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Answer :
The Central Limit Theorem tells us that for a population with any distribution, the distribution of the
sample means approaches a normal distribution as the sample size increases. The procedure in this
section forms the foundation for estimating population parameters and hypothesis testing.
From the given data we have,
Mean = 172 pounds
Standard Deviation = 30 pounds
Probability through the normal distribution is expresses as :
[tex]\begin{gathered} z=\frac{x-\mu}{\sigma} \\ \text{ where }\mu=\text{ mean \& }\sigma=\text{ Stanadard Deviation} \end{gathered}[/tex]a)
Probability for weighs over 200 pounds
x = 200
[tex]\begin{gathered} z=\frac{200-172}{30} \\ z=0.933 \end{gathered}[/tex]b) The probability that 30 randomly selected men have an average weight over 200 pounds is expresses as :
[tex]\begin{gathered} z=\frac{x-\mu}{\frac{\sigma}{n}} \\ \text{ where }\mu=\text{ mean, n = number \& }\sigma=\text{ Stanadard Deviation} \end{gathered}[/tex]So, here n =30
[tex]\begin{gathered} z=\frac{200-172}{\frac{30}{\sqrt[]{30}}} \\ z=\frac{28}{5.477} \\ z=5.11 \end{gathered}[/tex]Answer : z=5.11
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