Question
upstudy study bank question image url

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 =

Ask by Chan Morgan. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The approximate percentage of 1-mile long roadways with potholes numbering between 28 and 63 is 99.70.

Solution

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: \[ \text{ans} = 99.70 \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Bonus Knowledge

To find the approximate percentage of 1-mile long roadways with potholes numbering between 28 and 63 using the empirical rule, we first find how many standard deviations away these numbers are from the mean. The mean is 49 and the standard deviation is 7. Calculating the z-scores: - For 28: \( z = \frac{28 - 49}{7} = \frac{-21}{7} = -3 \) - For 63: \( z = \frac{63 - 49}{7} = \frac{14}{7} = 2 \) According to the empirical rule: - Approximately 68% of the data falls within 1 standard deviation of the mean (42 to 56). - About 95% falls within 2 standard deviations (35 to 63). - Roughly 99.7% falls within 3 standard deviations (28 to 70). Since 28 to 63 covers 3 standard deviations below the mean to 2 above, the approximate percentage of roadways with potholes numbering between 28 and 63 is around 99.7%. Thus, the answer is: ans = 99.70

Related Questions

Latest Statistics Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy