1. TMA 4255 Applied statistics
Spring 2010
Part 2

2. One-Way ANOVA

3. General setup – One-Way ANOVA

5. MINITAB analysis of Table 3.1

15. (MINITAB analysis of Table 13.1)

16. (Used in MINITAB)

17. 13.9 Randomized Complete Block Designs

20. Model with interactions between treatment and block:

23. Degrees of freedom:

27. Example 9
It is assumed that the maximal voltage of a storing battery depends on the
material used in the plates of the battery, and of the temperature in the room
where the battery is stored. To investigate this one takes four replicates for
each of three materials and three temperatures.

31. 13.12 Random Effects Model

33. s

35. MINITAB (Example 13.8)

37. Control Charts

41. Control Charts for Defects (Poisson)

44. Chi-square testing of model
H0: Data are from N(75,10^2)
n=60
Expected
Observed
Chi-square statistic: 19.1
Distribution under H0: Chi-Sq with df=7-1=6. Crit.value 5%: 12.59
P-value: P(Chi-Sq6 > 19.1) = 0.004

45. H0: Data are normally distributed
Estimated mean: 71.4, estimated st.dev. 10.7
Expected
Observed
Chi-square statistic: 6.6
Distribution under H0: Chi-Sq with df=7-1-2=4. Crit.value 5%: 9.49
P-value: P(Chi-Sq4 > 6.6) = 0.16

46. 10.15 Test for independence Eye color: 1=blue, 2=brown Hair color: light blond, dark blond, dark MINITAB:
Chi-Square Test: l.blond; d.blond; dark
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
l.blond d.blond dark Total
23, ,40 10,20
1, ,400 1,004
15, ,60 6,80
2, ,600 1,506
Total
Chi-Sq = 6,860; DF = 2; P-Value = 0,032

47. 10.16 Test for homogeneity Blood types: A, B, AB, O Populations: 1, 2, 3 MINITAB:
Chi-Square Test: A; B; AB; O
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
A B AB O Total
192,00 37,14 16, ,29
1,333 0,401 0, ,612
96,00 18,57 8, ,14
2,667 0,356 0, ,610
48,00 9,29 4, ,57
0,000 0,178 0, ,053
Total
Chi-Sq = 8,200; DF = 6; P-Value = 0,224
1 cells with expected counts less than 5.

48. From Gunnar Løvås: Statistikk for universiteter og høgskoler
From Gunnar Løvås: Statistikk for universiteter og høgskoler. Universitetsforlaget.
Simpson’s paradox
Sometimes the results of a statistical investigation may be misleading if
an important variable is overlooked or ignored. This is called Simpson’s
Paradox. Here is an example:
Example 258: A questionnaire about car damage has been sent to car
Drivers. The results could have been reported like this:
Damage No damage Total
Man
Woman
Total
Thus: A proportion 233/556 = 0.42 had car damage, while for women
this was 87/281 = 0.31.
Does this mean that women are better drivers than men?

49. Solution: A further study of the questionnaire reveal that a further categorization
can be made, namely the size of the cars. Then we could present the results as:
Big cars Small cars
Damage No damage Total Damage No damage Total
Man
Woman
Total
Now things have changed:
Big cars: 100*16/18 = 88% of women have damage, and 100*150/185=81% for men.
Small cars: 100*71/263= 27% for women; 100*83/371 = 22% for men
So women have more damage for both small and big cars!
How does this relate to the result on the previous slide?

52. Nonparametric statistics: Sign test
Example 1: Ratio height/length for 8 rectangular leather items found
at Shoshoni indians.
Will test H0: Golden ratio vs. H1: Not golden ratio, i.e.
H0: median = vs H1: median not equal to
MINITAB:
Sign Test for Median: h/l
Sign test of median = 0,6180 versus not = 0,6180
N Below Equal Above P Median
h/l , ,6670

53. Nonparametric statistics: Wilcoxon signed-rank test
Example 1: Ratio height/length for 8 rectangular leather items:
Will test H0: Golden ratio vs. H1: Not golden ratio, i.e.
H0: median = vs H1: median not equal to
MINITAB:
Wilcoxon Signed Rank Test: h/l
Test of median = 0,6180 versus median not = 0,6180
N for Wilcoxon Estimated
N Test Statistic P Median
h/l , , ,6600

54. Example 2: Fuel economy (km per liter for two types of tires)

55. Wilcoxon’s rank-sum test:
Comparison of nicotine content in two cigarette types A, B

59. Normal plots for each of the two cases