Answer:
(a) Yes, the sample size large enough to use the Central Limit Theorem for means.
(b) The mean and standard error of the sampling distribution are $68,000 and $2,800 respectively.
(c) The probability that the sample mean is more than $2800 away from the population mean is 0.1587.
Step-by-step explanation:
According to the Central Limit Theorem if we have a population with mean μ and standard-deviation σ and appropriately huge random samples (<em>n</em> > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.
Then, the mean of the distribution of sample means is given by,
And the standard deviation of the distribution of sample means is given by,
This is also known as the standard error.
Let <em>X</em> = income in a certain region in 2013.
The mean and standard deviation of the random variable <em>X</em> is:
A random sample of size, <em>n</em> = 100 residents are selected from the region.
(a)
The sample selected is quite large, i.e. <em>n</em> = 100 > 30.
The Central limit theorem can be applied to approximate the distribution of the sample mean income in that region.
(b)
Compute the mean of the sampling distribution of sample mean as follows:
Compute the standard error of the sampling distribution of sample mean as follows:
Thus, the mean and standard error of the sampling distribution are $68,000 and $2,800 respectively.
(c)
Compute the probability that the sample mean is more than $2800 away from the population mean as follows:
Thus, the probability that the sample mean is more than $2800 away from the population mean is 0.1587.