P Stats. Chapter 7 Sample Means.

Define: Distribution of sample means

The distribution of sample means consists of the sample means for all the possible random samples of a specific size (n) from a specific population.

Define: Expected value of M.

s the mean of the distribution of sample means, μ

Central Limit Theorem

For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity

All the possible random samples of size n =2 are selected from a population with μ = 40 and σ =25 and the mean is computed for each sample. Then all the possible samples of size are selected from the same population and the mean is computed for each sample. How will the distribution of sample means for compare with the distribution for ?

The variance for n=25 will be smaller than the variance for n=2 but the two distributions will have the same mean.

distribution of sample means

is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.

sampling error

is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.

If all the possible random samples, each with scores, are selected from a normal population with μ = 80 and σ= 18, and the mean is calculated for each sample, then what is the average value for all of the sample means?

80. Population mean and sampling mean are the same.

sampling distribution

is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

Define: Central limit theorem

specifies the basic characteristics of the distribution of sample means for any size samples from any population. Specifically, the shape will approach a normal distribution as the sample size increases, the mean is equal to the population mean, and the standard deviation (standard error) equals the population standard deviation divided by the square root of the sample size.

law of large numbers

states that the larger the sample size, n, the more probable it is that the sample mean will be close to the population mean.

expected value of M

the mean of the distribution of sample means is equal to the mean of the population of scores, μ.

Define: Standard Error of M

the standard deviation of the distribution of sample means. σ(M) = σ/√n

Describe the distribution of sample means (shape, mean, standard error) for samples of n = 64 selected from a population with a mean of μ = 90 and a standard deviation of σ = 32 .

The distribution will be normal because n > 30, with an expected value μ = 90 and a standard error of 32 / √64 = 4

If all the possible random samples of size are selected from a population with μ = 80 and σ =10 and the mean is computed for each sample, then what shape is expected for the distribution of sample means?

The sample means tend to form a normal-shaped distribution.

Standard error of M

The standard deviation of the distribution of sample means. he standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean μ.

Characteristics of the Distribution of Sample Means

1. The same means should pile up around the population mean. 2. The pile of same means tends to form a normal-shaped distribution. 3. The larger the sample size, the closer the sample means should be to the population mean, μ. .