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. Given the mean of 100 and a standard deviation of 15, the range of 85 to 115 corresponds to one standard deviation below and above the mean, thus encompassing about 68.30% of the scores.

A (68.30%) CORRECT

This choice accurately reflects the empirical rule of normal distributions, which states that roughly 68% of the data lies within one standard deviation (±15) from the mean (100). Hence, the scores between 85 (100 - 15) and 115 (100 + 15) include around 68.30% of the total data set.

B (95.40%) YOUR ANSWER

This percentage corresponds to the range that lies within two standard deviations from the mean. Specifically, it covers the interval from 70 (100 - 30) to 130 (100 + 30). Therefore, this choice overestimates the range specified in the question.

C (99.70%) YOUR ANSWER

This choice represents the percentage of data that falls within three standard deviations from the mean, covering the interval from 55 (100 - 45) to 145 (100 + 45). Since the question asks only for the range of 85 to 115, this option is not applicable.

D (100%) YOUR ANSWER

This choice suggests that all scores fall within the specified range, which is incorrect for any practical normal distribution. Normal distributions extend infinitely in both directions, so not every score can fall between 85 and 115.

Conclusion

In summary, the probability that a randomly selected vehicle safety score falls between 85 and 115, based on the properties of a normal distribution, is 68.30%. This reflects the empirical rule, confirming that approximately 68% of values lie within one standard deviation of the mean. The other options represent probabilities for broader ranges, thus not relevant to the specified interval in the question.

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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