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24 July, 08:12

The number of potholes in any given 1-mile stretch of freeway pavement in Pennsylvania has a Normal distribution. This distribution has a mean of 49 and a standard deviation of 9. Using the Empirical Rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 22 and 58?

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  1. 24 July, 11:15
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    83.85% of 1-mile long roadways with potholes numbering between 22 and 58

    Step-by-step explanation:

    The Empirical 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.

    In this problem, we have that:

    Mean = 49

    Standard deviation = 9

    Using the Empirical Rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 22 and 58?

    22 = 49 - 3*9

    So 22 is three standard deviations below the mean.

    Since the normal distribution is symmetric, 50% of the measures are below the mean and 50% are above the mean.

    Of those 50% which are below the mean, 99.7% of those are within 3 standard deviations of the mean, that is, greater than 22.

    58 = 49 + 9

    So 58 is one standard deviation of the mean.

    Of those which are above the mean, 68% are within 1 standard deviation of the mean, that is, lesser than 58.

    Then

    0.997*0.5 + 0.68*0.5 = 0.8385 = 83.85%

    83.85% of 1-mile long roadways with potholes numbering between 22 and 58
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