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

To solve this problem, we will use the empirical rule, which states that for a bell-shaped (normal) distribution:

1. Approximately 68% of the data falls within one standard deviation of the mean.
2. Approximately 95% of the data falls within two standard deviations of the mean.
3. Approximately 99.7% of the data falls within three standard deviations of the mean.

### Step 1: Identify the mean and standard deviation
- Mean ($$\mu$$) = 49
- Standard deviation ($$\sigma$$) = 7

### Step 2: Calculate the range for 1, 2, and 3 standard deviations
- **1 standard deviation**: 
  - Lower limit: $$\mu - \sigma = 49 - 7 = 42$$
  - Upper limit: $$\mu + \sigma = 49 + 7 = 56$$

- **2 standard deviations**: 
  - Lower limit: $$\mu - 2\sigma = 49 - 2(7) = 49 - 14 = 35$$
  - Upper limit: $$\mu + 2\sigma = 49 + 2(7) = 49 + 14 = 63$$

- **3 standard deviations**: 
  - Lower limit: $$\mu - 3\sigma = 49 - 3(7) = 49 - 21 = 28$$
  - Upper limit: $$\mu + 3\sigma = 49 + 3(7) = 49 + 21 = 70$$

### Step 3: Determine the range of interest
We are interested in the range between 28 and 63.

- The lower limit of 28 corresponds to 3 standard deviations below the mean.
- The upper limit of 63 corresponds to 2 standard deviations above the mean.

### Step 4: Apply the empirical rule
According to the empirical rule:
- Approximately 99.7% of the data falls within 3 standard deviations (between 28 and 70).
- Approximately 95% of the data falls within 2 standard deviations (between 35 and 63).

Since we are looking for the percentage of roadways with potholes numbering between 28 and 63, we can conclude that this range includes all values from 28 to 35 (which is part of the 99.7% range) and from 35 to 63 (which is part of the 95% range).

### Step 5: Calculate the percentage
Since the range from 28 to 63 includes all values from 28 to 35 (which is 0.15% of the total) and from 35 to 63 (which is 95% of the total), we can approximate the percentage as:

$$
\text{Percentage} \approx 99.7\%
$$

### Final Answer
Thus, the approximate percentage of 1-mile long roadways with potholes numbering between 28 and 63 is:

$$
\text{ans} = 99.70
$$
The approximate percentage of 1-mile long roadways with potholes numbering between 28 and 63 is 99.70.

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