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Psychological discovery 6 W12 Probability  The relationship between samples and populations are defined in terms of probability.  Goal of inferential statistics – to begin with a sample and then answer a general question about the population. o Identifying the types of samples that would probably be obtained from a specific population. o Reverse the probability rules to allow us to move from samples to populations. Using z-scores to describe sample means  To describe the relative standing of an entire sample.  Evaluate a sample mean in terms of its relative standing, o Compare the sample mean to the other sample means that occur in the situation. o Create a distribution showing these means – sampling distribution of means:  The frequency distribution of all possible sample means that occur when an infinite number of samples of the same size are selected from one raw score population.  Cannot infinitely sample a population but we know what it would look like because of the central limit theorem.  Z-score tells us if our sample mean is one of the more common means that is relatively close to the average sample mean, or if it is one of the few, higher or lower means that occur in this situation.  To compute the z-score for a sample mean, we need the SD of the sampling distribution. The central limit theorem  The mean of the sampling distribution equals the mean of the underlying raw score population from which we create the sampling distribution.  A sampling distribution is an approximately normal distribution.  The SD of the sampling distribution is mathematically related to the SD of the raw score population.  We can describe a sampling distribution without having to infinitely sample a population of raw scores. o We can create the sampling distribution of means for any raw score population. The standard error of the mean  The SD of the sampling distribution.  ‘average’ amount that the sample means deviate from the mean of the sampling distribution.  The true standard error of the mean: o When we know the true SD of the underlying raw score population, we will know the true SD of the sampling distribution. o Example: o Identify ___ and the N used to create the sample. o Compute square root of N. o Divide. o Indicates that the individual sample means differ from the sampling distribution of 500 by an average of 20 points when the N of each sample is 25.  Computing a z-score: o Compute ___ and identify the sample mean and sampling distribution. o Subtract ___ from average. o Divide. o A sample mean of 520 has a z-score of +1 on the sampling distribution means that occurs when N is 25. Describing the relative frequency of sample means  Z-score tells us our sampling mean’s relative location within the sampling distribution, and thus its relative standing among all means that occur in this situation.  Because a sampling distribution is always an approx. normal distribution, transforming all of the sample means in the sampling distribution into z-scores would produce a normal z-distribution.  Summary of describing a sample mean with a z-score: o 1: Envision the sampling distribution of means as a normal distribution with a ___ equal to the ___ of the underlying raw score population. o 2: Locate the sample mean on the sampling distribution by computing its z-score.  Using the ___ of the raw score population and your sample N, compute the standard error of the mean.  Compute z, finding how far your average is from the ___ of the sampling distribution, measured in standard error units. o 3: Using the z-table to determine the relative frequency of scores above or below this z-score, which is the relative frequency of sample means above or below your mean. Determining the probability of sample means  Using a sampling distribution of means, which is another type of probability distribution.  After computing z-score, you can determine the probability of obtaining means between 0 and +1. o The relative of such frequencies is 0.3413, so that the relative frequency of the sample means is also. o Therefore, the probability is 0/3413 that we will obtain a sample mean between 500 and 520 from this population.  The probability of selecting a particular sample mean is the same as the probability of randomly selecting a sample of participants who produce scores that result in that sample mean.  The procedure for using z-scores to compute the probability of sample means forms the basis for all inferential statistics. Deciding whether a sample represents a population  We begin with a known, und
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