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INFERENTIAL STATISTICS – Samples are only estimates of the population – Sample statistics will be slightly off from the true values of its population’s parameters Sampling error: – The difference between a sample statistic and a population parameter – Probability theory Permits us to estimate the accuracy or representativeness of the sample

The “Catch-22” of Inferential Statistics – When we collect a sample, we know nothing about the population’s distribution of scores We can calculate the mean (x-bar) & standard deviation (s) of our sample, but  and  are unknown The shape of the population distribution (normal or skewed?) is also unknown

Sampling Distribution (a.k.a. “Distribution of Sample Outcomes”) – “OUTCOMES” = proportions, means, obtained test statistics (z, t, F, chi-square) – Hypothetical, based on infinite random sampling, a mathematical description of all possible sampling event outcomes And the probability of each one – Permits us to make the link between sample and population… What is the likelihood that a sample findings accurately reflect the population? Is what’s true for the sample also true for the population?

1. Estimation ESTIMATION

Introduction to Estimation Estimation procedures – Purpose: To estimate population parameters from sample statistics – Using the sampling distribution to infer from a sample to the population – Most commonly used for polling data – 2 components: Point estimate (sample mean, sample proportion) Confidence intervals

Sampling Distributions: Central tendency – Sample outcomes (means, proportions, etc.) will cluster around the population outcomes – Since samples are random, the sample outcomes should be distributed equally on either side of the population outcome The mean of the sampling distribution for sample means (a bunch of x’s) is always equal to the population mean ( μ) The mean of the sampling distributions for proportions (infinite number of sample p’s), is equal to the population value P μ

Sampling Distribution: Dispersion

CONNECTION Probability theory tells us that outcomes plotted from repeated random samples will produce a normal distribution – We use z scores 95% of outcomes will fall within +/- 1.96 standard errors of the true population parameter 99% of outcomes will fall within +/-2.58 standard errors of the true population parameter

Calculate the Standard Error Based on Your Sample

Hypothesis Testing (intro) Estimation HYPOTHESIS TESTING

Hypothesis Testing & Statistical Inference We almost always test hypotheses using sample data – Draw conclusions about the population based on sample statistics – Therefore, always possible that any finding is due to sampling error  Are the findings regarding our hypothesis “real” or due to sampling error?  Is there a “statistically significant” finding?  Therefore, also referred to as “significance testing”

Testing a hypothesis 101 State the research & null hypotheses Set the criteria for a decision Alpha, critical regions for particular test statistic Compute a “test statistic” A measure of how different finding is from what is expected under the null hypothesis Make a decision REJECT OR FAIL TO REJECT the null hypothesis We cannot “prove” the null hypothesis (always some non-zero chance we are incorrect)

Sampling Distributions Again, HYPOTHETICAL distribution based on an infinite number of sample outcomes – Based on what would happen if we got an infinite number of sample outcomes from a population where THE NULL WAS TRUE by assuming the null hypothesis is correct. “OUTCOMES” are the test statistics (t, F, chi-square) – If the null was true, t should be close to zero (null says means are equal). – If the null was true, F should be less than (or close to) 1

Decision Making Since we assume that in the population the null is true… – Large observed “test statistics” indicate that our findings are odd, or rare, or quite different from what we would expect if our initial assumption was correct At some point, they will get large enough that we reject our initial assumption

Decision Making Part II Critical values – The value of a test statistic where the associated probability is the same as alpha Dictate “critical region” If observed test statistic is in critical region, we reject the null –this is a “significant” finding/relationship Sig values – The exact odds of obtaining any particular test statistic If the null is true, there is an ___% chance of obtaining this finding If it drops below alpha, we reject the null (significant finding)

Getting the right test statistic If one of the variables is I-R… – You can calculate a mean How many means are you comparing? – Only one sample mean (and a population mean) = one sample t-test – Dummy variable (sex) = two means = two sample (or independent sample) t-test – Nominal with more than two categories = F-test If both variables are nominal/ordinal – Only test is chi-square

Exam No multiple choice A few “short answer” Lots of interpretation/decision making Calculate (given formulas) – One sample t, chi-square – 95% and 99% confidence intervals (means, proportion) Interpret – t, F, chi-square

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