The National Highway Traffic Safety Administration analyzes the crash test data for new cars. One variable measured is the severity of the injury due to a collision. The higher the severity, the higher the points assigned. Assume that the injury rating is normally distributed with a mean (μ) of 605 points and a standard deviation (σ) of 185 points. One of the crash-test cars is randomly selected from the data and its rating is observed.

Find the probability that the rating will fall between 500 and 700 points.

Use z-scores.

Formula:

z = (x - mean)/sd

Find two z-scores, using 500 for x and also 700 for x. Mean = 605 and sd = 185.

Once you find the two z-scores, use a z-distribution table to determine your probability between the two z-scores.

.4987

.4988

To calculate the probability that the rating will fall between 500 and 700 points, we need to use the normal distribution and its properties.

First, let's standardize the values by converting them to z-scores. The z-score formula is (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

For 500 points:
z-score = (500 - 605) / 185 = -0.568

For 700 points:
z-score = (700 - 605) / 185 = 0.514

Now, we can use a standard normal distribution table or calculator to find the probability associated with these z-scores.

Using a standard normal distribution table or calculator, we can find the area to the left of -0.568 and the area to the left of 0.514.

The area from -∞ to -0.568 is approximately 0.2846.
The area from -∞ to 0.514 is approximately 0.6965.

Since we want the probability between 500 and 700 points, we subtract the smaller area from the larger area:

0.6965 - 0.2846 = 0.4119

Therefore, the probability that the rating will fall between 500 and 700 points is approximately 0.4119 or 41.19%.