Questions: David, 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, David models the boulders' weights using a normal distribution with a mean of 565 kg and a standard deviation of 100 kg. Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 493 kg and 681 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 %).

David, 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, David models the boulders' weights using a normal distribution with a mean of 565 kg and a standard deviation of 100 kg.

Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 493 kg and 681 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: David, 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, David models the boulders' weights using a normal distribution with a mean of 565 kg and a standard deviation of 100 kg . Use this table or the ALEKS calculator to find the percentage of boulders that weigh between 493 kg and 681 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 $98.23 \%$ ). $\square$ $\square \%$

Solution

Solution Steps

To find the percentage of boulders that weigh between 493 kg and 681 kg, we need to calculate the cumulative distribution function (CDF) for a normal distribution. First, we find the CDF value for 681 kg and subtract the CDF value for 493 kg. This will give us the probability that a boulder falls within this weight range. Finally, we convert this probability to a percentage.

Step 1: Define the Problem

We need to find the percentage of boulders that weigh between 493 kg and 681 kg, given that the weights follow a normal distribution with a mean ($\mu$) of 565 kg and a standard deviation ($\sigma$) of 100 kg.

Step 2: Calculate the Cumulative Distribution Function (CDF) Values

The cumulative distribution function for a normal distribution is used to find the probability that a random variable is less than or equal to a certain value. We calculate the CDF for the upper bound (681 kg) and the lower bound (493 kg).

CDF for 681 kg: $ \text{CDF}(681) = 0.8770 $
CDF for 493 kg: $ \text{CDF}(493) = 0.2358 $

Step 3: Calculate the Probability

The probability that a boulder's weight is between 493 kg and 681 kg is the difference between the CDF values for these weights:

\[ P(493 \leq X \leq 681) = \text{CDF}(681) - \text{CDF}(493) = 0.8770 - 0.2358 = 0.6412 \]

Step 4: Convert Probability to Percentage

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

\[ \text{Percentage} = 0.6412 \times 100 = 64.12\% \]