Inferring Population Parameters

z-Scores Standardizing scores by reporting them as distance from the mean in units of standard deviation Mean 1 Std. Deviation Test Score z = (score-mean)/sd

Gaussian distribution Continuous probability distribution Normal curve Normal distribution Gaussian distribution Continuous probability distribution Normal curve Characteristic curve Bell curve 0.13% 0.13% 2.14% 13.59% 34.13% 34.13% 13.59% 2.14% Standard Deviations (z scores) -3 -2 -1 +1 +2 +3 Cumulative Percentages .01% 2.3% 15.9% 50.0% 84.1% 97.7% 99.9% Percentile Equivalents 1 5 10 20 40 60 80 90 95 99 30 50 70 IQ Scores 55 70 85 100 115 130 145 SAT Scores (sd 209) 400 608 817 1026 1235 1444 1600

Percentile Rank (z-Scores) When the distribution is normal, identification of percent responses lower than a given score Mean 1 Std. Deviation Percentage of scores lower than target score Test Score z = (score-mean)/sd

Sample Mean When a sample group is being compared to the population, use a sampling distribution 1 Std. Deviation Group of 47 Mean Score z = (score-mean)/sd

Apply the central limit theorem Sample Mean When a sample group is being compared to the population, use a sampling distribution Mean Group of 47 Mean Score Apply the central limit theorem

Sample Mean When a sample group is being compared to the population, use a sampling distribution Standard Error (SD adjusted for group size) Percentage of group means lower than target group score Group of 47 Mean Score Compute a z-score

Probability that the group mean could have appeared randomly Sampling Distribution of the Mean represents distribution of randomly selected groups Mean Standard Error (SD adjusted for group size) Percentage of group means lower than target group score Group of 47 Mean Score Probability that the group mean could have appeared randomly

Probability that the group mean could have appeared randomly Sampling Distribution of the Mean Reporting the probability a group mean might appear randomly based on something that looks like a z-score Mean Standard Error (SD adjusted for group size) Group of 47 Mean Score Probability that the group mean could have appeared randomly

What if you do not know the SD of the population? A one sample t-test Mean Standard Error (sample SD adjusted for group size) Estimate the SD of the population using the sample SD Group of 47 Mean Score Probability that the group mean could have appeared randomly

One Sample t-Test To do this you need the population mean and the sample statistics You do not need to know anything about the population to which you are inferring except the mean score. EZAnalyze calls this the numerical test value (NTV) Group of 47 Mean Score

An example in Excel excel file: One Sample

OK, this was fun but you will almost never encounter a situation where you would do one sample tests. What we need to figure out is how to compare the means from two groups.

Central Limit Theorem When the means of randomly selected groups of a given size are plotted, a histogram approaching normal will appear. The almost normal distribution is called a sampling distribution of the mean. The population distribution does not have to be normal for this to work. The larger the group size the more normal the resulting distribution will be. This does not work very well for group sizes under 30.

Sampling Distribution of the Mean Remember what this is—a distribution of the mean of randomly selected groups of a given size. So, any give mean value on the distribution will represent how frequently that value was likely to appear randomly.

t-Test for Independent Samples Now the computer figures out the Sampling Distribution of the Mean for samples of x? Use the sample mean to estimate the population mean. Use the sample SD to estimate the population SD. Start with the mean and SD for sample of x? This starts the same way.

t-Test for Independent Samples Now the computer figures out the Sampling Distribution of the Mean for samples of x? Use the sample mean to estimate the population mean. Use the sample SD to estimate the population SD. Plot the mean of the comparison group on the sampling distribution What is the probability that the second mean could have appeared by chance in the sampling distribution.

Probability Testing Using the central limit theorem, a sampling distribution of the mean for groups the size of one of the samples is produced using estimates of the population parameters. The second group mean is plotted on the distribution to determine the probability it could have appeared randomly in the population the first sample represents. If the probability it could have appeared randomly is low, then the assumption is that the two samples came from different populations. Something made these two groups different.

An example in Excel Practice Problems: Performance Groups