Respuesta :
Answer:
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
The mean is given by:
[tex] \bar X = 13.6[/tex]
And the deviation is given by:
[tex]\sigma_{\bar X} =\frac{3}{\sqrt{100}}= 0.3[/tex]
Step-by-step explanation:
For this case we define the random variable X as "number of years of education of self-employed individuals in the U.S." and we know the following properties:
[tex] E(X) = 13.6 , Sd(X) = 3[/tex]
And we select a sample of n = 100
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
Solution to the problem
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
The mean is given by:
[tex] \bar X = 13.6[/tex]
And the deviation is given by:
[tex]\sigma_{\bar X} =\frac{3}{\sqrt{100}}= 0.3[/tex]
Answer:
a) Mean of the sampling distribution = 13.6 years
b) Standard deviation of the sampling distribution = 0.3
[tex]\bar {x} = N(13.6, 0.3)[/tex]
Step-by-step explanation:
Population mean, [tex]\mu = 13.6 years[/tex]
Population standard deviation, [tex]\sigma = 3.0 years[/tex]
Sample size, n = 100
a) Mean of the sampling distribution = mean of the normal distribution
[tex]\mu_{s} = \mu\\\mu_{s} = 13.6 years[/tex]
b) Standard deviation of the sampling distribution, [tex]\sigma_{s} = \frac{\sigma}{\sqrt{n} }[/tex]
[tex]\sigma_{s} = \frac{3}{\sqrt{100} } \\\sigma_{s} = \frac{3}{10} \\\sigma_{s} = 0.3[/tex]
