Question 5

The probability that a randomly selected vehicle safety score from the data set will be between 85 and 115 is 68.30%.

This probability corresponds to the empirical rule in statistics, which states that approximately 68% of data points in a normal distribution fall within one standard deviation of the mean. In this case, the mean is 100, and one standard deviation is 15, making the range from 85 to 115.

A) 68.30%

This option is correct because it accurately represents the probability of a data point falling within one standard deviation of the mean in a normal distribution. Since the mean is 100 and the standard deviation is 15, the range of 85 to 115 encompasses approximately 68% of the data.

B) 95.40%

This option is incorrect as it represents the probability of a data point falling within two standard deviations of the mean in a normal distribution. The range for two standard deviations would be from 70 to 130, which is broader than the specified range of 85 to 115.

C) 99.70%

This option is also incorrect. It indicates the probability of a data point falling within three standard deviations of the mean. The range for three standard deviations would be from 55 to 145, which includes a much larger range than the 85 to 115 interval.

D) 100%

This option is incorrect because it suggests that every data point falls within the specified range, which is not possible in a normal distribution. While most data points are present within the range of three standard deviations, not all will fall within one standard deviation.

Conclusion

The correct answer is 68.30% because it reflects the empirical rule for a normal distribution, which states that about 68% of data lies within one standard deviation of the mean. All other options fail because they refer to probabilities associated with wider ranges of data points, specifically those falling within two or three standard deviations, or they incorrectly assert that all data falls within the specified range.