Chapter 7

What constitutes as an extreme value in a normal distribution?

Any score that is z = -2 or +2

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.

For a population with of m = 40 and a standard deviation of s = 8, the standard error for a sample mean can never be larger than 8. (True or false.)

True. If each sample has n 5 1 score, then the standard error is 8. For any other sample size, the standard error is smaller than 8.

requirements for the standard error.

a. As sample size (n) increases, the size of the standard error decreases. (Larger samples are more accurate.) b. When the sample consists of a single score (n 5 1), the standard error is the same as the standard deviation (s M 5 s).

distribution of sample means

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

when n = 1, the standard error = oM is identical to the standard deviation = o. true or false

true

Characteristics of constructing a distribution of sample means?

1. The sample means should pile up around the population mean. Samples are not expected to be perfect but they are representative of the population. As a result, most of the sample means should be relatively close to the population mean. 2. The pile of sample means should tend to form a normal-shaped distribution. Logically, most of the samples should have means close to m, and it should be relatively rare to find sample means that are substantially different from m. As a result, the sample means should pile up in the center of the distribution (around m) and the frequencies should taper off as the distance between M and m increases. This describes a normal-shaped distribution. 3. In general, the larger the sample size, the closer the sample means should be to the population mean, m. Logically, a large sample should be a better representative than a small sample. Thus, the sample means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered.

The distribution of sample means is not always a normal distribution. Under what circumstances will the distribution of sample means not be normal

If the population from which the samples were selected was not normal or if outliers exist in a few or more sample

Shape of distribution of sample means

Most time perfectly normal especially of these conditions are satisfied 1. The population from which the samples are selected is a normal distribution. 2. The number of scores (n) in each sample is relatively large, around 30 or more.

Do you need to know all the possible sample means to find the probability of any specific sample mean?

YES

A population has a mean of m = 65 and a standard deviation of s = 16. a. Describe the distribution of sample means (shape, central tendency, and variability) for samples of size n = 4 selected from this population. b. Describe the distribution of sample means (shape, central tendency, and variability) for samples of size n = 64 selected from this population.

a The distribution of sample means has an expected value of m = 65 and a standard error of OM = 16/ Square root4 =8. The shape of the distribution is unknown because the sample is not large enough to guarantee a normal distribution and the population shape is not known. b. The distribution of sample means has an expected value of m = 65 and a standard error of OM = 16 square root64 = 2. The shape of the distribution is normal bc the sample is large enough

sampling distribution

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

inverse relationship between the sample size and the standard error

bigger samples have smaller error, and smaller samples have bigger error.

standard error of M

discrepancy, or error, between a sample statistic and the corresponding population parameter.

An example of sampling distribution?

distribution of sample means

Standard error defines the relationship between _____ and the accuracy with which M (sample mean) represent U (population mean)

sample size

Describe the relationship between the sample size and the standard error of M.

the larger the sample the smaller the standard error of M is