Question

Chang, 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, Chang 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%). % Х ?

Answer 1

To find the percentage of boulders that weigh between 245 kg and 449 kg according to the model, standardize the weights using the z-score formula and use a standard normal distribution table or calculator to find the percentage.

Step-by-step explanation:

To find the percentage of boulders that weigh between 245 kg and 449 kg according to the model, we need to find the area under the normal distribution curve between these two weights. First, we need to standardize the weights using the formula z = (x - mean) / standard deviation. For 245 kg, the z-score is (245 - 563) / 150 = -2.120. For 449 kg, the z-score is (449 - 563) / 150 = -0.760. We can then use a standard normal distribution table or a calculator to find the percentage between these z-scores. Using the table, we find that the percentage is 0.2257 or 22.57%.

Answer 2

About 20.66% of the boulders would weigh between 245 kg and 449 kg according to the normal distribution model.

How do we find the percentage of boulders?

The mean μ is 563 kg

The standard deviation σ is 150 kg

Standardized values for 245 kg and 449 kg are needed

(245 - 563)/150 = -2.12
(449 - 563)/150 = -0.76

The standardized values (z-scores) for the weights are approximately -2.12 for 245 kg and -0.76 for 449 kg.

Cumulative probability for z 245 kg which is -2.12 = 0.0170

Cumulative probability for z 449 kg which is -0.76 = 0.2236

0.2236 - 0.0170 = 0.2066

It can be said that about 20.66% of the boulders would weigh between 245 kg and 449 kg.