Answer:
a) X: number of years of education
b) Sample mean = 13.5, Sample standard deviation = 0.4
c) Sample mean = 13.5, Sample standard deviation = 0.2
d) Decrease the sample standard deviation
Step-by-step explanation:
We are given the following in the question:
Mean, μ = 13.5 years
Standard deviation,σ = 2.8 years
a) random variable X
X: number of years of education
Central limit theorem:
If large random samples are drawn from population with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], then the distribution of sample mean will be normally distributed with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex]
b) mean and the standard for a random sample of size 49
[tex]\mu_{\bar{x}} = \mu = 13.5\\\\\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} = \dfrac{2.8}{\sqrt{49}} = 0.4[/tex]
c) mean and the standard for a random sample of size 196
[tex]\mu_{\bar{x}} = \mu = 13.5\\\\\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} = \dfrac{2.8}{\sqrt{196}} = 0.2[/tex]
d) Effect of increasing n
As the sample size increases, the standard error that is the sample standard deviation decreases. Thus, quadrupling sample size will half the standard deviation.