Week 7 Notes - Psychology 211
Week 7 Notes - Psychology 211 Psychology 211
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This 7 page Class Notes was uploaded by Kennedy Patterson on Thursday February 26, 2015. The Class Notes belongs to Psychology 211 at University of Alabama - Tuscaloosa taught by Andre Souza in Spring2015. Since its upload, it has received 240 views. For similar materials see Psychology 211-003 Elem Statistical Methods in Psychlogy at University of Alabama - Tuscaloosa.
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Date Created: 02/26/15
Psychology 211 Probability Examples What is the top 5 95 9500 30 5 20 45 70 5 00 Week 7 2161522015 5 005 1005 9500 9505 on 2 table 165 z 165 X 2516520 x 6125 0 What is the bottom 25 25 975 9750 30 5 20 45 70 o What is the top 69 69 O69 30 5 20 45 70 25 0025 10025 9750 9750 on 2 table 196 have to make 196 negative because looking for left of mean 196 w 25 2519620 29 x 29 69 0690 06950 on 2 table 51 have to make 51 negative because looking for left of mean 51 25 512520 x x 725 Psychology 21 1 Week 7 2161522015 Hypothesis Testing Statistic as a random variable 0 Statistic number that describes a sample 0 Inferential Statistics make inferences about population parameters based on sample statistics 0 The sample statistic is the it is our best guess of the population parameter 0 The sample statistic is a random variable Different samples of the same population will have different statistics Example How conservative is the state of Alabama 0 You asked 5 students to each as this question to a sample of 50 people how conservative is Alabamaquot using a scale from 1 not very to 10 vary conservative 0 Each person collected the data and calculated the mean for their sample Kendra 658 Sophia 526 Naomi 427 Madeleine 319 Tiffany 740 0 Central limit theorem is the answer for Who has the most accurate mean Which one of these 5 means is closer to the actual population mean How do we measure weather each mean is a good representation of the population parameter 0 Central Limit Theorem 0 Given any population the means of random samples together will converge to a normal distribution 0 Regardless of the distribution of the underlying population random sampling of the mean approximate a normal distribution given a large enough sample size A Normal Distribution 0 Mean of sample distribution is the mean of the population u fflll l o39 o 0 Given a large enough sample random samples of the population mean will coverage to a normal distribution and the mean of the sampling distribution will coverage to the true population mean Psychology 21 1 Week 7 2161522015 0 Sampling Distribution Mean 0 Expected Value of a variable is the weighted average of all possible values the variable can take EX ZXPX u 0 Sampling Distribution Mean f l ZX n o The expected value of the sampling distribution mean which is a linear combination is Ei ElnX1X2X 1nEX1EX2EX 1nux1ux2uxn niln u Sampling Distribution Variance o The sampling distribution of the mean is a probability distribution 0 As such it has a central tendency which equals the population mean and has some variation 0 How do we measure the variation ie the spread of a distribution Variance Standard deviation 0 The expected variance of the sampling distribution is Vari Var 1nZX 1nVarX1VarX2VarX 1nn02 02n The Sampling Distribution of the Mean 0 The mean of the sampling distribution of the mean equals the population u o The standard deviation of the sampling distribution of the mean is 039 n o The sampling distribution standard deviation is known as standard error of the mean 0 As n increases the sampling distribution variance decreases that is the precision of the estimate of the statistics increases 0 Simulation 1000 sample size 10 1000 sample size 50 1000 sample size 100 1000 sample size 200 Psychology 211 r39cuucncy 51 1 ill Week 7 2161522015 Standard Error and Sample Size S 3 2 9 a l 5 l l N l 7 5 L L l I T 7 lquot l It iquot II I ll cam 2 nwarr Sampe10 39 lt 39nrr 0 F39vuuany sampo NCJFI39 What happens in real life 0 Q We collect only one sample We calculate the standard error of the mean using our sample standard deviation and the sample size The standard error tells us how much a sample mean would vary if we took several samples of the same size from the same population Remember the sampling distribution is normally distributed Sample means can be converted to zscores zIg 039 Sampling Distribution amp Sampling Error 0 To