Use the Empirical Rule to answer the question.A highway study of 6,000 vehicles that passed by a checkpoint found that their speeds were normally distributed, with a mean of 66 mph and a standard deviation of 7 mph.(a)How many of the vehicles had a speed of more than 73 mph? vehicles(b)How many of the vehicles had a speed of less than 45 mph?
Question
Use the Empirical Rule to answer the question.A highway study of 6,000 vehicles that passed by a checkpoint found that their speeds were normally distributed, with a mean of 66 mph and a standard deviation of 7 mph.(a)How many of the vehicles had a speed of more than 73 mph? vehicles(b)How many of the vehicles had a speed of less than 45 mph?
Solution
The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% falls within two standard deviations.
- About 99.7% falls within three standard deviations.
(a) The question asks for the number of vehicles with speeds more than 73 mph. The mean speed is 66 mph and the standard deviation is 7 mph. So, 73 mph is one standard deviation above the mean. According to the Empirical Rule, about 68% of the data falls within one standard deviation of the mean, which means about 32% falls outside this range. Since we're looking for the number of vehicles with speeds more than 73 mph, we're interested in the upper half of this 32%, which is 16%.
So, 16% of 6000 vehicles is 960 vehicles.
(b) The question asks for the number of vehicles with speeds less than 45 mph. This is more than three standard deviations below the mean (66 - 3*7 = 45). According to the Empirical Rule, about 99.7% of the data falls within three standard deviations of the mean, which means about 0.3% falls outside this range. Since we're looking for the number of vehicles with speeds less than 45 mph, we're interested in the lower half of this 0.3%, which is 0.15%.
So, 0.15% of 6000 vehicles is 9 vehicles.
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