According to a recent​ survey, the population distribution of number of years of education for​ self-employed individuals in a certain region has a mean of 13.5 and a standard deviation of 2.8. a. Identify the random variable X whose distribution is described here. b. Find the mean and the standard deviation of the sampling distribution of x overbar for a random sample of size 49. Interpret them. c. Repeat​ (b) for nequals196. Describe the effect of increasing n.

Respuesta :

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.

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