PY 211 BOOK NOTES ch. 5, 6, 7, 8
PY 211 BOOK NOTES ch. 5, 6, 7, 8 PY 211
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This 13 page Study Guide was uploaded by Allie Newman on Monday October 12, 2015. The Study Guide belongs to PY 211 at University of Alabama - Tuscaloosa taught by Rebecca Allen in Summer 2015. Since its upload, it has received 72 views. For similar materials see Elem Statistical Methods in Psychlogy at University of Alabama - Tuscaloosa.
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PY 211 Test 2 Notes from Book Chapters 5I 6 7I 8 Chapter 5 Probability o Allows us to make predictions regarding random events Symbolized by p 0 Frequency of times an outcome occurs divided by the total number of possible outcomes 0 Sample space outcome space total number of possible outcomes that can occur in an event 0 Probability varies between 0 and 1 0 and 100 and can never be negative 0 Improbable 0 Relative frequency of an outcome is the probability of observing that outcome probability of occurrence 0 Relative frequencies are often given in a gure Mutually Exclusive Mutually exclusive when two outcomes A and B cannot occur together 0 The probability of two mutually exclusive outcomes occurring together is 0 0 Example ipping a coin not possible to ip both a heads and a tails at one time 0 One or the other can occur but not both together 0 pA and B 0 o Additive Rule l pA and B pA pB o The sum of their probabilities is equal to the probability that one or the other outcome occurs Independent Outcomes Independent when the probability of one outcome does not affect the probability of the second outcome 0 Example when ipping a coin twice it is possible to ip heads both times 0 Multiplicative Rule l pA and B pA x pB o The probability that two independent outcomes occur is equal to the product of their individual probabilities Complementary Outcomes o Complimentary when the sum of their probabilities is equal to 100 0 Example if we ip a coin one time the probability of ipping heads or tails is 100 pA pB 100 0 When two outcomes are complimentary subtracting 1 from the probability of one outcome will give you the probability for the second outcome PM 1 93 or pB 1 pA Conditional Outcomes Conditional when the probability of one outcome is dependent on the occurrence of the other outcome 0 Example drawing a heart from deck of 52 cards second draw will be from 51 cards Conditional probabilities are stated as the probability of one event A given that another event B occurred Baye s TheoremLaw o pUP pPU x pU pP o Often applied to various conditional probability situations especially related to stats inference o A mathematical formula that relates the conditional and marginal unconditional probabilities of two conditional outcomes that occur at random Probability Distributions Probability distributions display the probability of each possible outcome for a given random variable 0 Probability distribution the distribution of probabilities for each outcome of a random variable 0 The sum of probabilities in a probability distribution is equal to 100 l sum of px 100 0 Random Variable not the actual outcome of a random experiment but rather describes the possible asyetundetermined outcomes in a random experiment 0 To construct a probability distribution we identify the random variable then distribute the probabilities of each outcome of that random variable Expected Value 0 The expected value of a given random variable is equal to the mean of the probability distribution of that random variable 0 To compute the mean value u multiply each possible outcome X by the probability of its occurrence p and then sum each product u sum of Xp Expected value can be used to assess risk by estimating the amount we can expect to gain or lose based on the known probabilities of outcomes for a given random variable 0 Expected value is the longterm mean o It is the average outcome for a random variable that is observed an in nite number of times Variance of a Probability Distribution 0 A measure of variability for the average squared distance that outcomes for a given random variable deviate from the expected value or mean of a probability distribution Standard Deviation of a Probability Distribution 0 While the expected value of the mean 5 the average outcome the standard deviation of a probability distribution is the distance that all other outcomes deviate from the expected value or mean 0 Standard deviation of a probability distribution is a measure of variability for the average distance that outcomes for a given random variable deviate from the expected value or mean of a probability distribution o It is calculated by taking the square root of the variance of a probability distribution 0 The computing formulas for the variance and standard deviation of a probability distribution do not require us to compute the squared deviation of each outcome from the mean Binomial Distribution 0 The distribution of probabilities for each outcome of a bivariate random variable 0 Only 2 possible outcomes for that event 0 The mean of a binomial probability distribution is the expected value of a random variable with two possible outcomes o The standard deviation of a binomial probability distribution is the average distance that bivariate outcomes deviate from the expected value or mean of a binomial probability distribution Chapter 6 Normal Distribution symmetrical bellshaped distribution 0 A theoretical distribution in which scores are symmetrically distributed above and below the mean the median and the mode at the center of the distribution 0 Characteristics 0 1 The normal distribution is mathematically de ned