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# Normal distribution probability - Central Limit Theorem

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Assume men's weights are normally distributed w/ a mean of 172 lb. and a s.d. of 29lb.

(a) If one man randomly selected, find the probability his weight is between 166 and 176 lb.

( b) If 16 men selected, find the probability that the mean weight is between 166 and 176 lb.

https://brainmass.com/statistics/central-limit-theorem/normal-distribution-probability-central-limit-theorem-26785

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This solution shows how to find the probability of an event related to a normal distribution and how to apply the central limit theorem to find the probability of an event related to a sample mean. Please see the attached Word document for details of how to solve this exercise.
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Assume men's weights are normally distributed w/ a mean of 172 lb. and a s.d. of 29lb.
If one man randomly selected, find P his weight is between 166 and 176 lb. Answer = .1389
b) If 16 men selected, find P mean weight between 166 and 176 lb. Answer = .5055

Recall what the Central Limit Theorem Tells us. It says that if you take a sample of size = n from a population with mean = and standard deviation of , then the sample mean, , is a random variable and has mean = and standard deviation = .
Solution:

The Central Limit Theorem helps us differentiate between how to handle X, a single observation taken from a distribution, and , the sample mean of n observations taken from that distribution.

If a single observation is taken from a Normal distribution with mean = and standard deviation = , you make inferences about X, one observation from that distribution by using .

If a sample of n observations is taken from that same distribution and you want to make inferences about , the mean of those n observations, by using .

(a)
The first question is: Find
Solve it by finding the Z scores using the distribution of X.

(b)
The second question is: Find .
Solve it by finding the Z scores using the distribution of .

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