00:01
In this question we have the speed limit is 65 miles per hour and the chief of the police department has their radar guns tested and we're told that the errors are normal.
00:23
By using the data and assuming that the radar guns provide unbiased measurements, so assuming unbiased, that means that the error is normal with a mean of zero and some variance sigma squared.
00:43
She issues a policy that officers should not stop cars unless the reading is at least 71 miles per hour, saying that this ensures that no more than 2 .5 % of cars driving at the speed limit will be pulled over for speeding.
00:57
So the probability that the error, so in that case the error would be 71, so 71 mph gives 2 .5 % of error leading to people at speed limit being pulled over.
01:32
So what does that mean? well what's the error in this case? this error, the critical error, is 71 minus 65 which is 6 mph.
01:45
So the probability that the error is at least 6 is 2 .5%.
01:50
But what we can do is we can work out a z score.
02:00
Z is going to be the error divided by sigma because it's mean zero.
02:07
So that means that the probability of having a z score greater than 6 over sigma is 2 .5%.
02:17
But what we can do then is we can say that 6 over sigma is the z score that encloses 2 .5 % above it, which we can look up in a table and it gives us 1 .9600.
02:35
Then sigma is just 6 over 1 .9600 over 0 which is 3 .0612.
02:48
So that's our standard error in miles per hour.
02:55
So that's the answer for part a.
02:57
Now part b, if the speed is 68 mph, how many will be pulled over? so what is the probability of being pulled over? well in this case to be pulled over, remember that the error would have to be at least, because we would need a speed of at least 71 to be pulled over...