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?

Rationale

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

In a normal distribution, approximately 68.30% of the data falls within one standard deviation of the mean. Since the mean is 100 and the standard deviation is 15, the range from 85 (100 - 15) to 115 (100 + 15) encompasses this central portion of the distribution.

A (68.30%) CORRECT

This choice accurately reflects the empirical rule for a normal distribution, which states that about 68.30% of observations lie within one standard deviation from the mean. In this case, the interval from 85 to 115 corresponds perfectly to one standard deviation below and above the mean of 100.

B (95.40%) YOUR ANSWER

This choice represents the percentage of data that falls within two standard deviations from the mean in a normal distribution. The range would be from 70 (100 - 2*15) to 130 (100 + 2*15), which is broader than the specified interval of 85 to 115.

C (99.70%) YOUR ANSWER

This choice corresponds to the percentage of data within three standard deviations from the mean. For this distribution, that would encompass the range from 55 (100 - 3*15) to 145 (100 + 3*15), which is much wider than the range specified in the question.

D (100%) YOUR ANSWER

This choice implies that all data points fall within the specified range, which is incorrect. In a normal distribution, it is impossible to capture 100% of the data within any finite range, as there will always be some probability in the tails of the distribution.

Conclusion

The empirical rule for normal distributions provides a clear framework for understanding probabilities associated with ranges of data. In this scenario, the correct probability of a randomly selected vehicle safety score falling between 85 and 115 is 68.30%, as it aligns with the expected distribution of values within one standard deviation of the mean. The other options reflect broader ranges that do not pertain to the specified interval.

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?

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

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