Infant mortality and economics
Is there any relationship?
Testing linear relationships
• We are interesting in establishing a
relationship between economic variables and
an important measure of health: infant
mortality
• We have data at the county level for North
Carolina; we want to know how the mean
infant mortality may change as, say, income
increases
Test hypothesis
• So the usual approach is to test the hypothesis
that the coefficient of income (slope) is equal
to zero
• To improve power, we want to do a one-sided
test, and reject the hypothesis of zero slope in
favor of a negative slope, if the t-statistic is
sufficiently negative
And if we fail to reject?
• Either the slope is really zero (or not far
enough away for us to care)
• Or the situation is really noisy:
• Noise variance is large
• Sample size is small
• Not much spread in explanatory variable
To show that the slope is small
• We need to show that if the slope were big
enough to care, we could have found it –
Pr(reject H | Hypothesis true) = level = .05 (say)
Pr(reject H | Alternative true) = power
Power will be a function of variance, sample
size, spread, and slope. If we keep the first
three as in our study, and show that the
power is LARGE for small values of slope, ….