An insurance company asks you to determine whether older drivers are safer than younger ones. Provide a directional hypothesis related to this study. Then, explain how you would need to change the hypothesis so that it would be non-directional. What happens to the rejection regions and why? Which of the two hypotheses do you think is more appropriate and why?
HO:p=R
Directional:
HA≥R
or
HA≤R
Nondirectional:
HA≠R
Directional hypotheses have alternate hypotheses that point in some direction as to having a value less than or greater than the quantity denoted by the null hypothesis. Nondirectional hypotheses simply state that the alternate hypothesized value is different than the null hypothesized value.
In your example...
HO:p=∂older−∂younger=0
HA:p≤0
where "∂" represents the total accident occurrence rate in set age groups (older = 31+; younger = 16-30) in the past year.
This is unidirectional because it hypothesizes that the trend of accident occurrences/driving safety rates in variant age groups tends toward one direction.
To make the hypothesis nondirectional, simply reword the hypothesis such that it doesn't indicate the general trend of accident occurrences/driving safety rates tending towards one direction. EXAMPLE: An insurance company asks you to determine whether a difference exists in the accident rates between younger and older drivers.
This should be more than enough information to answer the remaining few questions with.
A directional hypothesis related to this study could be: "Older drivers are safer than younger ones."
To change this hypothesis into a non-directional one, we would need to remove the directionality by stating that there is simply a difference between older and younger drivers in terms of safety, without specifying which group is safer. A non-directional hypothesis could be: "There is a difference in safety between older and younger drivers."
When comparing the rejection regions, a directional hypothesis has a one-tailed test, as it assumes a specific direction of effect (older drivers being safer). The rejection region will be concentrated on one side of the distribution, depending on the direction specified in the hypothesis. In this case, it would focus on the upper tail of the distribution.
On the other hand, a non-directional hypothesis utilizes a two-tailed test as it does not assume a specific direction of effect. The rejection region will be divided into two equal parts, one in each tail of the distribution. This allows for the possibility of either group being safer or less safe.
Considering the context of the study, a non-directional hypothesis would be more appropriate. Insurance companies seek to evaluate the overall safety of age groups rather than assuming that one age group is inherently safer than another. A non-directional approach allows for a more comprehensive analysis of the data and allows the study to account for all possible outcomes rather than just one specific scenario.