Statistical Inference Here we go…

Population mean = 100, SD = 15 Take 1 sample n=50

Take 10 samples of n=50

Take 100 samples of n=50

Take 200 samples of n=50

After 2000 repeated samples

Take 5000 samples of n=50

Sampling Error Ability testing is done on the sample, not on the population Random sampling implies that on average the sample mean will correspond to the population mean. ?

What is the Standard Error of the Mean? If many simple random samples of identical size are drawn from a population, and the same attribute is measured in each sample, the mean values will be normal distributed. The standard deviation of the normal distribution of the means of all SRSs of size n is the SEM.

Review: Making inferences about the population based on sample data There is no guarantee that a single sample can say anything about the population. There is a guarantee that the AVERAGE of many “single” samples can say something about the population. If infinitely many samples of size n are drawn from a population, and their means are calculated, those means will be normal distributed. The mean of the means will be the same as the population mean. The standard deviation of the normal distribution of the means of all samples is the SEM

REMEMBER: 2 kinds of variability SD  Person-to-person variability in the population SEM  Sample-to-sample variability among all possible samples

Using the Standard Error of the Mean The standard deviation of the means of all samples is the SEM. Therefore, 95% of all samples have a mean between {population mean – 1.96SEM} and {population mean + 1.96 SEM}

Populations and samples

Γ What if the sample is not a SRS?

APPLICATION TO PROBLEMS Can this sample be coming from a population with mean = μ? METHOD A: What kinds of samples can this population generate? METHOD B: What kind of a population could this sample be coming from? Same YES or NO answer

Problem 1 METHOD A Population mean IQ for individuals of normal intelligence is 100. Population SD is 15. Our sample mean is 112 Sample size is 85 Can we say that this sample is coming from a population with normal intelligence?

Problem 1 METHOD A - Solution Sample mean = 112, n=85 SD = 15 95% of the samples generated by the population with a mean of 100 have means between (100-(1.96*1.63)) = 96.8 and (100+(1.96*1.63)) = 103.2 It is unlikely that the sample is coming from a population with a mean of 100. CONCLUSION: This sample is probably NOT coming from a normal IQ population.

Problem 2 – METHOD A Is it possible that this sample is coming from a high school educated population?

Problem 2 – METHOD A - solution Is it possible that this sample is coming from a high school educated population?

Problem 2 – METHOD A - solution Is it possible that this sample is coming from a high school educated population? 95% of the samples from a high school educated population will have means between 10.8 and 13.2. Therefore, it is unlikely that this sample is coming from a population with a high school education.

Problem – Method A example Is it possible that this sample is coming from a population of students who averaged a GPA of 3.00 or higher?

Problem – Method A solution Is it possible that this sample is coming from a population of students who averaged a GPA of 3.00 or higher? 95% of the samples from a population of students who average 3.00 GPA will have means between 2.76 and 3.24. Therefore, it is likely that this sample is coming from a population of students who averaged a GPA of 3.00.

Problem – Method A Children who are given social skills training are expected to average a score of 67 on a social skills test. A sample of 25 children were tested on social skills and their average score was 59. The population SD is estimated to be 10. Is our sample of children in need of social skills training?

Problem – Method A Pop. Mean = 67 Sample Mean = 59 Pop SD = 10 N = 25 SEM = 10/5 = 2 95% of the time, the social skills trained population will generate samples with means between: 67- (1.96*2) = 63.08 and 67 + (1.96*2) = 70.92