Each score on xaxis and each frequency on yaxis The normal distribution is theoretical The mean median and mode are all located at the 50th percentile The normal distribution is symmetrical The mean can equal any value The standard deviation can equal any positive value The total area under the curve of a normal distribution is equal to 10 The tails of a normal distribution are asymptotic The tails of a normal distribution never touch the xaxis so it is possible to observe outliers in a data set that is normally distributed 0 Most behavioral data approximate a normal distribution rarely are they exactly normally distributed 0 In a normal distribution 50 of all data fall above the mean median mode and 50 fall below Proportions of area under a normal curve are used to determine the probabilities for normally distributed data OOOOOOO CDNOWU39lbULJN Standard Normal Distribution 0 Standard normal distribution 2 distribution is a normal distribution with a mean equal to 0 and a standard deviation equal to 1 0 Standard normal distribution is one of the in nite normal distributions o It has a mean of 0 and a variance of 1 o The standard normal distribution is distributed in z score units along the xaxis o The 2 transformation formula converts any normal distribution to the standard normal distribution with a mean equal to 0 and a variance equal to 1 0 Z x u o l for a population and Z x M o l for a sample 0 A z score is a value on the xaxis of a standard normal distribution 0 The numerical value of a z score speci es distance or the number of standard deviations that a value is above or below the mean 0 The mean in any normal distribution corresponds to a z score equal to 0 Unit Normal Table 0 The unit normal table 2 table is a type of probability distribution table displaying a list of z scores and the corresponding probabilities or proportions of area associated with each 2 score listed 0 Column A lists the z scores 0 The table lists only positive 2 scores Meaning that only 2 scores at or above the mean are listed in the table 0 Column B lists the area between a z score and the mean 0 The rst value for the area listed in column B is 00000 0 Column C lists the area from a z score toward the tail 0 The rst value for the area listed in column C is 050000 This is the total are above the mean 0 Normal distribution is used to determine the probability of a certain outcome in relation to all other outcomes in order to estimate probabilities under the normal curve 0 The unit normal table allows us to locate raw scores x and determine probabilities p for data that are normally distributed Locating Proportions The area at each 2 score is given as a proportion in the unit normal table 0 To locate the proportion and therefore the probability of scores in any normal distribution you 0 1 Transform a raw score x into a z score 0 2 Locate the corresponding proportion for the z score in the unit normal table 0 The total area is 0500 above the mean and 0500 below the mean in a normal distribution 0 In the standard normal distribution 2 scores above the mean are positive and z scores belowthe mean are negative 0 Because the normal distribution is symmetrical probabilities associated with positive 2 scores are the same for corresponding negative 2 scores 0 To nd the probabilities of scores in a normal distribution you must know the mean and standard deviation in that distribution Locating Scores 0 The unit normal table can used to locate scores that fall within a given proportion or percentile To nd the cutoff score for a given proportion you 0 1 Locate a z score associated with a given proportion in the unit normal table 0 2 Transform the z score into a raw score x Binomial Distribution 0 The distribution of probabilities for each outcome of a variable with only two possible outcomes 0 A binomial probability distribution is distributed with u hp and 0 square root npq o n sample size and p and q are the probabilities of each binomial outcome 0 Binomial distribution approximates a normal distribution 0 The larger the n sample size the more closely is approximates a normal distribution Normal Approximation to the Binomial Distribution 0 For a binomial distribution where hp and nq are greater than 10 the normal distribution can bed used to approximate the probability of any given outcome using these ve steps 0 Step 1 check for normality Step 2 compute the mean and standard deviation Step 3 nd the real limits Step 4 locate the z score for each real limit Step 5 nd the proportion located within the real limits 0 Real Limits 0 Real limits of the outcome of a binomial variable are the upper and lower values within which the probability of obtaining that outcome is contained 0 O O O o The real limits for a binomial outcome x are X 05 Chapter 7 Sampling Distribution 0 A distribution of all sample means or sample variances that could be obtained in samples of a given size from the same population 0 Researchers measure a mean and variance in some sample to gauge the value of the mean and variance in a population 0 Sampling Without Replacement o A method of sampling in which each participant or item selected is not replaced before the next selection The probability of each selection is conditionathey are not the same 0 This method of sampling is the most common method used in behavioral research 0 Sampling With Replacement o A method of sampling in which each participant or item selected is replaced before the next selection 0 This method of sampling is used in the development of statistical theory 0 Sample Design 0 A speci c plan or protocol for how individuals will be selected or sampled from a population Does the order