Question

Many studies on automobile safety suggest that when drivers of automobiles need to make emergency stops, the stopping distances follow an approximately normal distribution. Suppose that for one pickup truck traveling at 62 mph under typical conditions on dry pavement, the mean stopping distance is μ = 155 ft, with a standard deviation of σ = 3 ft. Sketch the probability distribution of X = the stopping distance for a randomly selected emergency stop. Label the mean and the points that are 1, 2, and 3 standard deviations from the mean.

Answer 1

To visualize the normal distribution of stopping distances for a pickup truck traveling at 62 mph, draw a symmetric curve around the mean of 155 ft, labeling points at one, two, and three standard deviations away. These represent common probabilities of occurrence according to the empirical rule.

Step-by-step explanation:

The question pertains to the normal distribution of stopping distances for a pickup truck traveling at 62 mph on dry pavement. The mean stopping distance is given as μ = 155 ft, and the standard deviation is σ = 3 ft. To sketch the probability distribution of X, the stopping distance for a randomly selected emergency stop, we would draw a symmetrical curve centered at the mean, 155 ft.

Label the mean on the curve. Then, label additional points to the right and left of the mean, representing one, two, and three standard deviations from the mean. These points would be at 158 ft (155 + 3), 161 ft (155 + 2*3), and 164 ft (155 + 3*3) to the right of the mean, and at 152 ft (155 - 3), 149 ft (155 - 2*3), and 146 ft (155 - 3*3) to the left of the mean.

Based on the empirical rule, approximately 68% of the stopping distances would fall within one standard deviation of the mean (± 1σ), 95% within two standard deviations (± 2σ), and 99.7% within three standard deviations (± 3σ). This implies that most emergency stops will result in stopping distances close to the mean with a lower probability of extreme values.