Questions: Felipe, a highway safety inspector, is concerned about the potential for accidents caused by boulders that fall down a sandstone cliff beside a main highway. As part of an accident simulator, Felipe models the boulders' weights using a normal distribution with a mean of 563 kg and a standard deviation of 150 kg. Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 245 kg and 449 kg according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23 %). % %

Felipe, a highway safety inspector, is concerned about the potential for accidents caused by boulders that fall down a sandstone cliff beside a main highway. As part of an accident simulator, Felipe models the boulders' weights using a normal distribution with a mean of 563 kg and a standard deviation of 150 kg. Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 245 kg and 449 kg according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23 %). % %
Transcript text: Random Variables and Distributions Normal distribution: Finding a probability, advanced Gabri Español Felipe, a highway safety inspector, is concerned about the potential for accidents caused by boulders that fall down a sandstone cliff beside a main highway. As part of an accident simulator, Felipe models the boulders' weights using a normal distribution with a mean of 563 kg and a standard deviation of 150 kg . Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 245 kg and 449 kg according to the model. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example $98.23 \%$ ). $\square \%$ $\square$ Explanation Check (c) 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy center| Accessibility
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Solution

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Solution Steps

Step 1: Define the Normal Distribution Parameters

The weights of the boulders are modeled using a normal distribution with the following parameters:

  • Mean (\( \mu \)): 563 kg
  • Standard Deviation (\( \sigma \)): 150 kg
Step 2: Calculate Z-scores

To find the probability that the boulders weigh between 245 kg and 449 kg, we first calculate the Z-scores for these values using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

For the lower bound (245 kg): \[ Z_{start} = \frac{245 - 563}{150} = -2.12 \]

For the upper bound (449 kg): \[ Z_{end} = \frac{449 - 563}{150} = -0.76 \]

Step 3: Calculate the Probability

Using the Z-scores, we can find the probability that a boulder weighs between 245 kg and 449 kg:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.76) - \Phi(-2.12) \]

From the calculations, we find: \[ P = 0.2066 \]

Step 4: Convert Probability to Percentage

To express the probability as a percentage, we multiply by 100:

\[ \text{Percentage} = P \times 100 = 0.2066 \times 100 = 20.66\% \]

Final Answer

The percentage of boulders that weigh between 245 kg and 449 kg is \\(\boxed{20.66\%}\\).

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