of selecting participants matter Determines how often people in a population can be selected Do we replace each selection before the next draw 0 Theoretical Sampling Sampling strategy used in development of statistical theory The order of selecting individuals matters Each individual selected is replaced before sampling again Total number of samples possible N n n sample size and N population size 0000 0 Experimental Sampling 0 Sampling strategy most commonly used in experimental research 0 The order of selecting individuals does not matter 0 Each individual selected is not replaced before selecting again 0 Total number of samples possible N n N n Unbiased Estimator Any sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter on average 0 The sample mean is an unbiased estimator because it equals the population mean on average When M sum ofx n then M u on average sample mean equal to population mean Central Limit Theorem 0 Explains that regardless of the distribution of scores in a population the sampling distribution of sample means selected at random from that population will approach the shape of a normal distribution as the number of samples in the sampling distribution increases CLT explains that the shape of a sampling distribution of sample means tends toward a normal distribution regardless of the distribution in the population 0 The probability distribution for obtaining a sample mean from a population is normal Standard Error of the Mean Standard Error 0 The standard deviation of a sampling distribution of sample means o It is the standard error or distance that sample mean values deviate from the value of the population mean Overview of Sample Mean 0 Sample mean is a good estimate of the value of the population mean 0 The sample mean is an unbiased estimator o On average the sample mean from a randomly selected sample will equal the population mean 0 A distribution of sample means follows the central limit theorem 0 Regardless of shape of distribution in a population distribution of sample means selected from the population will approach the shape of a normal distribution as number of samples increases 0 A distribution of sample means has minimum variance 0 The sampling distribution of sample means will vary minimally from value of population mean Variance On average the sample variance is equal to the population variance when we divide SS by df 0 SS sum of squared deviation of scores from their mean 0 df degrees of freedom n 1 This makes the sample variance an unbiased estimator of the population vadance The sampling distribution of sample variances tends toward a positively skewed distribution regardless of the distribution in the population as the number of samples increases 0 Although the sample variance equals the population variance on average the distribution of all other sample variances can vary far from the population variance when we divide SS by df Overview of Sample Variance Sample variance is a good estimate of the value of the population variance because it is unbiased The sample variance is an unbiased estimator o On average the sample variance we obtain in a randomly selected sample will equal the population variance when we divide SS by df where df n 1 o A distribution of sample variances follows the skewed distribution rule 0 Regardless of shape of distribution in population the distribution of sample variances selected will approach the shape of a positively skewed distribution as the number of samples increase 0 A distribution of sample variances has no minimum variance 0 The sampling distribution of sample variances will not vary minimally from the value of the population variance when we divide SS by df Standard Error of the Mean 0 To compute standard error of the mean divide the population standard deviation by the square root of the sample size 0 Larger values indicate greater sampling error or greater differences that can exist from one sample to the next 0 Sampling Error 0 The extent to which sample means selected from the same population differ from one another 0 This difference which occurs by chance is measured by the standard error of the mean 0 The standard error can increase or decrease depending on the sample size and the value of the population standard deviation 0 The largerthe standard deviation in the population the largerthe standard error 0 The largerthe sample size the smallerthe standard error The Law of Large Numbers Sample means deviate closer to the population mean as sample size increases Larger samples are associated with closer estimates of the population mean on average Transform and Convert o The 2 transformation is used to determine the likelihood of measuring a particular sample mean from a population with a given mean and variance Chapter 8 lnferential Statistics 0 Hypothesis 0 A statement or proposed explanation for an observation a phenomenon or a scienti c problem that can be tested using the research method A hypothesis is often a statement about the value for a parameter in a population 0 Hvoothesis Testind significant testind o A method for testing a claim or hypothesis about a parameter in a population using data measured in a sample Systematic way to test claims or ideas about a group or population 0 In this method we test a hypothesis by determining the likelihood that a sample statistic would be selected if the hypothesis regarding the population parameter were true Method of testing whether hypotheses about a population parameter are likely to be true 0 4 Steps to Hypothesis Testing 0 1 Identify a hypothesis or claim that we feel should be tested 0 2 Select a criterion upon which we decide that the hypothesis