understand hypothesis testing you need to know the concepts of sampling distribution and sampling error Sampling distribution represents the distribution of sample statistics Sampling error represents the variability of those statistics from one sample to another In statistics error means random variability and not a mistake or carelessness Imagine a population A with 10000 people in it Each person in this population has a certain number of siblings For this population the average amount of siblings per person is u 543 Psychology 211 Week 7 2161522015 H K 543 Error residual 0 Sampling Distribution 0 If we collect a random sample of 50 people from this population A we 0 expect the mean of this sample to be as close as possible to the true mean u 543 Imagine that Lauren Andre Stephanie Molly Rebecca Anny and Karen decide to each collect a different sample of 50 people from this population AEach one will calculate the mean for hisher sample Laurenl 3L 49 Andre I2 54 Stephanie I2 58 Molly I43 61 Rebecca I5 51 Anny l 6 68 Karen I7 32 The distribution of all these sample means I1 I2I7 is known as sampling distribution of the mean 0 Sampling Error 0 0 Note that each person was off by a little bit compared to the true population meanp 543 For each person I u Lauren 49 543 O53 Andre 54 543 003 Stephanie 58 543 067 Molly 61 543 067 Rebecca 51 543 O33 Anny 68 543 137 Karen 32 543 223 These differences are called sampling error Sampling distributions tells us what values we might expect to obtain for a particular statistic under a set of predefined conditions Note that this is conditional probability the probability of something happening if something is true Variance average variability Variance average of all the residuals Psychology 211 Week 7 2161522015 Hypothesis Testing 0 O The objective of science is to check whether a set of data agrees with a certain prediction This prediction is called hypothesis A hypothesis is a statement about a population A hypothesis is a prediction that a parameter takes a particular numerical value or falls within a particular interval A significance test uses data to summarize the evidence about a hypothesis It does this by comparing sample statistics point estimates of parameters to the values predicted by the hypothesis Example In 2011 a local newspaper in Austin claimed that male professors make more money than female professors A hypothesis is a statement about the population Let Wwomen be the mean salary of female professors and Wmen the mean salary for male professors Then the hypothesis is that Wwomen lt Wmen Hypothesis testing is really a systematic way to test claims or ideas about a group of population The objective of hypothesis testing is to determine the likelihood that a sample statistic would be selected if the hypothesis regarding the population parameter were true There are four basic steps to do hypothesis testing Step 1 state the hypothesis Step 2 set a criteria for decision aka choose an OiIevel Step 3 compute the test statistic Step 4 make a decision Hypotheses 0 Every significance test has two hypotheses about the value of a parameter Null hypothesis Alternative hypothesis The null hypothesis H0 is a statement about the value of a population parameter The alternative hypothesis H1 or Ha is a statement that the population parameter falls in an alternative range of values than those stated in the null hypothesis Usually the null hypothesis states no effect In the example about the salaries of male amp female professors 39 H0 39 Wmen Wwomen 39 H1 39 Wmen Wwomen Psychology 21 1 Week 7 2161522015 0 Example 0 Someone thinks that men save more money because they are more responsible You think that women save more because they are awesome We know from the Census research that women save on average 103 of their monthly income with a standard deviation of O 37600 0 You asked 32 of your guy friends what percentage of their salary they save every month and the average was 1121 0 Question do men really save money differently than women 0 Step 1 state your hypothesis 39 H0 39 JOOmen JOOwomen 39 H1 39 JOOmen IJwomen 0 Step 2 set the criteria for decision 0 level 005 Critical value 196 0 Step 3 compute the test statistic z 137 0 137 on ztable 9147 1 9147 0853 853amp of getting 1121 of higher Z VT Z 1121 103 091 137 376 066 W 0 Step 4 Make your decision Because the zvalue is not larger than zcritical we fail to reject H0 HI IQ
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