Answer :
Final answer:
About 8 men in a group of 329 weigh less than 128 pounds, about 52 men weigh more than 206 pounds, and approximately 314 men would weigh within 2 standard deviations of the mean.
Explanation:
This problem requires you to use the rules of a normal distribution, where the mean (average) is 180 pounds and the standard deviation is 26 pounds.
A. To find how many men had a weight of less than 128, you need to first calculate the z-score, which is the number of standard deviations a data point is from the mean. The z-score for 128 pounds is (128-180)/26 = -2. Therefore, we are looking for the proportion of men who fall below 2 standard deviations under the mean. According to a standard normal distribution table, about 2.28% men will weigh less than 128 pounds. So, in a group of 329 men, about 0.0228*329 = approximately 8 men would weigh less than 128 pounds.
B. Similarly, for those who weigh more than 206 pounds, we calculate the z-score as (206-180)/26 = 1. Hence we're looking for the proportion of men who fall above 1 standard deviation above the mean. The standard table shows that this area is about 15.87% or 0.1587. So, in a group of 329 men, around 0.1587*329 = about 52 men would weigh more than 206 pounds.
C. Those who weigh within 2 standard deviations of the mean are those who are within the range of (180 - 2*26) and (180 + 2*26), or between 128 and 232 pounds. The normal distribution table shows that about 95.44% or 0.9544 men fall within this range. So, in a group of 329 men, about 0.9544 * 329 = approximately 314 men would weigh within 2 standard deviations of the mean.
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