56. A normally distributed data index of vehicle safety ratings has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected vehicle safety score from the data set will be between 85 and 115?
Answer: A
The probability that a randomly selected vehicle safety score from the data set will be between 85 and 115 is 68.30%.
The range of 85 to 115 encompasses one standard deviation below and above the mean in a normal distribution. In this case, approximately 68.30% of the data lies within one standard deviation of the mean.
A) 68.30%
This option is correct because, according to the empirical rule, about 68.30% of the values in a normal distribution fall within one standard deviation of the mean. Since the mean is 100 and the standard deviation is 15, the interval from 85 (100 - 15) to 115 (100 + 15) captures this percentage of the distribution.
B) 95.40%
This option is incorrect because it represents the percentage of data that falls within two standard deviations from the mean in a normal distribution. The range for this would be from 70 (100 - 2*15) to 130 (100 + 2*15), which is much wider than the specified range of 85 to 115.
C) 99.70%
This option is also incorrect as it refers to the percentage of data that falls within three standard deviations from the mean. This would cover the range from 55 (100 - 3*15) to 145 (100 + 3*15), which exceeds the bounds of 85 to 115.
D) 100%
This option is incorrect because it suggests that 100% of the data falls within the specified range, which is not true for any range in a normal distribution. In reality, there will always be some data points that fall outside of any finite range.
Conclusion
The correct answer, 68.30%, accurately reflects the portion of data within one standard deviation from the mean in a normal distribution. The other options either refer to wider ranges or do not apply to the specified interval, thereby confirming that they are not suitable answers.