Question

Suppose a simple random sample of size nequals=6464 is obtained from a population with mu equals 84μ=84 and sigma equals 16σ=16. ​(a) Describe the sampling distribution of x overbarx. ​(b) What is Upper P (x overbar greater than 87.6 )P x>87.6​? ​(c) What is Upper P (x overbar less than or equals 79.2 )P x≤79.2​? ​(d) What is Upper P (81.3 less than x overbar less than 87.6 )P 81.3

Answer 1

Answer:

a. $\bar X$ is distributed $N(84;4)$

b. $P(\bar X \geq 87.6) = 0.03593$

c. $P(\bar X \leq 79.2) = 0.00820$

d. $P(\79.2 \leq \bar X \leq 87.6) = 0.95587$

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 $\µ$ and variance $\frac{\sigma^2}{n}$, then, the sample mean is distributed as a normal random variable with means $\mu_{\bar X}=\mu=84$ and variance $\sigma^2_{\bar X}=\frac{\sigma^2}{n}=\frac{16^2}{64}=4$.

b.

$P(\bar X \geq 87.6) = 0.03593$

c.

$P(\bar X \leq 79.2) = 0.00820$

d.

$P(\79.2 \leq \bar X \leq 87.6) = 0.95587$