Answer :
To find the minimum weight of the middle 95% of the players, we need to find the corresponding z-scores for the 2.5th and 97.5th percentiles of the normal distribution.
Using a standard normal distribution table or calculator, we find that the z-score for the 2.5th percentile is -1.96 and the z-score for the 97.5th percentile is 1.96.
Then, we can use the formula:
z = (x - mean) / standard deviation
Rearranging this formula to solve for x, we get:
x = z * standard deviation + mean
Substituting in the values for z, standard deviation, and mean, we get:
x = (-1.96)(25) + 200 = 151
and
x = (1.96)(25) + 200 = 249
Therefore, the minimum weight of the middle 95% of the players is between 151 and 249 pounds.
In summary, we used the normal distribution, z-scores, and the formula for converting z-scores to raw scores to determine the minimum weight of the middle 95% of football players with a normally distributed weight distribution. By finding the z-scores for the 2.5th and 97.5th percentiles and using the formula x = z * standard deviation + mean, we calculated that the minimum weight is between 151 and 249 pounds. The concept of deviation was also used to determine how far away from the mean the data points are in terms of standard deviations, which allowed us to use the z-scores to find the raw scores. This is a useful statistical technique for understanding and analyzing data that follows a normal distribution.
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