CHAPTER 8 SAMPLING DISTRIBUTIONS8.1 Distribution of the Sample Mean

Obj: Describe the distribution of a sample mean from normal and not normal populations

A SAMPLING DISTRIBUTION

Let the students in this class be the population. List the number of siblings that each of you have.

List each sample of size 2.

Find each sample mean.

Show the sampling distribution of the sample mean.

LAW OF LARGE NUMBERS

As additional observations are added to the sample, the difference between the sample mean, x, and the population mean μ approaches zero.

MEAN AND STANDARD DEVIATION OF THE SAMPLING DISTRIBUTION

If a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ, then the sampling distribution of x will have mean μx = μ and standard deviation σx = .

The standard deviation of the sampling distribution of x is called the standard error of the mean and is denoted σx.

n

CENTRAL LIMIT THEOREM

Regardless of the shape of the population, the sampling distribution of x becomes approximately normal as the sample size n increases.

If the population is normally distributed, the sample will be normally distributed. If the population is not normal or if it is not known, then a sample size of greater than or equal to 30 is required.

EXAMPLE

Suppose a simple random sample of size n = 49 is obtained from a population with μ = 80 and σ = 14.

a. Describe the sampling distribution of x.

b. What is P(X > 83)?

c. What is P(X < 75.8)?

d What is P(78.3 < X < 85.1)?

YOUR TURN Old Faithful has a mean time between

eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, A) What is the probability that a randomly

selected time interval is longer than 95 minutes? B) What is the probability that a random sample

of 20 time intervals between eruptions has mean longer than 95 minutes?

C) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?

D) What effect does increasing the sample size have on the probability?

ASSIGNMENT

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