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 1.8 seconds and standard deviation 0.16 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.

a) What percentage of drivers have a reaction time more than 2.12 seconds? %

b) What percentage of drivers have a reaction time less than 1.64 seconds? %

c) What percentage of drivers have a reaction time less than 1.96 seconds? %

Explanation / Answer

as it falls in
68% fall within (plus and minus) 1 standard deviation
95% fall within (plus and minus) 2 standard deviations
99.7% fall within (plus and minus) 3 standard deviations

a)
(2.12-1.8)/0.16 = 2 Standard Deviations
95% fall within (plus and minus) 2 standard deviations
This leaves 5% that have a reaction time lower than -2 SD or higher than 2 SD.
The percentage of drivers who have a reaction time more than 2.12 seconds is 2.5%.

b)
(1.64-1.8)/0.16 = -1 Standard Deviation
68% fall within (plus and minus) 1 standard deviation
This leaves 32% that have a reaction time lower than -1 SD or higher than 1 SD.
The percentage of drivers who have a reaction time lower than 1.64 seconds is 16%.

c)
(1.96-1.8)/0.16 = 1 Standard Deviation
68% fall within (plus and minus) 1 standard deviation
34% represents those from 1 Standard Deviation to the Mean (1.8).

50% of them are below the Mean.
50%+34% = 84%

The percentage of drivers who have a reaction time lower than 1.96 seconds is 84%.

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