
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
"phat" discussed in class and the text. Sorry
about that: to include the "phat" 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? Email 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(1p)
both exceed 10.)

 Since np =
350(0.13) = 45.5 and n(1p) = 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 ??????????

