Exam 2 notes
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Date Created: 02/09/15
EXAM 2 09232013 NORMAL CURVE Bellshaped Unimodal Symmetrical Ubiquitous curve shows up everywhere 0 Representative sample are more likely to show the curve 0 So consistent can be used to catch cheaters Eg Chicago school teachers French soldiers height 0 Normal distribution 0 Sample size 0 mean 0 standard deviationsquared 0 much of inferential statistics is built on the assumption that the data are normally distributed 0 the larger the sample the better it represents the population STANDARDIZATION What if we want to compare variables tat are on different scales 0 Height inches vs weight lbs 0 We need a method to convert the raw scores of inches and lbs onto the same standard scale Standardization converts raw scores into standard scores for which we know the percentiles 0 We will convert raw scores into 2 scores 0 Z scores follow a Zdistribution o It is also a normal distribution Zscores can be used to calculate percentiles 0 Given a certain z score what percentile is this observation in Theoretically 100 of the population is represented under the normal curve 0 midpoint is the 50th percentile 2 distribution and percentiles o 68 within 1 SD of the mean 0 96 within 2 SD s 0 gt99 within 3 SD s Z SCORES To compute a z score we need to know the population mean and standard deviation 0 Z scores can be positive or negative 0 Positive above the mean 0 Negative below the mean 0 Z scores are scaled in terms of standard deviation 0 Eg z 078 means you are 78 standard deviation above the mean Ex Hours slept by students on Thursday night 0 population mean 7 0 SD 15 What are Z scores for Z1 5715 133 Z2 9715 133 CONVERT Z SCORES TO RAW SCORES Population mean 53 SD 20 o What are the raw scores X for Z 215 l 2152 53 1 Z 135 l 1352 53 8 Z 85 Z 040 COMPARISONS USING Z SCORES Because we have placed raw scores on a standard scale we can now make comparisons between scores 0 Ex 2008 Michael Phelps vs 1972 Mark Spitz 2008 Phelps had 8 Gold Medals 1972 Spitz had 7 Gold Medals 0 the two swam many of the same events using raw scores you can compare them ZSpitz 11278 12033456 166 ZPhelps 102961094318 203 CENTRAL LIMIT THEOREM 0 States a distribution of sample means is a more normal distribution than a distribution of scores even when the population is not normal 0 Distribution of sample means approaches normality as sample size increases 0 Even when the population is normally distributed 0 Distributions of around 30 sample means are large enough 0 Distribution of means is less variable than a distribution of raw scores 0 Limits the effects of outliers o Is a more precise estimate of the population mean Ex heights in inches of 30 college students from a statistics class Distribution of scores 0 Randomly select students and plot their heights Distribution of means Randomly select 3 students and plot the mean of their heights STANDARD ERROR o a distribution of scores is characterized by population mean u population standard deviation 0 o a distribution of means is characterized by W mean of all possible samples of a given size from a population of individual scores o M standard error of mean a standard deviation of a distribution of means 0 mean of the raw scores is the same as the mean of averages O39M 0 65 6 N1O N 200 GM 5M DISTRIBUTION OF MEANS 0 Can calculate zscores for the means of the distribution of SUMMARY 0 means Zstatistics Tells us how many standard errors a sample mean is from the population mean Ch 7 MuM OM Z The normal curve In uence of sample size Standardization Why Raw scores z scores Z raw scores Making comparisons across scales Central limit theorem Standard errors THE Z TABLE 0 A table of scores with 2 values and the given percentages Not values are only present for positive 2 cores 0 Recall percentile rank is the percentage of score below the observed score USING THE Z TABLE 0 Positive 2 scores 0 To nd the percentile of a positive score nd the percentage between the score and the mean and add 50 0 Negative 2 scores 0 To nd the percentile of a negative 2 score nd the percent beyond the score ie the tail of the distribution 0 to nd above the negative score nd the percentage between the score and the mean and add 50 0 at least as extreme o to nd the percentage more extreme in either direction nd the percentage beyond the score and multiply it by 2 this tells us the percentage of scores that are least as extreme as a given 2 score ex before working out percentages it helps to draw a normal curve what is the percentile for a zscore of 123 o 390750 8907 what is the percentage beyond a zscore of 123 0 1093 what percentage of scores are more extreme 123 o 10932 2186 for an IQ test p 1000 15 Fred has an IQ of 105 what percentile is Fred in ZFred X p O 105100 115 33 ile Fred 1295 50 6223 what percent of individuals have lQ s higher than Fred 3707 What percent of individuals have scores at least extreme as Fred 37072 7414 suppose the average US female height u 645 in o 25 in how tail is a female in the 68th percentile X z o p X z 25 645 6568 47 25 645 found 2 score by 6850 and then looking at mean to z for 18 then using that z score How tail is a female in the 25th percentile because its negative percentile under 50 looking for in tail 6283 67 25 645 Z TABLE AND DISTRIBTION OF MEANS 0 Typically we are interested in samples instead of individuals 0 We can use the mean of our sample and compare it to the population mean 0 This is known as 2 test compares samples to population Assumptions a characteristics we ideally require the population from which we are sampling to have so that we can make accurate inferences o Parametric test inferential stats based on set of assumption about population what well do in this course 0 Nonparametric test inferential stats based on less assumptions about the population beyond this course A statistic is robust if produces a fairly accurate result eve when the data suggests that the population might not meet some of the requirements 3 ASSUMPTIONS 1 the DV is asses using a continuous eg scale measure 0 interval and ratio 2 participants are randomly selected 3 distribution of the population of interest is approximately normal HYPOTHESIS TESTING STEP 1 0 Identity populations distributions and assumptions for test to be used 0 Check that its okay to proceed with the planned hypothesis test 0 check PowerPoint slide for more info 0 STEP 2 0 State the null and research hypotheses Both hypotheses are about the population we are trying to generalize too and not the sample collected 0 