Question

Jan 12,8:03:54PM When Brooklyn runs the 400 meter dash, her finishing times are normally distributed with a mean of 76 seconds and a standard deviation of 3 seconds. Using the empirical rule, what percentage of races will her finishing time be between 70 and 82 seconds?

Answer 1

Using the empirical rule for normal distribution, it is estimated that 95% of Brooklyn's races will have finishing times between 70 and 82 seconds, which is within two standard deviations from her mean time of 76 seconds.

Step-by-step explanation:

The question pertains to the normal distribution and the application of the empirical rule (also known as the 68-95-99.7 rule) which states that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% falls within two standard deviations.
• Approximately 99.7% falls within three standard deviations.

In Brooklyn's case, her mean finishing time is 76 seconds, and the standard deviation is 3 seconds. Therefore:

1. One standard deviation from the mean is the interval from 73 to 79 seconds (this covers 68% of her races).
2. Two standard deviations from the mean is the interval from 70 to 82 seconds (this covers 95% of her races).
3. Three standard deviations from the mean would be from 67 to 85 seconds (covering 99.7% of her races).

Given the question, we're interested in the percentage of races with times between 70 and 82 seconds, which falls within two standard deviations. Using the empirical rule, it is estimated that 95% of her races will have finishing times in this range.