invest_3ed.pdf

# Because we rejected the null hypothesis we are also

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variables that help to explain the larger increase in fatality rates in states that increased their speed limit. Because we rejected the null hypothesis, we are also interested in examining a confidence interval to estimate the size of the treatment effect. We first approximate the t * critical value for say 95% confidence, again using min (19 ± 1, 32 ± 1) = 18 as the degrees of freedom.

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Chance/Rossman, 2015 ISCAM III Example 4.2 312 > iscaminvt(.975, 18, "below") the observation with 0.975 probability below is 2.101 Then the 95% confidence interval can be calculated, 32 21 19 31 10092 . 2 69 . 13 53 . 8 2 2 ² r ± ± = ± 22.2 r 16.85 We are 95% confident that the true “treatment effect” is in this interval or that the mean percentage increase in traffic fatality rates is between 5.4 percentage points to 39.1 percentage points higher in states that increase their speed limit compared to states that do not increase their speed limit (continuing to be careful not to state this as a cause and effect relationship). Before we complete this analysis, it is worthwhile to investigate the amount of influence that the outlier (the District of Columbia) has on the results, especially because D.C. does have different characteristics from the states in general. The updated R output (two-sided p-value) is below: As we might have guessed, the mean increase in fatalities for the “No” group has increased so that the difference in the group means is less extreme. This leads to a less extreme test statistic and a larger p-value (one-sided p-value = 0.01785/2 = 0.0089) so somewhat weaker evidence against the null hypothesis in favor of the one-sided alternative hypothesis. (e) The other technical condition is that we have independent random samples or random assignment to groups. We do not have either in this study, because we are examining the population of all states (and D.C.), and the states self-selected whether they changed their speed limit. Thus, any p-value we calculate is in a sense hypothetical because we have all the states here, we might ask the question: would the two groups look this different if whether or not they increased their speed limit had been assigned at random? So the above p-value measures how often we would see a difference in group means at least this large based on random assignment to the two groups if there were no true treatment effect. Even though this p-value is hypothetical, we still have some sense that the difference observed between the groups is larger than we would expect to see “by chance” even in a situation like this where it is not feasible to carry out a true randomized experiment. This gives some information that can be used in policy decisions but we must be careful not to overstate the attribution to the speed limit change.
Chance/Rossman, 2015 ISCAM III Example 4.3 313 Example 4.3: Distracted Driving? Try these questions yourself before you use the solutions following to check your answers.

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