Null hypothesis Ho No changeno difference 0 Research hypothesis H1 changedifference STEP 3 0 Determine the characteristics of the comparison distribution What parameters and statistics have to be know in our data set in order to complete the hypothesis test 0 Z test Mean of means standard error Basically computing a 2 statistic STEP 4 0 Determine critical values or cutoffs Decide before running any tests what scores will constitute a signi cant difference between our populations 0 Critical values the test statistic values beyond which we will reject the null hypothesis 0 Critical region area under the distribution curve in which if the test statistic falls we will reject the null hypothesis Also known as alpha 0 Common practice is to set a 05 means that we are going to reject the null if the z stat we calculate is equal or greater than 5 of data 0 Corresponds to the set of scores in the 2 distribution that are higher then that value we would reject our null hypothesis if we got a 2 statistic if there were fewer than 5 of that cut off point 0 196 two tailed test 0 164 one tailed test STEP 5 0 Calculate the test statistic Use the information from step 3 to nd the value of our STEP 6 0 Make a decision Compare your test statistic to the critical values determined in step 4 Decide if you should reject or fail to reject the null hypothesis based on the location under the distribution curve of your test statistic o Statistically signi cant pattern in the data differs from what we would expect by chance Reject the null hypothesis ZTEST EXAMPLE DATA 0 Consideration of Future Consequence CFC Scale 0 How able are you to look ahead and acknowledge the impact of the decision you are making now 0 From a national database of incoming college freshmen 0 Population distribution u 351 o 61 0 We allow students to volunteer and be in a career discussion group and then measure them with the VFV 0 Career discussion group N 45 mean 37 0 Does this discussion group differ from the population on the CFC Step 1 0 Identify populations distributions and assumptions for test to be used 0 Comparison pop national database 0 Pop Of interests career discussion group 0 Distribution 2 dist Of means Step 2 Assumptions 0 Continuous DV Yes scores on the CFC scale 0 Random selection Students elect to take career discussion course 0 Normal distribution Sample size of 45 Step 2 0 State the null and research hypothesis Null there will be no difference between the CFC scores from the quotcareer discussionquot group and the national database 0 Null statistically looks like u pm 0 Research there will be a difference between the CFC scores from the quotcareer discussionquot group and the national database 0 Research statistically looks like u NOT um Step 3 0 Determine the characteristics of the comparison of the comparison distribution dist Of means 0 Need the population parameters u 351 o 61 0 Need distribution of means statistics um 351 om osq Root N 61 sq Root 45 09 Step 4 0 Determine critical values or cutoffs 0 Use alpha 05 0 We have a twotailed hypothesis no direction so well use the extreme 25 of space under both tails o This will equate to zscores at the 25 and 975 percentiles o bc its twotailed zcrit196 alpha 05 o onetailed test hypothesis test in which the research hypothesis is direction positioning either a decrease of an increase in the DV but not both as a result of the IV 0 two hypothesis test in which the research hypothesis does not indicate a direction of difference Step 5 0 Calculate the test statistic 0 Calculate the zscore for the sample mean within a distribution of means zstatistic o 2 37 35109 211 Step 6 0 Make decision 0 211 is higher than the zcrit so we reject the null p lt 05 o the focus group has a higher mean than the gen population does CONFIDENCE INTERVALS Cl Interval estimates Calculating Cl s with z distributions EFFECT SIZE 0 Effect of sample size on statistical signi cance o What effect size is Cohen s d STATISTICAL POWER 0 Five factors that affect statistical power 0 Importance of statistical power GENDER DIFFERENCES IN MATH Some studies have shown gender differences in mathematicssexuaitysefesteemetc o they rejected their null hypothesis 0 when we conduct a hypothesis test we only have two outcomes 0 reject o fail to reject an accurate understanding of gender differences may not be found in significant effects 0 each distribution has variability each distribution has overlap Hyde Fennema amp Lamon 1990 Conducted a metraanalvsis the statistical analysis of a collection of results from individual studies for the purpose of integrating ndings Concluded 0 Mean gender differences in mathematical reasoning were very small 0 When extreme tails of the distribution were removed differences were smaller and reversed direction favored girls 0 The gender differences depended on task Females were better with computation Males were better with problem solving 0 gender difference in mathematics performance very similar ovenap HYPOTHESIS TESTING Statistically signi cant does not always mean quotvery importantquot 0 There is a real measurable difference says nothing about the size or reliability of the difference 0 Knowing two means differ is only part of the story 0 We also should know how much two distributions overlap 0 Notice how much men and women overleaped on the mathematics performance 0 Testing whether one number that s supposed to characterized a population parameter is different from population parameter we are comparing it against 0 quotAre these two the samequot hypothesis testing provides us with a point estimate 0 may want interval estimate 0 real life election polls Mitt Romney will get 51 of votes give or take 3 CONFIDENCE INTERVAL CI 0 Con dence interval interval estimate we would expect for a sample statistic a certain percentage of the time if we were able to sample repeatedly for ztests well create a con dence interval for the sample mean Mower 39 20M Msampe Mupper 20M Msampe o 2 critical value 0 0M standard error measure of precision smaller the oMthe smaller the interval EX Does posting calories on the menu at Starbucks reduce the amount of calories a customer consumer o u247o201 Sampled 1000 people who were provided a menu with calories listen M 232 o The calorie group was signi cantly lower than the Starbuck s population 2 236 Compared to 196 it is a big difference 196 Responds to alpha at 05 o What is the 95 con dence interval calories consumed by sample of participants given menus with calories listed Step 1 lt7 d7 0 O W o
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