Running times for 400 meters are Normally distributed for young men between 18 and 30 years of age with a mean of 93 seconds and a standard deviation of 16 seconds. Thus, 99.7% of running times are approximately between
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45 and 141 seconds
you can play around with Z table stuff here:
http://davidmlane.com/hyperstat/z_table.html
For this problem, you find that 99.7% of the area lies less than 2.748 ? above the mean.
To find the range within which 99.7% of the running times fall, we need to calculate the z-scores associated with the percentiles.
First, we need to find the z-score corresponding to the lower tail. Since we want to include 99.7% of the data, the lower tail will be (100% - 99.7%) / 2 = 0.15%.
Using a standard normal distribution table or a z-score calculator, we find that the z-score for a lower tail of 0.15% is -3.00 (approximately).
Next, we calculate the z-score corresponding to the upper tail. Since we want to include 99.7% of the data, the upper tail will also be 0.15%.
Using the same z-score table or calculator, we find that the z-score for an upper tail of 0.15% is +3.00 (approximately).
Now we can calculate the actual running time values associated with these z-scores using the formula:
x = μ + (z * σ)
where:
x = running time
μ = mean running time = 93 seconds
z = z-score
σ = standard deviation = 16 seconds
For the lower tail:
x_lower = 93 + (-3 * 16)
= 93 - 48
= 45 seconds (approximately)
For the upper tail:
x_upper = 93 + (3 * 16)
= 93 + 48
= 141 seconds (approximately)
Therefore, approximately 99.7% of running times are between 45 seconds and 141 seconds.
To find the running times that represent 99.7% of the data, you will need to calculate the range within which the data falls.
First, consider that a normal distribution is symmetric, which means the mean is the middle point of the distribution. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
Given that the mean is 93 seconds and the standard deviation is 16 seconds for the running times, you can calculate the range as follows:
1. Calculate the lower bound: Mean - (3 * Standard Deviation)
Lower Bound = 93 - (3 * 16) = 93 - 48 = 45 seconds
2. Calculate the upper bound: Mean + (3 * Standard Deviation)
Upper Bound = 93 + (3 * 16) = 93 + 48 = 141 seconds
Therefore, approximately 99.7% of the running times will fall between 45 seconds and 141 seconds.