Chapter 10 – Sampling Distributions

Math 22

Introductory Statistics

The Sampling Distribution of a Sample Statistic The sampling distribution of a

sample statistic is the distribution of values for a sample statistic obtained from repeated samples.

The Sampling Distribution of the Sample Mean

Let be the mean of a sample of size n from any population that has mean and standard deviation .

For all sample sizes n, the sampling distribution of the sample mean:

Is exactly normally distributed.

x

The Sampling Distribution of the Sample Mean Is centered at , the mean of the

population. Has a standard deviation of ,

where is the standard deviation of the population.

Note: We don’t know what the true population mean and population standard deviation are.

n/

Central Limit Theorem (CLT)

The sampling distribution of sample means will become normal as the sample size increases.

The Probability of the Sample Mean Old z score

Note: This z score is used to calculate the probability of a single observation.

xx

zx

x

The Probability of the Sample Mean New z score

Note: This z score is used in calculating the probability of the sample mean.

n

xxz

x

x

/

The Sampling Distribution of the Sample Proportion

CLT Applied to the Sample Proportion.If the sample size n, is sufficiently large (both and are at least 5), then the sampling distribution of the sample proportion:

Is approximately normally distributed Is centered at , the true proportion of

success in the Bernoulli population. Has a standard deviation of

( )1 n

n )1( n