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Feb. 13
• Chapter 12, Try 1-9
• Read Ch. 15 for next Monday
• No meeting Friday
Quiz from end of last time
• 40 of 100 men have high blood pressure 50 of 200 women have high blood pressure
• For men, risk of high b.p. = 40/100 = .40
• Relative risk for men compared to women = .40 / (50/200) = .40 / .25 = 1.6
• For women odds of high b.p. = 50 to 150, which could be reduced to 1 to 3
Simpson’s Paradox
• Nature of relationship is different for whole group than it is for each subgroup
• CAUSE – Confounding effect of a third variable
Graduate Admissions Example
• In graduate academic program A:– 400 of 650 men applicants admitted (61.5%)– 50 of 75 women applicants admitted (66.7%)
• In graduate academic program B:– 50 of 350 men applicants admitted (14.3%)– 125 of 425 women applicants admitted (29.4%)
• Women had higher acceptance rate in both programs
Total of programs A and B
• Men: (400+50) / (650+350) = 450/1000 = 45% admitted
• Women: (50+125) / (75+425) = 175/500 = 35% admitted
• Overall, acceptance rate is higher for men even though women had higher acceptance in each program.
• What’s going on?
Confounding
• Program B is harder to get into
• Most women apply to program B
• Program A is easier to get into
• Most men apply to program A
Stat 200 survey question
• Have you ever driven under the influence of alcohol or drugs?
Approximate Results by Gender
Have Have Not Total
Male 208 (52%) 192 400
Female 242 (40%) 358 600
Total 450 (45%) 550 1,000
A Research Question
• Is there a “statistically significant” relationship?
• Does the relationship observed in the sample also hold in the population?
Chi-Square Procedure
• A Chi-square test is used to analyze statistical significance.
The idea of Chi-Square
• Chi-square measures the difference between the observed counts and “expected counts”
• Expected counts = the counts that would occur if there were no relationship.
Properties of Expected Counts
• Same row and column totals as observed counts
• Row percentages are the same in each row.
Expected Counts
Have Have Not Total
Male 400
Female 600
Total 450 (45%) 550 1,000
Expected Counts
Have Have Not Total
Male 180 (45%) 220 400
Female 270 (45%) 330 600
Total 450 (45%) 550 1,000
Chi-square Statistic
• Sum of (obs.-exp)2/exp where sum is over all cells.
• For our example, Chi-square=13.2
Chi-Square and Statistical Significance
• Guideline: A chi-square value is statistically significant if it is over 3.84
• Why? – Values over 3.84 will occur less than 5% of the time “just by luck”
In our example -
• 13.2 is larger than 3.84
• CONCLUDE= there is a statistically significant relationship
• So, we believe there is a relationship in the larger population
Example
• In Stat 200, students classified by handedness and gender.
• 56 of 545 (10.3%) of females are left-handed
• 43 of 355 (12.1%) of males left-handed.• Not significant, Chi-square=0.741 • So, apparently no relationship between
handedness and gender
Example
• In Stat 200, students classified by whether they smoke cigarettes and whether they’ve smoked marijuana in last 6 months.
• 217 of 735 (29.5%) of non-smokers of cigs have smoked marijuana
• 109 of 160 cig smokers (68%) have smoked marijuana.
• Significant, Chi-square=84.5 ; So, there is a relationship between cig and marijuana smoking