10: Cross-tabulated counts (Key odd)

Review Questions

p = population proportion; = sample proportion
(a) 6.63 (b) 10.83 (c) 5.99 (d) 7.81 (e)
true.
5
df = (4-1)(3-1) = 6
(a) 0.05 < P < 0.10 by using the table; P = 0.082 by computer (b) 0.01 < P < 0.025 by using the table; P = 0.018 by computer
False. Chi-square distributions are asymmetrical with a long right tails.
H₀: p₁ = p₀ or H₀: "no association between the row variable and column variable in the population"
(a) 0.20 (b) 0.10 (c) 0.01
The size of the shaded region is between 0.20 (chi-square landmark 4.64) and 0.15 (chi-square landmark 5.32). The precise area determined with StaTable is 0.156379.

Exercises

10.1 Tobacco use in high school students.

(A) Calculate the prevalence of tobacco use by gender. _male = 39 / 318 = 0.1226 = 12%; _females = 26 / 336 = 0.0774 = 8%
(B) Test the difference in proportions for significance with a chi-square test. H₀: no association between gender and tobacco use in the population vs. H₁: association in the population. X²_stat = [(39 - 31.606)² / 31.606] + [(279 - 286.395)² / 286.395] + [(26 - 33.394)² / 33.394] + [(310 - 302.605)² / 302.605] = 1.73 + 0.19 + 1.64 + 0.18 = 3.74 with df = 1; 0.05 < P < 0.10 (P = 0.053); the evidence against H₀ is marginally significant

[Here's are results by other testing methods: X²_{stat,
continuity-corrected} = 3.250, df = 1, P = 0.071; Fisher's exact (two-tailed) P = 0.067; exact Mid-P P = 0.058. All methods suggest that evidence against H₀ is marginally significant.]

10.3 Do seatbelt laws prevent injury?

(A) Calculate the conditional distribution of injuries before and after enactment of the law. See below. What type of association is seen? Injuries became less severe after enactment of the law (e.g., "no injury" increased from 90.5% to 92.6%).

	No injury	Minimal injury	Minor injury	Major injury	Total
After	92.6%	4.6%	2.5%	0.3%	100.0%
Prior	90.5%	5.5%	3.6%	0.4%	100.0%
Total	90.8%	5.3%	3.4%	0.4%	100.0%

(B) Test the association for statistical significance.

H₀: no association in the population between level of injury and time period vs. H₁: "association"

X²_stat = 6.644, df = 3

P = 0.084

The evidence against H₀: is marginally significant [Note The lack of statistical significance does not imply unimportance.]

(C) Describe the trend OR^₁ = 1.00; OR^₂ = 0.83; OR^₃ = 0.68; OR^₄ = 0.69 severity of injury decreased after enactment
...and test it for significance.; z_{stat, trend} = 2.55, P = 0.011

10.5 Vitamin C and the common cold . Are these proportions significantly different? H₀: p₁ = p₀ (no association) vs.H₁: p₁ p₀ (association) Test statistic: X²_stat = [(36 - 39.29)² / 39.29] + [(21 - 17.71)² / 17.71] + [(35 - 31.71)² / 31.71] + [(11 - 14.29)² / 14.29] = 0.28 + 0.61 + 0.34 + 0.76 = 1.99 with df = 1. P = 0.159; The evidence against H₀ is not significant.

Observed	Cases	Non-cases
Vit C+	36	21
Vit C-	35	11

Expected	Cases	Non-cases
Vit C+	39.29	17.71
Vit C-	31.71	14.29

[Yates's correction chi-square = 1.429, P = 0.232]

10.7 Frequency of problems at community mental health centers.

(A) Compare the distribution of problems within centers. See conditions distributions below. "Other problems" were most common at Community Center 1 (49%) and Community Center 3 (41%), while "Problems with living" was most common at Center 2 (40%); see table below.

(B) Conduct a chi-square test of association. H₀: no association between center and problem type in population against H₁: "association"; X²_stat= 9.54 with 4 df (calculations shown below); P = 0.049; the evidence against H₀ is significant

Observed and expected counts
X²_stat= [(25 - 26.21)² / 26.21] + [(20 - 26.85)² / 26.85] + [(43 - 34.94)² / 34.94]
          + [(24 - 21.44)² / 21.44] + [(29 - 21.97)² / 21.97] + [(19 - 28.59)² / 28.59]
          + [(32 - 33.35)² / 33.35] + [(34 - 34.18)² / 34.18] + [(46 - 44.47)² / 44.47]
          = 9.54
df = (3-1)(3-1) = (2)(2) = 4
P-value: Landmarks: c²_4,.95 = 9.49 (right tail = 0.05) and c²_4,.975 = 11.14 (right-tail = 0.025). 0.025 < P < 0.05 (P = 0.049)
There is a significant difference in the types of problems seen at the centers. The most common problem seen at center 1 and center 2 were "other problems" (49% and 41% of totals, respectively). The most common problem seen at center 2 was "problems with living" (40% of total).

10.9 Drove when drinking alcohol.

(A) Proportions

_W = 243 / 2154 = 0.113; SE = 0.00682; margin of error = 0.013
_B = 25 / 508 = 0.049; SE = 0.00958; margin of error = 0.019
_H = 55 / 526 = 0.105; SE = 0.01337; margin of error = 0.026

(B) X²_stat= 18.359, df = 2, P = 0.00010

10.11 Anger and heart disease (hard outcome, normotensives).

(A) _low = 0.00997 (about 1.0%); _moderate = 0.01332 (about 1.3%); _high = 0.02844 (about 2.8%).
(B) H₀: no association between anger-trait and the coronary outcomes in the population, Pearson chi-sq. = 13.763, df = 2, P = 0.0010; the evidence against the null hypothesis is highly significant.
(C) The study shows a significant association between high- anger groups and CHD risk.
[Advanced users: Mantel test for trend chi-sq. = 9.894, df = 1, P = 0.0017]

10.13