Question

The speed limit on the freeway is 65 mph. The chief of the police department has their radar guns tested: from repeated measurements, the errors in the readings seem to follow a normal distribution. Using the data and assuming the radar guns provide unbiased measurements of the true speeds, the chief issues a policy that officers should not stop cars unless the reading is at least 71 mph. She says this policy ensures that no more than 2.5% of cars driving at the speed limit will be pulled over for speeding.

(a) What is the SD of the radar gun readings?

(b) Under the chief’s assumptions, what percentage of cars whose actual speed is 68 mph will be pulled over?

(c) Suppose that, unbeknownst to the chief, due to a factory production error the radar guns actually have a bias of +2 mph – that is, on average they will report a speed that is 2 mph higher than the actual speed. Under the chief’s policy, what percentage of cars whose actual speed is 68 mph will be pulled over?

Answer 1

a)since for top 2.5% values , critical z =1.96

SD=(x-mean)/z score =(71-65)/1.96=3.06

b)

for normal distribution z score =(X-μ)/σ

probability =P(X>71)=P(Z>(71-68)/3.06)=P(Z>0.98)=1-P(Z<0.98)=1-0.8365=0.1635

c)due ot baise expected speed=68+2 =70

probability =P(X>71)=P(Z>(71-70)/3.06)=P(Z>0.33)=1-P(Z<0.33)=1-0.6293=0.3707