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Answer :
Final answer:
By applying the Central Limit Theorem and calculating the Z-score, you can determine the probability of randomly selecting a 20-29 year old man with a weight of 163 pounds or less from a given population.
Explanation:
In order to answer your question, we can employ the Central Limit Theorem, which states that regardless of the original distribution, the sampling distribution of the mean of a large sample drawn with replacement will be approximately normally distributed.
Given the population average (mean) weight of 166 pounds and a standard deviation of 22 pounds, you are seeking the probability of selecting 49 men (N=49) with an average weight of 163 pounds or less.
First, calculate the standard error by dividing the standard deviation by the square root of N (22/ sqrt(49)). This results in a standard error of 3.1429 pounds.
Next, compute the Z score using the formula Z = (X- μ)/σ. In this scenario, X is the weight we are comparing to the mean (163 pounds), μ is the population mean (166 pounds), and σ is the standard error (3.1429 pounds).
Performing the calculation gives us Z = (163 - 166)/3.1429, resulting in Z = -0.954.
Looking up -0.954 in a standard normal Z-table gives you the probability of randomly selecting a man with a weight of 163 pounds or less from this population.
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