A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 50 seconds. Between what times do we expect most (approximately 95%. of the boys to run the mile?
To find the range of times during which we can expect approximately 95% of the boys to run the mile, we need to determine the interval that covers the middle 95% of the distribution.
Since the distribution is normal, we can use the Empirical Rule, also known as the 68-95-99.7 rule. According to this rule, in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean
- Approximately 95% of the data falls within two standard deviations of the mean
- Approximately 99.7% of the data falls within three standard deviations of the mean
In this case, we want to find the time range for the middle 95% of the boys, so we will consider two standard deviations from the mean.
Step 1: Calculate the range
Lower Range = Mean - (2 * Standard Deviation)
Upper Range = Mean + (2 * Standard Deviation)
Lower Range = 460 - (2 * 50) = 460 - 100 = 360
Upper Range = 460 + (2 * 50) = 460 + 100 = 560
So we expect most (approximately 95%) of the boys to run the mile between 360 seconds to 560 seconds.