Avoiding an accident while driving can depend on reaction time. That time, measu

Question

Avoiding an accident while driving can depend on reaction time. That time, measured from the time the driver first sees the danger until the driver gets his/her foot on the brake pedal, can be described by a normal model with mean 1.9 seconds and standard deviation 0.13 seconds. Use the 68-95-99.7 rule (NOT a z table) to answer the following questions. The pictures of the 68-95-99.7 rule at this link might help.
http://www.oswego.edu/~srp/stats/6895997.htm

What percentage of drivers have a reaction time more than 2.16 seconds?

________%

What percentage of drivers have a reaction time less than 1.77 seconds?
________%

What percentage of drivers have a reaction time less than 2.03 seconds?
________%

Explanation / Answer

The 68-95-99.7 Rule tells us about the approximate probability that is found within a certain number of standard deviations from the population mean.

• First, the 68-95-99.7 Rule says that the probability within 1 standard deviation from the mean is approximately 68%.

This means that:

P(X+) 0.68

In this case, we have that:

  = 1.90.13 = 1.77

+ = 1.9+0.13=2.03

so then

Pr(1.77 X 2.03) 0.68

Then, since the probability inside of the interval (1.77, 2.03) is 0.68, then using the Law of Complement, the probability OUTSIDE of 1.77,2.03 is 1 - 0.68 = 0.32. Also, since the normal distribution is symmetric, then we also conclude that half of that probability (0.32/2 = 0.16) goes to each of the two tails. This means that

P(X1.77) 0.16

Thus, percentage of drivers having reaction time less than 1.77 seconds is 16%

P(X2.03) 10.16 = 0.84

Thus, percentage of drivers having reaction time less than 2.03 seconds is 84%

• Similarly, the 68-95-99.7 Rule says that the probability within 2 standard deviations from the mean is approximately 95%. This means that:

(2 X +2) 0.95

In this case, we have that:

2 = 1.92×0.13 = 1.64

+2 = 1.9+2×0.13 = 2.16

so then

Pr(1.64X2.16) 0.95

Then, since the probability inside of the interval (1.64,2.16) is 0.95, then using the Law of Complement, the probability OUTSIDE of 1.64,2.16 is 1 - 0.95 = 0.05. Also, since the normal distribution is symmetric, then we also conclude that half of that probability (0.05/2 = 0.025) goes to each of the two tails. This means that

P(X1.64) 0.025

P(X>2.16) 0.025

Thus, percentage of drivers having reaction time more than 2.16 seconds is 2.5%

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