2. Extensive study

7. I assume the mean age of this class is not 23.5!

10. Ho: u=80, but the value of u is really equal to 82 score 80 82

13. Decision-making always exists risk

14. Prob. of sample ? 2.5% Prob.=Significance level

20. Judgment in a relaxed manner

25. Z a/2 z

26. Z p/2 a/2 z

38. Critical limit

39. When n≥30, t distribution is almost equal to standard normal distribution. So we also use z-test.

40. s

42. EXCEL

43. Introduction to analysis of variance Suppose you are interested in determining whether certain situations produce differing amounts of stress. You know the amount of the hormone corticosterone (HC) circulating in the blood is a good measure of how stressed a person is. You randomly assign 15 students into three groups of 5 each. The students in group 1 have their HC levels measured immediately after returning from vacations (low stress). The students in group 2 have

44. their HC levels measured after they have been in class for a week (moderate stress). The students in group 3 are measured immediately before final exam week (high stress). All measurements are taken at the same time of day. You record the data shown in Table 7.1. Scores are in milligrams of HC per 100 milliliters of blood. Null hypothesis: the different situations affect stress equally. (H 0 : u 1 =u 2 =u 3 ) Can we develop a technique to solve the problem easily? ----F-test

45. Table 7.1----sample data group1 group2 group3 X 1 X 2 X 3 2 10 10 3 8 13 7 7 14 2 5 13 6 10 15 4 8 13

46. The distribution of statistic F F fundamentally is the ratio of two independent variance estimates of the same population variance σ 2 Like the t distribution, F distribution varies with degrees of freedom. However, F distribution has two values for degrees of freedom, one for the numerator and the other for the denominator. df 1 =n 1 -1 df 2 =n 2 -1

47.  F (n 1 -1, n 2 -1) 0 F 1 The distribution of statistic F

48. The analysis of variance ------ANOVA It is a statistical technique employed to analyze multigroup experiments. Using the F-test allows us to make one overall comparison that tells whether there is a significant difference between the means of the groups. If the groups are divided by a factor (for example: situation) to be investigated in the experiment, then the analysis of variance is called as one-way’s.

49. The assumption of ANOVA The population of each group is of normal distribution The population of each group is of equal variance, σ 1 2 = σ 2 2 = …… = σ k 2 = σ 2 The sampling from different group is of independent

50. SST SSW and SSB

52. Solution of Table 7.1 K=3, n=15, n j =5 SSB=203.333 SSW=54 df B =k-1=2 df W =n-k=12 SB 2 =SSB/df B =203.333/2=101.667 SW 2 =SSW/df W =54/12=4.5 F=SB 2 /SW 2 =22.59 F crit =3.88 (α=0.05) Therefore, H 0 is rejected  F F  F  F

53. Multiple comparisons We concluded that the three situations were not the same in the stress levels they produced. This means that at least one condition differs from at least one of the other. To determine which conditions differ, multiple comparisons between pairs of group means are usually made. Here we shall discuss a priori comparisons

54. A priori comparisons Definition of statistic t

55. Solution of Table 7.1(α=0.05)

56. Two-way analysis of variance The two-way analysis of variance allows us in one experiment to evaluate the effect of two independent variable (factors) and the interaction between them suppose a professor in physical education conducts an experiment to compare the effects on nighttime sleep of different intensities of exercise and the time of day when the exercise is done. For this example, let’s assume that there are two levels of exercise (light and heavy) and two times of

57. day (morning and evening). The experiment is shown diagrammatically in Figure 7.1. Figure 7.1 Factor B, exercise intensity b 1,lightb 2,heavy Factor A, time of day a 1,morning a 1 b 1 ; sleep scores of subjects who do light exercise in the morning a 1 b 2 ; sleep scores of subjects who do heavy exercise in the morning a 2,evening a 2 b 1 ; sleep scores of subjects who do light exercise in the evening a 2 b 2 ; sleep scores of subjects who do heavy exercise in the evening

58. Here there are two factors and each factor has two levels. Thus, the design is referred to as a 2×2 design. For example, if factor A has three levels, then the experiment would be called a 3×2 design. There are three analyses done in this design: First, we want to determine whether factor A has a significant effect on sleep, disregarding the effect of factor B. Second, we want to determine whether factor B has a significant effect on sleep, without considering the effect of factor A. Finally, we want to determine whether there is an interaction between factors A and B in their effect on sleep.

59. Intensity of exercise lightheavy No significant time of day or intensity of exercise effects sleep morning evening

60. Intensity of exercise lightheavy sleep morning evening Significant time of day effect; no other effect Intensity of exercise lightheavy sleep morning evening Significant time of day effect; no other effect

61. lightheavy Intensity of exercise sleep morning evening Significant intensity of exercise effect; no other effect lightheavy

62. lightheavy Intensity of exercise sleep morning evening Significant intensity of exercise and time of day effects; no interaction effect lightheavy

63. lightheavy Significant interaction effect; no other effect Intensity of exercise sleep morning evening lightheavy

64. lightheavy Intensity of exercise sleep morning evening Significant intensity of exercise and time of day effects; no interaction effect lightheavy Intensity of exercise lightheavy

65. B ∑ b1b1 b2b2 bcbc A a1a1 cell 11 cell 12 …cell 1c ∑ row1 X a2a2 cell 21 cell 22 …cell 2c ∑ row2 X …………… arar cell r1 cell r2 …cell rc ∑ rowr X ∑∑ col1 X∑ col2 X∑ colc X

66. SSW SSR SSC and SSRC Not affected by A or B Where: p= the number of data in each cell

67. SSW SSR SSC and SSRC Only affected by A

68. SSW SSR SSC and SSRC Only affected by B

69. SSW SSR SSC and SSRC Only affected by interaction

71. Data lightmoderateheavy Morn- ing 6.5 7.4 7.3 7.2 6.6 6.8 7.4 7.3 6.8 7.6 6.7 7.4 8.0 7.6 7.7 6.6 7.1 7.2 ∑x=129.20 ∑x 2 =930.50 n=18 =6.97 =7.20=7.37=7.18 Even- ing 7.1 7.2 7.6 7.5 8.2 7.6 7.4 8.0 8.1 7.6 8.2 8.0 8.2 8.7 8.5 9.6 9.5 9.4 ∑x=147.20 ∑x 2 =1212.68 n=18 =7.67=7.88=8.98=8.18 ∑x=90.50 ∑x 2 =645.30 n=12 =7.32 ∑x=90.50 ∑x 2 =685.07 n=12 =7.54 ∑x=98.10 ∑x 2 =812.81 n=12 =8.18 ∑x=276.4 ∑x 2 =2143.18 N=36

75. 1.For the row effect F=SR 2 /SW 2 =9.000/0.186=48.42>F crit (0.05,1,30)=4.17 We reject H 0 with respect to the A variable, which in this experiment is time of day. 2.For the column effect F=SC 2 /SW 2 =2.377/0.186=12.78>F crit (0.05,2,30)=3.32 We reject H 0 with respect to the B variable, which in this experiment is amount of exercise. 3.For the row×column interaction effect F=SRC 2 /SW 2 =0.856/0.186=4.60>F crit (0.05,2,30)=3.32 We reject H 0 regarding the interaction of variables A and B. Homework: S402 12.11, 12.12, 12.22, 12.36 EXCEL