Canadian Leadfoots?

Are Canadians unfairly targeted for speeding tickets?

Note: Your browser cannot display some mathematical notation as text. Because of this, the symbol p* should be equated with the "p-hat" discussed in class and the text. Sorry about that: to include the "p-hat" would require creating a graphic for each instantiation; this would be tedious (on my part) and would slow your browser. Furthermore, the graphic would not align with the remainder of the text. Complaints? E-mail Netscape and Microsoft.


Observational studies over a long time show that 13% of all vehicles that travel on Interstate 81 through Adams, NY, are Canadian registered. Assume that state troopers do not discriminate when they ticket drivers for speeding. If so, then 13% of all tickets should given to Canadian vehicles. Many drivers are ticketed each day, sifting through all the troopers’ records will be time consuming and tedious. Instead we take an SRS of 350 tickets and compute p*, the proportion of sampled tickets that were given to Canadian vehicles.

Questions in red; answers follow.

Identify the response variable being measured. Is it categorical or quantitative?
 
The response is the nationality of the vehicle that was ticketed (Canadian or US—we may safely ignore all others as their occurrence is very infrequent). This is a categorical variable.
 
Identify the population and the parameter of interest.
 
The population is all tickets in the troopers’ records. (This is a large population.) The population is not the drivers, nor the vehicles. Since we are sampling from the records, the population is all records. The parameter is assumed to be 0.13—we are assuming 13% of all ticketed vehicles are Canadian vehicles.
 
Identify the sample and the statistic.
 
The sample is the 350 selected tickets. The statistic is p*, the proportion of sampled tickets that were given to Canadian vehicles.
 
The statistic has a distribution; what is the mean of this distribution? Is the statistic an unbiased estimate of the parameter? What is the standard deviation of the statistic? (Only use the formula for the standard deviation of p* when the population is at least 10 times the size of the sample; otherwise answer "I don’t know, I’d have to ask an expert.")
 
The mean of the distribution is 0.13. p* is an unbiased estimate. Since the population is certainly at least 10 times the size of this sample, we may use the formula for the standard deviation—it is 0.0180 (keep 4 decimal places here).
 
Does the statistic have approximately the normal distribution? (Only if np and n(1-p) both exceed 10.)
 
Since np = 350(0.13) = 45.5 and n(1-p) = 350(0.87) = 304.5 an approximate normal distribution applies.
 
What is the probability that p* > 0.18 ( is greater than 0.18)? That is, how likely is it that the sample proportion of tickets given to Canadians exceeds 0.18 (the percentage exceeds 18%).
 
0.0027. Take the reciprocal (1/x) to get about 1 in 370. The answer is 0.0027, but you should use the 1 in 370 to attach some meaning to the probability.
 
Suppose 63 of the sampled tickets were given to Canadian vehicles. What can we conclude?
 
63/350 = 0.180. We’ve seen that if 0.13 of all tickets go to Canadians then there’s only a 1 in 370 chance of observing a sample proportion this high. While it’s possible such a sample might be drawn when 0.13 of all tickets are given to Canadians, it is highly unlikely. A more reasonable conclusion might be that the parameter’s value is actually greater than 0.13! (The only way we could find out for sure is by examining the entire population of tickets.) We conclude that Canadians get ticketed more than they should (0.0027 measures of the reliability of this conclusion).
 
A newspaper reports: "Police unfairly target Canadian drivers for speeding tickets." Why is such a conclusion unfounded?
 
?????????? What do you think ??????????