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

Question

Avoiding an accident while driving can depend on reaction time.That time, measured from the time the driver first sees the dangeruntil the driver gets his/her foot on the brake pedal, can bedescribed by a normal model with mean 1.9 seconds and standarddeviation 0.19 seconds. Use the 68-95-99.7 rule (NOT a ztable) to answer the following questions.

What percentage of drivers have a reaction time morethan 2.28 seconds?
x%

What percentage of drivers have a reaction time lessthan 1.71 seconds?
x%

What percentage of drivers have a reaction time lessthan 2.09 seconds?
x%

What percentage of drivers have a reaction time morethan 2.28 seconds?
x%

What percentage of drivers have a reaction time lessthan 1.71 seconds?
x%

What percentage of drivers have a reaction time lessthan 2.09 seconds?
x%

Explanation / Answer

For the second question, let's note that 1.9 - .19 = 1.71. That is, 1.71 is one standard deviation below the mean. We knowfrom the Rule that 68% of the data is within one standarddeviation, so this leaves 32% that is outside one standarddeviation. We only want all the ones less than 1.71 seconds,however. The curve is symmetric, so this leaves 16% on either side.Hence, 16% of the data are less than 1.71seconds. For question number three, note that 1.9 + .19 = 2.09, so once again we're looking at one standard deviation. However,this is a slightly different case, as we want the percentage below2.09. We know that 50% of the data are on either side of the mean,so just below the mean, we have 50%. Now we need to figure out whatpercentage is between 1.9 and 2.09. Well, we know 68% of the datais within one std. dev. around the mean, and the curve is symmetric(this plays a big role in normal distributions, most of the time),so that means 34% of the data is on each side of the mean. Thismeans that 34% of the data is between the mean and one standarddeviation in either direction. Thus, 34% of the data is between 1.9and 2.09. So, we take the 50% we had before and tack on the 34%, and thatgives us 84% of the data being less than 2.09 seconds.

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