Module 10: Inference for Means
Hypothesis Test for a Population Mean (3 of 5)
Hypothesis Test for a Population Mean (3 of 5)
Learning outcomes
- Under appropriate conditions, conduct a hypothesis test about a mean for a matched pairs design. State a conclusion in context.
Another common use of the t-test for a population mean is in “before and after” situations. In this situation, we have two quantitative measurements from a single sample of individuals. This is an example of a matched-pairs design.
Example
Drinking and Driving
The Centers for Disease Control and Prevention (CDC) website cites studies from the National Highway Traffic Safety Administration to support this statement: “Alcohol use slows reaction time and impairs judgment and coordination, which are all skills needed to drive a car safely. The more alcohol consumed, the greater the impairment.” All states in the United States have adopted a blood alcohol concentration of 0.08% (80 mg/dL) as the legal limit for operating a motor vehicle. The CDC website continues, “Note: Legal limits do not define a level below which it is safe to operate a vehicle or engage in some other activity. Impairment due to alcohol use begins to occur at levels well below the legal limit.”
It is this last statement that may be surprising to drivers. Interviews with drunk drivers who were involved in accidents reveal that drunk drivers do not realize how drunk they are. “I only had one or two drinks – I am okay to drive” is a common sentiment. Suppose a college conducts a study to call attention to this issue. Researchers use a random sample of 20 college students to examine the effect of drinking two beers on reaction times. They use a driving simulator to measure each student’s reaction time before and after drinking two beers. The reaction time is the time it takes the student to hit the brakes in the simulator when an obstacle appears in the road.
In this situation, we have two quantitative measurements for each student. To measure the effect of the two beers, we subtract the two reaction times to create one measurement of “change” or “effect.” This controls the effects of individual characteristics that could influence reaction time, such as driving experience or natural quickness.
Here is a partial list of the data. We define the difference as “before minus after.” If the “after” reaction time is longer, then the difference is negative. A negative value means the reaction time is slower after drinking.
Note: It is common to define the difference in measurements as “before minus after.” But we could also define the difference the other way around as “after minus before.” In this definition, if the “after” reaction time is longer, then the difference is positive, so a slower reaction time after drinking corresponds to a positive value. This makes less intuitive sense to us. We want “negative” to mean “drinking has a negative effect,” so we used the other definition, “before minus after.”
Step 1: Determine