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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 interval from 85 to 115 is exactly one standard deviation below and above the mean, since the mean is 100 and the standard deviation is 15. In a normal distribution, the empirical rule states that approximately 68 percent of observations fall within one standard deviation of the mean, about 95 percent fall within two standard deviations, and about 99.7 percent fall within three standard deviations. Because the range 85 to 115 corresponds to mean 1 standard deviation, the probability of selecting a score in that range is about 68 percent. Among the available options, 68.8 percent is the correct choice and best represents this probability. The other values correspond to wider intervals: 95.4 percent is associated with two standard deviations and 99.7 percent with three. A value of 100 percent would imply every possible score lies in that range, which is not true for a normal distribution. Therefore, the correct answer is 68.8 percent because the question describes the one-standard-deviation interval around the mean.