Inferential Statistics

Population Curve Mean Mean Group of 30

Population Curve Mean Computing the means of randomly selected groups of 30 Theoretically, a new frequency distribution will appear representing the means of groups of 30.

Sampling Distribution of the Mean Population Mean Population Distribution Sampling Distribution Mean (n = 30) Sampling Distribution for random groups of 30

Sample Sizes As the sample size increases the Sampling Distribution of the Mean gets narrower. Population Distribution n=30 n=100 n=1000 If the sample size was really large then it would be unlikely to get sample group means that were very far from the population mean. If the sample size was really small then sample group means would be distributed similarly to the population.

Standard Error of the Mean The standard deviation of a Sampling Distribution is called the Standard Error. Population Distribution n=30 n=100 n=1000 For any distribution the larger the sample size the smaller the numerical standard error.

Standard Error Sampling Distribution Mean Sampling Distribution (n = 30) 1 Standard Error The relationship of standard error to a sampling distribution is the same as standard deviation to a normal distribution

Standard Error Sampling Distribution Mean Sampling Distribution (n = 30) 1 Standard Error Remember this is not a distribution of scores on the test. It is a distribution of the means of randomly selected groups of

Measuring Group Means Against the Sampling Distribution Sampling Distribution (n = 30) 1 Standard Error At any given Standard Error it is possible to compute the likelihood that a higher or lower group mean could have occurred by chance (remember the relationship of z scores to percentile rank) Probability of higher mean 15.87% (.16)

Measuring Group Means Against the Sampling Distribution Sampling Distribution (n=30) 2 Standard Errors At any given Standard Error it is possible to compute the likelihood that a higher or lower group mean could have occurred by chance (remember the relationship of z scores to percentile rank) Probability of higher mean % (.02)

Probability How unlikely does the occurrence of a group mean have to be before we would say that it is so unlikely that it couldn’t have happened by chance? It must have occurred for some other reason.

Significance If the probability that a given group mean would occur by chance in a sampling distribution is very small then the occurrence of that group mean is said to be significant. Probability of higher mean

Significance To use the z-score analogy when the percentile ranking of a group mean is really high (or really low) then it is significant. It is significant because it is unlikely to occur randomly. Probability of higher mean

Significance Most social science research declares that group means occurring by chance less than 5% of the time are significant. Most medical research uses 1% or much less. 1% (.01) 5% (.05)

Our pre/post test example. 1% (.01) 5% (.05) Pre-test mean Post-test mean Probability (p) that this score or higher could have appeared by chance is.010 (1% or 1 in 100 times) substantially less than 0.5 (5 % or 1 in 20 times) With a z-score we used a table to look up percentile rank. Since the normal distribution in now a sampling distribution based on a specific group size the z- table won’t work. A new table for each possible group size needs to be generated to test the “percentile rank” of the group mean comparison. Fortunately the computer does this for you.

t-Critical The sampling distribution of the mean is a different shape for every sample size. Therefore the cut-off for rejecting the null has a different value for every sample size. Jackie had you look up the cut-off value by determining the df (identifying the distribution) and then looking up on a table of “critical values” whether the difference was significant. Now you can read the p value directly to do this.