Answer:
The mean total taxable earnings of all wage earners in a county is same as the population mean, $28.29.
The probability that the mean taxable wages in James' sample of 56 counties will be less than $33 million is 0.8508.
The probability that the mean taxable wages in James' sample of 56 counties will be greater than $30 million is 0.3520.
Step-by-step explanation:
Let <em>X</em> = total taxable earnings of workers.
The expected value of the random variable <em>X</em> is:
E (X) = <em>μ</em> = $28.29
The standard deviation of the random variable <em>X</em> is:
SD (X) = <em>σ</em> = $33.493.
The data was collected from 56 American counties.
The sample mean of a random variable is the an unbiased estimator of the population mean.
If repeated samples are collected from a population and the mean for each sample is computed then the expected value of the sample means is same as the population mean.
So the mean total taxable earnings of all wage earners in a county is same as the population mean, $28.29.
According to the Central limit theorem if we have a population with mean μ and standard deviation σ and take appropriately huge random-samples (n ≥ 30) from the population with replacement, then the distribution of the sample-means will be approximately normally distributed.
Then, the mean of the sample means is given by, .
And the standard deviation of sample means is,
Compute the value of P (\bar X < 33) as follows:
Thus, the probability that the mean taxable wages in James' sample of 56 counties will be less than $33 million is 0.8508.
Compute the value of P (\bar X > 30) as follows:
Thus, the probability that the mean taxable wages in James' sample of 56 counties will be greater than $30 million is 0.3520.