being tested is true or not 0 3 Select a random sample from the population and measure the sample mean 4 Compare what we observe in the sample to what we expect to observe if the claim being tested is true 0 1 State the hypotheses 2 Set the criteria for a decision 3 Compare the test statistic 4 Make a decision 0 Note The null and alternative hypotheses must encompass all possibilities for the population mean Null Hypothesis HO step 1 o A statement about a population parameter such as population mean that is assumed to be true 0 The null hypothesis is a starting point 0 We will test whether the value stated in the null hypothesis is likely to be true 0 The key reason we are testing the null hypothesis is because we think it is wrong 0 We state what we think is wrong about the null hypothesis in an alternative hypothesis Alternative Hypothesis H1 step 1 o A statement that directly contradicts a null hypothesis by stating that the actual value of a population parameter is less than greater than or not equal to the value states in the null hypothesis Level of Signi cance step 2 o A criterion ofjudgment upon which a decision is made regarding the value stated in a null hypothesis 0 The criterion is based on the probability of obtaining a statistic measured in a sample if the value stated in the null hypothesis were true 0 ln behavioral science the criterion or level of signi cance is typically set at 5 0 When the probability of obtaining a sample mean would be less than 5 if the null hypothesis were true then we reject the value stated in the null hypothesis Test Statistic step 3 o A mathematical formula that identi es how far or how many standard deviations a sample outcome is from the value stated in a null hypothesis o It allows researchers to determine the likelihood of obtaining sample outcomes if the null hypothesis were true 0 The value of the test statistic is used to make a decision regarding a null hypothesis Making a Decision step 4 o The decision is based on the probability of obtaining a sample mean given that the value stated in the null hypothesis is true 0 If probability of obtaining a sample mean is less than or equal to 5 when null hypothesis is true then the decision is to rejectthe null hypothesis 0 If the probability of obtaining a sample mean is greaterthan 5 when null hypothesis is true then the decision is to retain the null hypothesis Reiect Null Hvbothesis 0 Sample mean is associated with a low probability of occurrence when null hypothesis is true 0 Reject if p value is less than or equal to 005 Retain Null Hvbothesis 0 Sample mean is associated with a high probability of occurrence when null hypothesis is true 0 Retain if p value is greater than 005 0 Probability of obtaining a sample mean is stated by the p value 0 P Value o The probability of obtaining a sample outcome given that the value stated in the null hypothesis is true 0 The p value for obtaining a sample outcome is compared to the level of signi cance 0 It varies between 0 and 1 and can never be negative 0 Signi cance the decision to either reject or retain the null hypothesis 0 Signi cance describes a decision made concerning a value stated in the null hypothesis 0 When the null hypothesis is rejected we reach signi cance 0 When the null hypothesis is retained we fail to reach signi cance FEE Lle ad fr t l l M walla ail latitihe Samplea and amplify A Review l the Natalia T LE Eisndar ie riatin il put Distributions piling listri utitm Eiharaijteristi Fl l ill t ll 7 7 Iiilean ll lulan Ful lL trainee g 3r quot 51 emfmg a W 55 gmzi J biennium vr39r la REVi1 53 the Hay i asreneee etwa Popuation plea and Sampling Distributions TA LE BEE N Fmpuiatiun ULETF Lllllllfi l l mi am l llistriihlitimi Eamlallistriutitirl Means is if ames of all Societies of a E39EtliE th poaaibe sample malaria arena in a Eil il39i of JEFEEHE that Earl be salaried giggle Fj JLlil from the popuatin a E ail l EEJM plE aim a it Typi allgpi no Teak Algae Wall is the Could a any Gaul be any Normal rdiatritmtim shape ahapa sharia Types of Error TALK33 Four OUtD MEE for Making a magic Decision ll lllj j t llests Reject the Null Htlljil th sls Truth in the True 0 RRECT j EROR Population 1 a u o Idf l l n can be either correct frreotlgr reject r retain the Hull hllP thEEE SE39 W Willing lim 39rt EC l39El39E l Ir retain the null hypotheaiaj 0 Type II error is the probability of incorrectly retaining the null hypothesis beta error 0 Type II retaining a null hypothesis that is actually false 0 Researchers directly control for the probability of a Type I error by stating an alpha level 0 Type I error incorrect decision is to reject a true null hypothesis Probability of rejecting a null hypothesis that is actually true We control for type I error by stating a level of signi cance Alpha level the largest probability of committing a Type I error that we will allow and still decide to reject the null hypothesis 0 a 005 0 To make a decision we compare the alpha level to the p value the actual likelihood of obtaining a sample mean if the null were true 0 The correct decision is to reject a false null hypothesis false when it is indeed false 0 Power the probability of rejecting a false null hypothesis It is the probability that a randomly selected sample will show that the null hypothesis is false when it is indeed false OneSample Z Test 0 A statistical procedure used to test hypotheses concerning the mean in a single population with a known variance 0 Need to know mean and standard deviation in a single population Nondirectional Tests H1 does not equal 0 Nondirectional tests are used to test hypotheses when we are interested in any alternative