1 POS 3713 Midterm 3 Study Guide 1 What is the difference between a sample and a population and why and to what extent do we care about each a Population data for every possible relevant case b Sample subset of cases that is drawn from an underlying population i Random sample draw a sample on the basis of randomness ii Sample of convenience non random c We are interested in the sample as far as it helps us to learn about the underlying population we do not care about the sample itself d Statistical inference use what we know to be true about one thing sample to infer what is likely to be true about another thing the population e Samples are good because they avoid costs of surveying the entire population they are practical and they help with feasibility we cannot survey the population every time some phenomenon happens i Always includes some degree of uncertainty 2 What is a normal distribution and what information can we learn from its mean and standard deviation a It is a bell shaped statistical distribution that can be entirely characterized by its mean and standard deviation i ii It is symmetrical about its mean mode median mean are the same It has a predictable area under the curve within specified distances of the mean 68 95 99 rule page 135 1 The symmetric bell curve allows us to estimate the of cases that fall within a given distance of the mean 3 What is the central limit theorem How does it allow us to calculate the uncertainty of our estimates of population parameters a Central limit theorem a fundamental result from statistics indicating that if one were to collect an infinite number of random samples and plot the resulting sample means those sample means would be distributed normally around the true population mean The unofficial sovereign of probability theory i ii We can estimate the population parameter using the sample statistic b Insight of central limit theorem infinite number of random samples and plot our sample means to each of these random samples the samples would be distributed normally i Sampling distribution would be normally shaped the frequency distribution is not normally shaped c Take random sample and calculate its mean repeat step infinite times and plot the sample means 4 What is a random sample and how does it differ from a convenience sample 2 a Random sample is a sample of a population made randomly a sample such that each member of the underlying population has an equal probability of being selected b Convenience sample non random 5 Why does the central limit theorem only apply to random samples a A non randomly selected sample of convenience does very little to help us build bridges between sample and the population about which we want to learn We cannot use the central limit theorem to construct a sampling distribution and create a confidence interval in a sample of convenience non random 6 How do you interpret a 95 confidence interval a Confidence interval a probabilistic statement about the likely value of a population characteristic based on the observations in a sample b Recall 68 95 99 rule c Population mean 2 x standard error of the mean 7 How does sample size affect the standard error of the mean a Larger sample sizes will reduce the size of the standard errors b Smaller sample sizes will increase the size of the standard error c If we have a large sample then it is easier to make inferences about the population of interest d The smaller our standard error the tighter our resulting confidence 8 What is a p value Be sure to understand what it tells you as well as what it intervals will be does not tell you a p value the probability that we would see the relationship that we are finding because of random chance Tells us the probability that we would see the observed relationship between the two variables in our sample data if there were truly no relationship between them in the unobserved population i It ranges from 0 to 1 b The lower the p value the greater confidence we have that there is a relationship between the variables c The more data on which the measurement is made the lower our p value will be the larger the sample size the more confident we are that our sample accurately represents the population d Limitations i Subject to manipulation 1 Higher sample size will reduce p value by reducing standard error ii Not comparative with other p values 1 Because p value varies with population size iii Smaller p value does not mean that one relationship is stronger than the other iv Silent as to the validity of our measurements 1 Was sample selected randomly The less random it is the less confidence we have in the p value 3 v Not conclusive evidence for causality 1 Statistical significance simply detects whether relationship is due to chance 9 What is a null hypothesis and how does it relate to the p value What is an alternative hypothesis a Null hypothesis a theory based statement about what we would b expect to observe if our theory were incorrect If theory states that there is covariation between X and Y the null states that there is no covariation c 1 minus p value tells us the level of confidence with which we can reject the null hypothesis If we reject the null we accept the hypothesis d e Alternative hypothesis hypothesis that sample observations are influenced by some non random cause 10 What is the difference between one and two tailed tests and how do we choose the appropriate one a One tailed i Ho u1 u2 or u1 u2 ii Ho u1 u2 or Ha u1 u2 b Two tailed i Ho u1 u2 ii Ha u1 u2 c Choose tail according to theory d Positive one tailed test tail to the right e Negative one tailed test tail to the left f Two tailed test tail on both left and right 11 What is statistical significance What does it mean if a relationship is not statistically significant a Statistically significant relationship A conclusion based on the observed data that the relationship between two variables is not due to random chance and therefore exists in the broader population b Most scientist use standard If p is less than 05 then the relationship is statistically significant c That a relationship is statistically significant does NOT necessarily mean that the relationship between X and Y is strong or causal 12 What are bivariate hypothesis tests and how do we determine which one is most appropriate to test any given hypothesis a Bivariate hypothesis helps us answer the question are X and Y related i Bivariate two variables b Categorical variables variables for
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