Question

Fill in the blanks to correctly complete the sentence below. Suppose a simple random sample of size n is drawn from a large population with mean mu and standard deviation sigma. The sampling distribution of x overbar has mean mu Subscript x overbarequals​______ and standard deviation sigma Subscript x overbarequals​______.

Answer 1

For this case we select a sample size of n from a large population (n>30) and we know the following properties for the random variable X:

`E(X)= \mu , Sd(x) =\sigma`

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 sampling distribution of x overbar has mean
`\mu_(\bar X) =\mu` and standard deviation
`\sigma_(\bar X)= (\sigma)/(√(n))`.

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".

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".

Solution to the problem

For this case we select a sample size of n from a large population (n>30) and we know the following properties for the random variable X:

`E(X)= \mu , Sd(x) =\sigma`

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 sampling distribution of x overbar has mean
`\mu_(\bar X) =\mu` and standard deviation
`\sigma_(\bar X)= (\sigma)/(√(n))`.