Question

The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has a bell-shaped distribution. This distribution has a mean of 42 and a standard deviation of 11. Using the empirical rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 31 and 64? Do not enter the percent symbol.

Answer 1

The empirical rule states that, in a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.

• Approximately 95% falls within two standard deviations of the mean.

• Approximately 99.7% falls within three standard deviations of the mean.

In your case, you're given that the mean number of potholes is 42, and the standard deviation is 11. Let's break this down step by step:

1. First, find the range of values that fall within one standard deviation of the mean:

• Lower limit: Mean - 1 * Standard Deviation = 42 - 11 = 31

• Upper limit: Mean + 1 * Standard Deviation = 42 + 11 = 53

2. Next, find the range of values that fall within two standard deviations of the mean:

• Lower limit: Mean - 2 * Standard Deviation = 42 - 2 * 11 = 20

• Upper limit: Mean + 2 * Standard Deviation = 42 + 2 * 11 = 64

Now, you want to find the approximate percentage of 1-mile-long roadways with potholes numbering between 31 and 64.

The range from 31 to 64 falls within two standard deviations of the mean, according to the empirical rule.

Therefore, approximately 95% of the 1-mile-long roadways will have potholes within this range.

So, the approximate percentage is 95%.

Explanation:

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