PSYCH 240 1st Edition Lecture 8Section 5: Probability cont.Normal Distribution-We have been working with distributions built up from a set of observed scores-"Formal" Distributions: we can define "idealized" distributions that follow a particular mathematical function-One famous formal distribution is the normal, or Gaussian, distributionoThis is a special distribution that is seen for many natural variables Normal Curve--High probability density means that a lot of scores are near the value of the variable-Low probability density means that few scores are near the value of the variable-The normal curve is:oSymmetricaloBell shapedoUnimodal -Because the normal curve is defined mathematically, we can also mathematically work out the proportion of scores in a range of values-For tests and homework just now roughly how many scores fall between standard deviation intervalsoWhen a variable is normally distributed, the following proportion of scores fall between standard-deviation intervalsThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.o oEx. Samples and Populations-Populations consist of all possible scores of a variable-Sample: smaller set of scores actually available to a researcher-Statistic: index that is computed from the sample data-- regular lettersoSample mean = MoSample standard deviation = SoEmpirical probabilities are sample statistics. We have a sample of some limited number of attempts, and we calculate the proportion of times a certain outcome occurs-Parameter: characteristic of a population as a whole-- Greek lettersoPopulation Mean = μ oPopulation Standard deviation = σ oTheoretical probabilities are population parameters. The population is every possible attempt, and there is no limit to how many attempts you can take, so it's the proportion of outcomes you get with an infinite number of attempts-Law of Large Numbers: sample estimates will tend to converge to population values as sample size increasesoSays that the estimated empirical probability will tend to converge to the true theoretical probability as sample size increases-Populations don't have to be infinite-Instead of asking everyone, researchers usually select a smaller sample and use it to estimate the population value Sample Size-The sample mean will converge to the true population mean as sample size increases-With very large samples, it becomes very unlikely that the scores in the sample will be consistently above or consistently below the mean, so you probably get close to the right value Sampling Bias-Ideally, samples should be selected from the population entirely at random-If samples are not random, sample statistics give biased estimates of population values. That is, the sample value tends to come out consistently lower or consistently higher than the population value-Truly random sampling is often not possible, but we try to avoid sampling techniques that exclude population members with certain characteristics Objectives--Understand the difference between empirical probabilities and theoretical probabilities, and know the definition of both. Know both the objective and subjective interpretations of probability. --Understand how probabilities can be used to communicate how certain or uncertain we are that a claimor hypothesis is true – in other words, how confident we are in our beliefs. --Be able to use the addition and multiplication rules for combining probabilities, and know the conditions for which each is appropriate.--Know how to compute percentage change given the old and new values of a variable and know how tocompute the new value of a variable given the old value and percentage change. --Understand why large percentage changes can occur even when the absolute change in a variable is small. --Know the difference between marginal and conditional probabilities, and be able to recognize the standard notation for indicating marginal and conditional probabilities. --- Be able to compute marginal and conditional probabilities from a contingency table. --- Be able to recognize a normal (or Gaussian) distribution, and know the three critical properties of a normal distribution. Understand what a low versus a high probability density means. --- Understand the type of process that naturally produces a normal distribution. --- For a normal distribution, know approximately what proportion of scores fall between the mean and 1 standard deviation (SD) away from the mean, between 1 and 2 SDs away from the mean, and beyond 2 SDs away from the mean. --Be able to answer questions about what proportion of scores fall between two values, higher than a single value, or lower than a single value for a normally distributed variable. (I will always use numbers that correspond to an exact z score so you can use the proportions that you memorized for the above objective.) --- Know the difference between populations and samples, and be able to define each term. --- Know the difference between a statistic and a parameter. --- Know that sample statistics are usually indicated with regular letters and population parameters are usually indicated with Greek letters. --- Know what the “law of large numbers” states. --- Understand why larger samples tend to produce better estimates of a population (theoretical) proportion and a population mean. --Understand what sampling bias is, and know why we try to get random samples from a population. --Know what it means to say that a statistic is