Answer:
a) The mean is 0.8.
b) The standard deviation is 0.0365.
c) Normal
Step-by-step explanation:
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
80% of Americans over the age of 25 have earned a high school diploma.
This means that [tex]p = 0.8[/tex]
Suppose we take a random sample of 120 Americans
This means that [tex]n = 120[/tex].
Question a:
The mean is:
[tex]\mu = p = 0.8[/tex]
The mean is 0.8.
Question b:
The standard deviation is:
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.8*0.2}{120}} = 0.0365[/tex]
The standard deviation is 0.0365.
Question c:
By the Central Limit Theorem, approximately normal.
