Example 5.2: Speed Limit Changes
In 1995, the National Highway System Designation Act
abolished the federal mandate of 55 miles per hour maximum speed limit and
allowed states to establish their own limits.
Of the 50 states (plus
(a) Identify the observational units and response variable of interest. Is this a randomized experiment or an observational study?
(b) Produce numerical and graphical summaries of these data and describe how the two groups compare.
(c) Are the technical conditions for a two-sample t-test met for this study? Explain.
(d) Carry out a two-sample t-test to determine whether the average percentage change in interstate
highway traffic fatalities is significantly higher in states that increased
their speed limit. If you find a
significant difference, estimate its magnitude with a confidence interval.
(e) Discuss what the p-value in (d) measures.
Analysis:
(a) The observational units are the 50 states (and the
(b) The following graphical display is dotplots of the percentage change in traffic fatalities for each state (and D.C.) in the two groups on the same scale:
Since the distributions are reasonably symmetric, it makes sense to report the means and standard deviations as the numerical summaries:
No increase 
no = -8.56% sno =
31%
Increase yes = 13.75% syes =
21%
These results indicate that there is a tendency for the percentage change in traffic fatalities to be higher in those states that increase their speed limits. This tendency is also seen in stacked boxplots:
The boxplots also reveal an
outlier, the
These summaries also reveal that the two sample distributions are reasonably similar in shape and spread.
(c) In considering the technical conditions, we see that the
sample sizes (19 and 32) are reasonably large.
Coupled with the normal shaped sample distributions, the normality/large
sample size conditions appears to be satisfied for us to use the t distribution.
The other technical condition is that we have independent random samples or randomization. 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 would be hypothetical. Since 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? Thus, we will proceed as if this was a hypothetical experiment.
(d) Let d represent the true “effect” of increasing the speed limit on the traffic fatality rate (states that didn’t change speed limit – states the did change speed limit)
H0: d = 0 there is no true effect from increasing the speed limit
Ha: d < 0 increasing the speed limit leads to an increase in traffic fatalities (higher average percentage change with increase in speed limit)
In theory, we can apply the two-sample t procedure to model the hypothetical randomization distribution. In this case, the test statistic will be
= -2.78
If we approximate the degrees of freedom by min(19-1, 32-1) = 18, then we find the one-sided p-value in Minitab to be:
These calculations are confirmed by the Test of Significance Calculator applet and by Minitab:
Note: Minitab uses a more exact method for determining the degrees of freedom. Our “by hand” method (also used in the applet) is conservative in that the p-value found will be larger than the actual p-value as seen here.
Such a small p-value (.005 < .01) reveals that we would
observe such a large difference in group means by random assignment alone if
there was no treatment effect only about 5 times in 1000, convincing us that
the observed difference in the group means is larger than what we would expect just
from randomization. We have strong
evidence that something
other than “random chance” led to this difference. However, we cannot attribute the difference
solely to the speed limit change since this was not actually a randomized
experiment. Since the states
self-selected, there could be confounding variables that help to explain the
larger increase in fatality rates in states that increased their speed limit.
Since we rejected the null hypothesis, we are also be 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.
Then the 95% confidence interval can be calculated,
= -22.4 + 16.90
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 5.5% to 39.3% higher in states that increase their speed limit compared
to states that do not increase their speed limit (continuing to be careful not
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
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. While we would still reject the null hypothesis at the 5% level of significance, we would not at a 1% level of significance. Thus, we would now say we have “moderate” evidence against the null hypothesis.
(e) 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 was no true treatment effect. However, since this was not a randomized study, this p-value should be considered hypothetical. Still, we 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.