Transcript Chapter 1
CHAPTER 1 EVERYTHING YOU EVER WANTED TO KNOW ABOUT STATISTCS Building Statistical Models • Discovering some thing about a phenomena • We explain a Phenomena by collecting data from real world and then draw conclusion about what i being studied • Analogy: Building a Bridge across a river Figure 1.1 • Model is an accurate represenation of the real world. • Social Scientists build real world models to predict how these processes operate under certain conditions • Simple Statistics Models: The Mean, sums of squares & Std Deviation • Mean: Represents a summary of data. 5 Stat lectures and measured the number of friends they had i.e 1,2,3,3 and 4 Mean is 2.6 (Fig 1.2) • Total error = sum of deviances • • • • • • Sum of Squared Errors (SS) – measures accuracy of model Variance: Measures how well the model fits the data = SS / N-1 Std Deviation: Measures how well the mean represents the data. Large and small standard deviation Model Outcome= Model + Error Figure 1.2 Model over estimates Red line is the model Error in the model Model over estimates SS: Sum of squared Errors, Variance and Standard Deviation Figure 1.3 Mean is a good fit of data Mean is a poor fit of data Frequency Distributions In an ideal world data is distributed symmetrically around the mean: Normal distribution Figure 1.4 Properties of frequency distributions Skewness Positively skewed (left) Negatively skewed (right) Figure 1.5 In a normal distribution the value is zero Kutosis: Distributions vary in their pointyness Leptokurtic Platykurtic Figure 1.7 In a normal distribution the value is zero 1.5.3 Standard Normal Distribution Beachy Head example Z scores Z=X–X /s Important Z Scores Z = +-1.96 i.e 95% of z scores lie between this range Z = +- 2.58 i.e 99% of z scores lie between Z = +- 3.29 i.e 99.9% of z scores lie between Figure 1.8 Is my sample representative of population Sampling distribution tells us the behaviour of samples Figure 1.9 Linear models Figure 1.11 Confidence Intervals • Lower boundary of confidence intervals = X – (1.96 x SE ) • Upper boundary of confidence intervals = X – (1.96 x SE ) How can we tell if our model represents the real world • • • • • • • • • • Generate a Hypothesis Collect useful data Fit a statistical model to the data Assess this model to see whether it supports initial predictions Prediction made by researcher Experimental Hypothesis (Alternative Hypo) Status Quo is Null Hypothesis Example: Hamburgers make you fat Fischer – Statistically Significant : When we are 95% certain that a result is genuine (i.e not a chance finding) should we accept it as being true. Or I f there is only a 5 % probability of some thing occuring by then we accept that it is a true finding. We say it is a statistically significant finding. Test Statistics Two Types of Variance • Systematic Variation Due to some genuie effect i.e that can be explained by the model that we have fitted to the data. • Unsystemativ Variation This variation is not due to the effect in which we are interested. i.e that can not be explained by the model that we have fitted to the data. Whteher a model is a reasonable representation of what is happening in the population • Calculate Test Statistics ( A test stat is simply a stat that has known properties i.e we know how frequently different values of this stat occur) • By knowing this we can calculate the probability of obtaining a particular value. • The test statistics are t, F & chi square. They represent the same thing test statistics = Variance explained by the model / Variance not explained by the model One and two tailed tests Figure 1.13 Type I & Type II Errors What we did beleive from the experiement Fail to Reject H0 Reject H0 True state of nature:What is in reality H0 true (Null is True) Correct decision 1- Type I error P(Rejecting the null hypothesis, when it is true) H0 false Null is false) Type II error P(Acceptinng the Null Hypothesis when it is fls) Correct Decision 1- Power Effect Sizes • R = .10 Small effect i.e Explains 1% of the total variance • R = .30 Small effect i.e Explains 9% of the total variance • R = .50 Small effect i.e Explains 25% of the total variance Statistical Power • The ability of a test to detect an effect of that size is known as statistical power. 1- Power