Elementary Statistics Chapter 8 Section 8.1/2

Sampling Distribution

is a probability distribution for all possible values of the static computed from a sample of size n.

Suppose a simple random sample of size n is drawn from a large population with mean "μ" and standard deviation "σ". The sample distribution of "x overbarx" has mean "μ Subscript x overbarx" =​______ and standard deviation σ Subscript x overbarσx =​______.

"μ", "σ overbar square root of n" The sample mean matches the population​ mean, but the standard error of the mean depends on the sample size. Notice that as "n​" increases, sigma x over barσx decreases.

A simple random sample of size n=64 is obtained from a population with μ=64 and σ=44. Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? What is the sampling distribution of x overbar? What is the sampling distribution of x overbar​?

No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x overbarx becomes approximately normal as the sample​ size, n, increases. The sampling distribution of x overbar is normal or approximately normal with μ Subscript x overbar = 64 and σ Subscript x overbar = 0.5.

The standard deviation of the sampling distribution of x overbarx​, denoted σ Subscript x overbar​, is called the standard error of the mean.

The sampling distribution of the sample mean "x overbarx" is the probability distribution of all possible values of the random variable "x overbarx" computed from a sample of size n from a population with mean muμ and standard deviation σ. Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of "x overbarx" will have mean "μ Subscript x overbar = μ" and standard deviation σ Subscript x overbar = σ Over Square root of n. The standard deviation of the sampling distribution of "x overbarx​", σ Subscript x overbar, is called the standard error of the mean.

Is the statement below true or​ false? The distribution of the sample​ mean, x overbar, will be normally distributed if the sample is obtained from a population that is normally​ distributed, regardless of the sample size.

The statement is true. The sampling distribution of the sample mean x overbarx is the probability distribution of all possible values of the random variable x overbarx computed from a sample of size n from a population with mean muμ and standard deviation sigmaσ. Suppose that a simple random sample of size n is drawn from a large population with mean muμ and standard deviation sigmaσ. The sampling distribution of x overbarx will have mean mu Subscript x overbarμxequals=muμ and standard deviation sigma Subscript x overbar Baseline equals StartFraction sigma Over StartRoot n EndRoot EndFractionσx= σ n. If a random variable X is normally​ distributed, the distribution of the sample​ mean, x overbarx​, is normally distributed.

To cut the standard error of the mean in​ half, the sample size must be increased by a factor of four.

To cut the standard error of the mean in​ half, the sample size must be increased by a factor of four.

Sampling Distribution of the Sample Mean

X is the probability distribution of all possible values of the random variable X computed from a sample of size n from a population with mean "μ" and standard deviation "σ".