estimation and hypothesis testing now the real fun begins
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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
oNeed to recruit a (hopefully representative) sample to get started
oSampling methods:oSimple random samplingoEveryone in population has = chance of being recruited
oSystematic samplingoSelecting participants from a preexisting list using a sampling
fraction (Ex: every 3rd person on a dorm roster)oCluster samplingoGroups of people are recruited as single units (ex: classrooms,
sororities, etc.)oGroup performance evaluated rather than individual scores
oStratified random samplingoFirst define subgroups (strata)
o Ex: college majorsoRecruit 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 distributionStandard 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 sampleHypothesis = educated guess, reasonable
prediction about some phenomenon at work in the population
Null hypothesis (H0) = 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, Ha)◦It is still hardly “proven”
Coming up with a hypothesis
InterestFeasibilityRelevanceFalsifiability Replicating previous workOperational 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 H0
1. State the hypothesis2. Set the criterion for rejecting H0
3. Compute the test statistic4. Decide whether to reject H0.
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
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