to the null hypothesis Nondirectional tests also called twotailed tests are hypothesis tests in which the alternative hypothesis is stated as not equal to a value stated in the null hypothesis 0 The researcher is interested in any alternative to the null hypothesis 0 For twotailed tests the alpha is split in half and placed in each tail of a standard normal distribution 0 Critical Values 0 A cutoff value that de nes the boundaries beyond which less than 5 of sample means can be obtained if the null hypothesis is true Sample means obtained beyond a critical value will results in a decision to rejectthe null hypothesis A critical value marks the cutoff for the rejection region Rejection Region 0 The region beyond a critical value in a hypothesis test 0 When the value of a test statistic is in the rejection region we decide to reject the null hypothesis Otherwise we retain the null hypothesis 0 Z Statistic 0 An inferential statistic used to determine the number of standard deviations or z scores in a standard normal distribution that a sample mean deviates from falls above or below the population mean stated in the null hypothesis Obtained Value o Is the value of a test statistic o This value is compared to the critical vaues of a hypothesis test to make a decision 0 When the obtained value exceeds a critical value we decide to rejectthe null hypothesis Otherwise we retain the null hypothesis 0 A nondirectional test is conducted when it is impossible or highly unlikely that a sample mean will fall in the direction opposite to that stated in the alternative hypothesis Twotailed tests are more conservative and eliminate the possibility of committing a Type III error quot 7 r 7 39 Iw rc a39led lasts lhree 39 l V l as for One and l Commonly Used Levels of Significance l f neialled 05 1645 or i 645 1 96 m 233 or 7139 001 309 or E309 i330 A Directional Tests H1 is greater than or less than 0 Directional tests onetailed tests are hypothesis tests in which the alternative hypothesis is stated as greater than or less than a value stated in the null hypothesis 0 The researcher is interested in a speci c alternative to the null hypothesis 0 For onetailed tests the alpha level is placed in a single tail of a distribution Onetailed tests are associated with a greater power assuming the value stated in the null hypothesis is false 0 Type III Error o A type of error possible with onetailed tests in which a decision would have been to rejectthe null hypothesis but the researcher decides to retain it because the rejection region was located in the wrong tail The quotwrong tailquot refers to the opposite tall from where a difference was observed and would have otherwise been signi cant Measuring the Size Effect Cohen39s d o A decision to rejectthe null hypothesis means that an effect is signi cant o Hypothesis testing determines whether or not an effect exists in a population 0 Effect 0 For a single sample an effect is the difference between a sample mean and the population mean stated in the null hypothesis In hypothesis testing an effect is not signi cant when we retain the null hypothesis It is signi cant when we reject the null hypothesis 0 Effect Size 0 A statistical measure of the size of an effect in a population which allows researchers to describe how far scores shifted in the population or the percent of variance that can be explained by a given variable Effect size measures the size of an observed effect from small to large 0 One advantage to knowing the effect size 0 is that its value can be used to determine the power of detecting an effect in hypothesis testing Poweris the likelihood of detecting an effect 0 This lets the researcher now the probability that a randomly selected sample will lead to a decision to reject the null hypothesis if it is false As effect size increases power increases 0 Increasing sample size also increases power by reducing standard error Thus increasing the value of the test statistic in hypothesis testing Increasing alpha level the probability of a type I error also increases power 0 The larger the rejection region the greater the likelihood od rejecting the null hypothesis and the greater the power will be 0 O O 0 Increase power by Increase effect size sample size and alpha Decrease beta standard error and population standard deviation Summary of Factors That Increase Power the probability l eorease 039 Effect size Type ll error miSampie size I 39 ype I error GM Standard error Cohe n s d o A measure of effect size in terms of the number of standard deviations that mean scores shifted above or below the population mean stated by the null hypothesis The larger the value of d the larger the effect in the population Cohen39s d M u o Smalllj dis less than 02 Medium dis greater than 02 but less than 08 0 Large dis greater than 08 Cohen s Effect Size Conventions COO 0 Standard rules for identifying small medium and large effects based on typical ndings in behavioral research Value bei n measured type ef distributien is the test based upsn What does the test measure earl lee inferred frem thetest Clan iiiis test stand aims in research reports HVl thESiS Significance Tasting p 1snrquotalue Sampling distrillsatien The prebalaility ef attaining a measured sample mean Mather an effect eaists a sealatlas Year the test statistie can be repeated without an effect size 7 MERLE I Distinguishing Characteristiss fer Hypethesis Testing and Effect Size Effect Size Bahamas a Pgeuletien distr izbatieri The size of a measured effect in the Pepulatien lire Tn 153 all ea Ne effth size is aimest reperted with a test stati always stir
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