greater than the critical t = 2.306, we reject the null hypothesis. So, there is a significant relationship between the two variables. 44 This Photo by Unknown Author is licensed under CC BY-SA

45. EXAMPLES 6. Summarize the results Solution: There is a sufficient evidence to conclude that there is a significant relationship between number of height of the father and height of the son. Thus, we will proceed to regression analysis . 45 This Photo by Unknown Author is licensed under CC BY-SA

46. EXAMPLES 7. Compute the values of a and b in the regression equation Y = bX + a using the following formula: 46 This Photo by Unknown Author is licensed under CC BY-SA

47. EXAMPLES 8. Form the regression equation. Solution: Substitute the values of a and b in the equation Y = bX + a Y = 0.78X + 16.55 47 This Photo by Unknown Author is licensed under CC BY-SA

48. EXAMPLES 9. Predict the height of the son if the height of the father is 78 inches. Solution: Find the value of Y when X=78 in the regression equation. Y = 0.78X + 16.55 Y = 0.78(78) + 16.55 Y = 77.39 inches So, the predicted height of the son whose father is 78 inches is 77.39 inches. 48 This Photo by Unknown Author is licensed under CC BY-SA