Question

a sample random sample of size ( n = 64 ) is obtained from a population that is skewed right with ( mu = 88 ) and ( sigma = 32 ).
(a) describe the sampling distribution of ( overline{x} ).
(b) what is ( p(overline{x}>95.4) )?
(c) what is ( p(overline{x}leq78.6) )?
(d) what is ( p(88<overline{x}<97.8) )?
(a) choose the correct description of the shape of the sampling distribution of ( overline{x} ).
a. the distribution is approximately normal.
b. the distribution is uniform.
c. the distribution is skewed left.
d. the distribution is skewed right.
e. the shape of the distribution is unknown.
find the mean and standard deviation of the sampling distribution of ( overline{x} ).
( mu_{overline{x}}=)
( sigma_{overline{x}}=)
(type integers or decimals. do not round.)

Explanation:

Step1: Use Central Limit Theorem

According to the Central Limit Theorem, for a sample of size \(n = 64\) (\(n\geq30\)), the sampling distribution of \(\bar{x}\) is approximately normal. The mean of the sampling distribution \(\mu_{\bar{x}}=\mu\). Given \(\mu = 88\), so \(\mu_{\bar{x}}=88\).

Step2: Calculate the standard deviation of the sampling distribution

The formula for the standard deviation of the sampling distribution (standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 32\) and \(n = 64\), then \(\sigma_{\bar{x}}=\frac{32}{\sqrt{64}}=\frac{32}{8}=4\).

Answer:

\(\mu_{\bar{x}} = 88\)
\(\sigma_{\bar{x}} = 4\)