Question

Suppose the selling price of homes is skewed right with a mean of 350,000 and a standard deviation of 160000 If we record the selling price of 40 randomly selected US homes what will be the shape of the distribution of sample means what will be the mean of this distribution what will be the standard deviation of this distribution

Answer 1

From the central limit theorem we know that the distribution for the sample mean
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

The mean would be:

`\mu_(\bar X) =350000`

And the standard deviation would be:

`\sigma_(\bar X) =(160000)/(√(40))= 25298.221`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the selling price of a population, and for this case we know the following info

Where
`\mu=350000` and
`\sigma=160000`

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
`\bar X` is given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

The mean would be:

`\mu_(\bar X) =350000`

And the standard deviation would be:

`\sigma_(\bar X) =(160000)/(√(40))= 25298.221`

Answer 2

The distribution will be approximately normal, with mean 350,000 and standard deviation 25,298.

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Population:

Suppose the selling price of homes is skewed right with a mean of 350,000 and a standard deviation of 160000

Sample of 40

Shape approximately normal

Mean 350000

Standard deviation
`s = (160000)/(√(40)) = 25298`

The distribution will be approximately normal, with mean 350,000 and standard deviation 25,298.