Sample Distributions STAT 1010
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This 3 page Class Notes was uploaded by Susannah Gilmore on Wednesday March 23, 2016. The Class Notes belongs to STAT 1010 at University of Virginia taught by in Spring 2016. Since its upload, it has received 49 views. For similar materials see Introduction to Statistics in Statistics at University of Virginia.
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Date Created: 03/23/16
Sample Distributions (Chapter 10) Important definitions explained in these notes: Sample space Event Parameter Statistical inference The Law of Large Numbers Discrete Probability Models Continuous Probability Models Random Variable Random Variable X bar or xx Central Limit Theorem An unbiased estimator The Rules of Calculating Probabilities: All values between 0 and 1 If S is sample space of a probability model, then P(s)=1 Addition rule for disjoint events: if two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities a. P(A or B) = P(A) + P(B) For any event A, P(A does not occur) = 1-P(A) Probability Models Definition: A probability model with a finite space To assign probabilities, you just list the probabilities of all individual outcomes a. The probability of each individual must be between 0 and 1 b. The sum of all probabilities must equal one c. The probability of an event is the sum of the probabilities of the outcomes making up the event Discrete Probability Models This means that each variable is discrete, or that each number is a finite value, for example, the set (1,2,3,4,5) would be discrete. A set with numbers like (1.2,3.45,1/3,4.5) would NOT be discrete. All qualitative variables are discrete. Continuous Probability Models This means that each variable can have infinitely many values, such as (3.33333) It assigns all probabilities as areas under a density curve No part of a density curve can be negative The probability of an event is the area under the curve between a and b a. X is a random variable o The probability of any individual outcome is 0! Random Variable: a variable with a value that is a numerical outcome of a random phenomenon, usually denoted by X o Uses capital letters o The probability distribution of a random variable (represented by X) tells us what values X can take and how to assign probabilities to those value Random Samples o When you have a population, usually you (as an experimenter) will take random samples to test a variable. o If we take multiple random samples from the same population, we can assume that the different random samples will yield slightly different statistics. After taking many samples however, the values will become increasingly similar. This is known as the- a. Sampling Variability: the value of a statistic in repeated random sampling b. X bar, or xx̄, is the mean of the sample taken from a population c. So, if we know x bar, we can make inferences about the mean of the population, because we assume that the sample is representative of the population The Law of Large Numbers: This law states that as the number of observations drawn increases, the sample mean (x bar) of the observed values gets closer and closer to the mean of the population. So the more observations in our sample, the more accurate the sample mean is at representing the true population mean. Realistically, when you do an experiment, you want to have large samples with many observations, to get an accurate representation of the population mean. o When comparing the distribution of the population mean (σ) and the sample mean (xx), the sample mean distribution will be skinnier Central Limit Theorem: This states that whenever the shape of the population distribution is large, or when the number of trials (n) is large or n≥30. o The mean of a sample distribution is the same as the population mean: o The standard deviation of a sample distribution is: o You can only use these formulas if the number of trials (n) is greater than or equal to 30 o An unbiased estimator is an estimator that is correct on the average in many samples a. (The mean of its sampling distribution is equal to the true value of the parameter being estimated) Other definitions to know: o Sample space: an exclusive list of all possible outcomes that are mutually exclusive o Event: a subset of the sample space o Parameter: a number that describes the population o Statistical inference: using sample data to draw conclusions about the entire population
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