Estimation and Hypothesis Testing Now the real fun begins

Inferential Statistics Making inferences about the population based on sample measurements Requires interpretation ◦Not merely observing and describing

Sampling o Need to recruit a (hopefully representative) sample to get started o Sampling methods: o Simple random sampling o Everyone in population has = chance of being recruited o Systematic sampling o Selecting participants from a preexisting list using a sampling fraction (Ex: every 3 rd person on a dorm roster) o Cluster sampling o Groups of people are recruited as single units (ex: classrooms, sororities, etc.) o Group performance evaluated rather than individual scores o Stratified random sampling o First define subgroups (strata) o Ex: college majors o Recruit so as to ensure all strata represented in sample

Chain of Reasoning in Inferential Statistics 1. To draw inferences about the parameter based on an estimate from a sample, sample must have been created with random selection 2. Sample estimate must be compared to an underlying distribution of estimates (all possible outcomes)from all other samples of the same size that could be selected from the population 3. We can draw reasonable conclusions about the parameters based on such comparisons and the probability of outcomes achieved using ramdon sampling

Sampling Distribution of the Mean Distribution of all possible sample means for all samples of a given size randomly selected from population In practice, we deal with theoretical sampling distributions based on central limit theorem ◦bigger sample size  sampling distribution more closely resembles normal distribution Standard deviation of sampling distribution = standard error of the mean

Generalizations about Sampling Distribution of the Mean 1. As sample size gets larger, one observes less variability in the sampling distribution of the mean (standard error decreases) 2. Even in a non-normally distributed population, the sampling distribution still more closely resembles the normal distribution with a larger sample size

Hypotheses We get to use inferential statistics for hypothesis testing ◦Drawing conclusions about the populations based on observations of a sample Hypothesis = educated guess, reasonable prediction about some phenomenon at work in the population Null hypothesis (H 0 ) = what we actually test with our statistical procedures ◦States that there is no relationship between the variables or no difference between the groups

The dreaded P Word So, despite common language (even in “science” shows or articles), we do not PROVE a hypothesis Instead, we set out to disprove or refute the null hypothesis Refuting the null allows us to say we have support for our hypothesis (the alternative hypothesis, H a ) ◦It is still hardly “proven”

Coming up with a hypothesis Interest Feasibility Relevance Falsifiability Replicating previous work Operational definitions

Research hypothesis versus statistical hypothesis Research hypothesis: educated prediction about relationships between study variables based on past research Statistical hypothesis: expected result of your specific statistical test Sometimes, much work is needed to get from a research hypothesis to a statistical hypothesis Need precise operational definitions!

Errors in Hypothesis Testing Type I Error ◦Rejecting the null hypothesis when it is indeed true ◦Interpreting your results as supporting your hypothesis when there really is no relationship or difference ◦False positive Type II Error ◦Failing to reject the null when it is indeed false ◦Interpreting your results as showing no support for your hypothesis when it is actually a sound hypothesis ◦False negative, failing to detect a genuine phenomenon

Level of Significance When can you reject the null? How different do the scores have to be before you can say you’ve found some evidence for your hypothesis? Conventionally, we go by the level of significance or alpha(α) level ◦Probability of making a Type I error ◦Lower probability = Higher certainty you can trust findings that support the alternative hypothesis ◦Standard in the field:.05 or.01

Steps in testing H 0 1. State the hypothesis 2. Set the criterion for rejecting H 0 3. Compute the test statistic 4. Decide whether to reject H 0.

t Distributions t distributions: A family of distributions that are symmetrical, bell-shaped, and centered on the mean, but changes for each sample of a certain size Degrees of freedom (df): number of obs minus number of restrictions ◦Freedom to vary ◦As number df increases, difference between t distribution and normal distribution decreases

Statistical Significance versus Practical Significance A finding can be “significant” but not necessarily interesting, important, or meaningful in real-world settings