Canadians unfairly targeted for speeding tickets?
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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 USwe 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 0.13we are
assuming 13% of all ticketed vehicles are
- 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
dont know, Id have to ask an
- 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
deviationit is 0.0180 (keep 4 decimal
- Does the statistic
have approximately the normal distribution? (Only
both exceed 10.)
- Since np =
350(0.13) = 45.5 and n(1-p) = 350(0.87)
= 304.5 an approximate normal distribution
- 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. Weve
seen that if 0.13 of all tickets go to
Canadians then theres only a 1 in 370
chance of observing a sample proportion this
high. While its 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
parameters 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
- ?????????? What do you