Notes

• At 0.01 confidence level, we can reject the null hypothesis and conclude that the test result is
statistically significant. In this case, we interpret this as a statistically significant relationship between
smoking and platelet aggregation.
Inference about the Relationnship Between Two Bınary Variables
• A common way to analyze the relationship between binary (in general, categorical) variables is to use
contingency tables. Contingency tables are a tabular representations of the frequencies for all possible
combinations of the two variables.
• We refer to our estimate of SD12 as the standard error and denote it as SE 12 :
1
1
• Using the point estimate p12 along with the standard error SE12, we can find confidence intervals for
µ12 as follows:
,
where z crit is obtained for a given confidence level c as before
• We test the null hypothesis, H0 : µ 12 = 0, using the two-sample z-test as follows.
o First, we obtain the z-score,
o Then, we calculate the p-value, which is the observed significance level:
if H A : µ12 > 0, p obs = P(Z ≥ z),
if H A : µ12 < 0, p obs = P(Z ≤ z),
if H A : µ12 ≠ 0, p obs = 2×P(Z ≥ |z|).
Example
• As an example, suppose that we want to investigate whether smoking during pregnancy increases the
risk of having a low birthweight baby.
• We denote the population proportion of low-birthweight babies for nonsmoking mothers as µ 1 , and
the population proportion of low-birthweight babies for smoking mothers as µ 2 . We use µ 12 to denote
the difference between these two proportions:
µ 12 = µ 1 – µ 2
• If smoking and low birthweight are related, we expect the two population proportions to be different
and µ 12 to be away from zero. Therefore, we can express our hypothesis regarding the relationship
between the two variables as H A : µ 12 ≠ 0. we use the two-sided alternative. The corresponding null
hypothesis is then H 0 : µ12 = 0.
• For our example, p 1 = 0.25 and p 2 = 0.40. These are the point estimates for µ 1 and µ 2 , respectively. As
a result, the point estimate of µ 12 is p 12 = 0.25− 0.40=−0.15.
0.251 0.25 0.401 0.40
0.07
115
74
• The 95% confidence interval of µ 12 is 0.15 2 0.07, 0.15 2 0.07 0.29, 0.01