Respuesta :
Answer:
a) Yes
b) Yes
c) Yes
d) 0.6
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.
A large population has skewed data with a mean of 70 and a standard deviation of 6.
This means that [tex]\mu = 70, \sigma = 6[/tex]
Samples of size 100
This means that [tex]n = 100[/tex]
a) Will the distribution of the means be closer to a normal distribution than the distribution of the population?
According to the Central Limit Theorem, yes.
b) Will the mean of the means of the samples remain close to 70?
According to the Central Limit Theorem, yes.
c) Will the distribution of the means have a smaller standard deviation?
According to the Central Limit Theorem, the standard deviation of the population is divided by the sample size, so yes.
d) What is that standard deviation?
[tex]s = \frac{\sigma}{\sqrt{n}} = \frac{6}{\sqrt{100}} = \frac{6}{10} = 0.6[/tex]
So 0.6.
