The random variable $X$ has mean $25$ and standard deviation $4.5$. A random sample of $500$

Question

The random variable $X$ has mean $25$ and standard deviation $4.5$. A random sample of $500$ observations is taken. Find the value of $a$, correct to $3$ decimal places, such that $\mathrm{P}\left( \overline{X}>a \right)= 0.75$. Explain whether it is necessary to apply the Central Limit Theorem in this case.

Accepted Answer

$a=24.864\text{ (3 d.p.)}$. The Central Limit Theorem is required because the distribution of $X$ is not stated to be normal.

Sampling - Central Limit Theorem

Sampling - Central Limit Theorem

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