Answer:
a. [tex]\bar X[/tex] is distributed [tex]N(84;4)[/tex]
b. [tex]P(\bar X \geq 87.6) = 0.03593[/tex]
c. [tex]P(\bar X \leq 79.2) = 0.00820[/tex]
d. [tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]
Step-by-step explanation:
a.
The central limit theorem states that, for large n, the sampling distribution of the sample mean is approximately normal with mean [tex]\µ[/tex] and variance [tex]\frac{\sigma^2}{n}[/tex], then, the sample mean is distributed as a normal random variable with means [tex]\mu_{\bar X}=\mu=84[/tex] and variance [tex]\sigma^2_{\bar X}=\frac{\sigma^2}{n}=\frac{16^2}{64}=4[/tex].
b.
[tex]P(\bar X \geq 87.6) = 0.03593[/tex]
c.
[tex]P(\bar X \leq 79.2) = 0.00820[/tex]
d.
[tex]P(\79.2 \leq \bar X \leq 87.6) = 0.95587[/tex]
