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 49 and a standard deviation of 7 . Using the empirical rule (as
presented in the book), what is the approximate percentage of 1 -mile long roadways with potholes
numbering between 28 and 63 ?
(Round percent number to 2 decimal places.)
Do not enter the percent symbol.
ans

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UpStudy AI · Accepted Answer

Let the number of potholes per mile be approximately normally distributed with mean $$ \mu=49 $$ and standard deviation $$ \sigma=7 $$.

The given interval is from 28 to 63. Notice that

$$
28 = 49 - 3(7) \quad \text{and} \quad 63 = 49 + 2(7)
$$

The empirical rule tells us that approximately:
- $$95\%$$ of the data lie between $$\mu-2\sigma$$ and $$\mu+2\sigma$$, and
- $$99.7\%$$ of the data lie between $$\mu-3\sigma$$ and $$\mu+3\sigma$$.

This means:
- From $$49-2(7)=35$$ to $$49+2(7)=63$$ we have about $$95\%$$ of the data.
- The additional data from $$49-3(7)=28$$ to $$35$$ (i.e. from $$-3\sigma$$ to $$-2\sigma$$) represents approximately half of the extra $$100\% - 95\% = 5\%$$ that lie outside $$\mu\pm2\sigma$$. This portion is approximately

$$
2.5\% - \text{(a small adjustment since actually }99.7\%\text{ are within }3\sigma\text{)}
$$

A more refined estimate using the empirical rule is as follows:  
Since $$99.7\%$$ are within $$\mu\pm3\sigma$$, the amount in each tail beyond $$3\sigma$$ is about $$0.15\%$$. Thus, from $$-3\sigma$$ to $$-2\sigma$$ the proportion is

$$
2.5\% - 0.15\% = 2.35\%.
$$

So the total percentage of roadways having potholes between 28 and 63 is approximately:

$$
95\% + 2.35\% = 97.35\%.
$$

Thus, the approximate percentage (rounded to two decimal places) is

$$
\boxed{97.35}
$$
Approximately 97.35% of 1-mile long roadways have potholes numbering between 28 and 63.

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