POS3713 – Political Science Research Midterm 3 Five-Step Model for Hypothesis Testing1. Make assumptions and determine if they are met -Typically we assume random sampling (central limit theorem)2. State the null hypothesis - Typically: there is no difference; exception: if we have a directional research hypothesis3. Determine the sampling distribution and critical region4. Calculate the test statistic - The sampling distribution and the test statistic will depend on the particular hypothesisyou are testing5. Interpret the results- Do we reject the null hypothesis? (is there a statistically significant relationship?) - Compare your test statistic to the critical value from the sampling distribution - Remember, you need to calculate the degrees of freedom to determine the critical value - If the test statistic is larger, then reject the nullSamples vs. Population – Sample (known data) to population (unknown data); Sample is the subset within population hopefully chosen randomly; population are all occurrences of your phenomenon of interest; sample is a subset of the population of interest; we use sample to infer things about the population; we take samples because of the feasibility, costs, and practicality because if done well, sample parameters (traits) will accurately reflect population parameters (traits). Statistical Inference – The bridge from what we know about the sample to what we believe, probabilistically, to be true about the broader population; we have a sample (sample mean) and we make inference on population based on that sample; we use what we know to be true about one thing (the sample) to infer what is likely to be true about another thing (the population); we use types of data gathered about the sample (a subset of cases that are drawn from an underlying population) to infer certain causations of the population (data for every possible relevant case). Simple Random Sample – Ideal standard for a sample; every case in your population of interest has an equal chance of being selected as part of the sample; on average, parameters from a simple random sample will reflect the population parameters; every case having an equal chance of appearing in the sample.Sampling Distribution – Taking a sample of a random # an infinite number of times; it is a hypothetical distribution of sample means because scientists almost never actually draw more than one sample from an underlying population at one given point in time; if we took those sample means and plot them the key outcome is: the sampling distribution would be normally shaped, even though the underlying frequency distribution is clearly not normally shaped (sets up the Central Limit Theorem). In other words, distribution for a calculated statistic taken from a repeated sample within a population an infinitenumber of time. Central Limit Theorem – For any trait or variable, event those that are not normally distributed in the population, if repeated random samples of size N are drawn from any population, with mean m and standard deviation s, then, as N becomes large, the sampling distribution of sample means will approachnormality with mean µ and standard deviation ^σ√N- Generally by “N becomes large” a rule of thumb is N > 100- The central limit theorem says that the sampling distribution will be normally distributedIn other words,1) the sampling distribution is normally distributed 2) mean (X ¿ from sample is a good estimate of the true mean (μ)3) The standard deviation of sampling distribution gets smaller as sample size gets biggerStandard Error of the Mean – it is the measure of uncertainty; our measure of our standard deviation ofour sampling distribution; tells us the uncertainty in our estimate of the mean without pulling hundreds of repeated samples; standard error of the mean equals the standard deviation of the sample over the square-root of the sample size ^σ× = ^σ√N .The formula for the standard error has the sample size in the denominator, and as the denominator increases, the fraction becomes smaller. This means, the standard error decreases as the sample size increases. Smaller standard error means greater confidence in our estimate. Tells us that as sample size increases, the standard error decreases. 95% Confidence Interval of the Mean – 95% of the sampling distribution lies plus or minus 1.96 standard deviation of the sampling distribution (this assumes a large sample size); to calculate the confidence interval of the mean we need to know the mean and the standard deviation, N-Size (or standard error). C.I. =X ± 1.96^σ√n; the 95% confidence interval tells us the range of values in which the population mean is likely to fall. It is essentially trying to give an interval of how confident we think the mean is (what the highest and lowest the mean can be/is). The 1.96 from the equation comes from the critical values of .05=1.96 (critical value of .10=1.64) this is assuming a large sample size.The Different Distributions 1. Standard Normal Distribution – Bell shaped and symmetric; mean, median, and mode are all the same; predictable curve line under the curve 68-95-99 Rule – a normal distribution curve; 68% of cases fall within one standard deviation of the mean, 95% of cases fall within two standard deviation of the mean, and 99% of cases fall within three standard deviation of the mean2. t-Distribution –a normal distribution except it correct s for cases that has small sample sizes. Comes from the quality controlled tests for beer. T-critical values assuming large sample size. Also uses the p-value of .05 and .1 (critical values of 1.96 and 1.64)- Critical Values: the threshold you need to pass for your X2 to assume there is a relationship and it was not caused by chance. 2. χ2 Distribution – somewhat of a normal distribution that goes along with the Tabular Analysis table (given to you on the test if anything). The Research Hypothesis – H1: the research hypothesis is the difference we expect to see: a. the difference we expect to see or b. the difference between our estimate and the hypothesized parameter. The relationship we expect to see with 2 variables. [Up to now all the research hypothesis we’ve done have been that 2 variable are related]The Null Hypothesis – H0: the null hypothesis; any difference observed is just the result of random chance; our goal is to reject the null hypothesis, that is, we want to eliminate the possibility that any
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