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10 March, 15:00

Avoiding an accident while driving can depend on reaction time. Suppose that reaction time, measured from the time the driver first sees the danger until the driver gets his/her foot on the brake pedal, is approximately symmetric and mound-shaped with mean 1.9 seconds and standard deviation 0.12seconds. Use the 68-95-99.7 rule to answer the following questions. What percentage of drivers have a reaction time more than 2.14 seconds?%What percentage of drivers have a reaction time less than 1.78 seconds?%What percentage of drivers have a reaction time less than 2.02 seconds?%

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  1. 10 March, 15:03
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    2.5% of drivers have a reaction time more than 2.14 seconds

    16% of drivers have a reaction time less than 1.78 seconds

    84% of drivers have a reaction time less than 2.02 seconds

    Step-by-step explanation:

    The Empirical Rule (68-95-99.7 rule) states that, for a normally distributed random variable:

    68% of the measures are within 1 standard deviation of the mean.

    95% of the measures are within 2 standard deviation of the mean.

    99.7% of the measures are within 3 standard deviations of the mean.

    The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

    In this problem, we have that:

    Mean = 1.9s

    Standard deviation = 0.12

    What percentage of drivers have a reaction time more than 2.14 seconds?

    2.14 = 1.9 + 2*0.12

    So 2.14 is two standard deviations above the mean.

    Of the 50% of the measures above the mean, 95% are within 2 standard deviations of the mean, so, below 2.14. The other 5% is above.

    0.05*0.5 = 0.025

    2.5% of drivers have a reaction time more than 2.14 seconds

    What percentage of drivers have a reaction time less than 1.78 seconds?

    1.78 = 1.9 - 0.12

    So 1.78 is one standard deviation below the mean.

    Of the 50% of the measures that are below the mean, 68% are within one standard deviation of the mean, that is, greater than 1.78.

    100 - 68 = 32

    0.32*50 = 0.16

    16% of drivers have a reaction time less than 1.78 seconds

    What percentage of drivers have a reaction time less than 2.02 seconds?

    2.02 = 1.9 + 0.12

    So 2.02 is one standard deviation above the mean.

    Of the measures that are below the mean, all are below 2.02.

    Of those that are above, 68% are below 2.02.

    0.5 + 0.68*0.5 = 0.84

    84% of drivers have a reaction time less than 2.02 seconds
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