Descriptive Statistics Statistical Inference 3 24 14 9 12 PM Midterm 3 March 26 2014 Study Guide Five Step Model for Hypothesis Testing 1 Make assumptions and determine if they are met 2 State the null hypothesis 3 Determine the sampling distribution 4 Calculate the test statistic 5 Interpret the results o Compare your test statistic to the critical value from the sampling distribution Remember you need to calculate the degrees of freedom to determine the critical value If the test statistic is larger then reject the null Descriptive Statistics Samples vs Population o Population any well efined set of unites of analysis o Variables Nominal described based on frequency categories Ordinal quantities that have a natural ordering ranking rating Interval Similar to ordinal except intervals between each value are equally split temperature Ratio Interval data with a natural zero point time o Deviations The difference between an observed value and the mean Tell us how spread out the data is Find mean of data set then how far each point is from the mean deviations The total sum of deviations is zero some values negative some positive In order to get rid of negative sign square the deviations sum of squared errors The sum of the squared errors divided by the number of Typically tells us how much a data point differs from the o Variance data points mean o Standard Deviation Square root of the variance o This all applies if looking at population If looking at sample we divide the sum of the squared errors by the sample size n minus one Samples are drawn from a theoretically constituted population o Sample parameters are estimates of population parameters AKA our real goal Larger sample smaller sampling error estimates of population parameters more precise o Standard error of the mean equals the standard deviation of the sample over the square root of the sample size Statistical Inference Simple Random Sample o A sample is chosen from a population o Each individual is chosen randomly and entirely by chance such that each has a probability of being chosen at any stage during the sampling process unbiased Sampling Distribution o Any sample we take is one of a number of samples we could have taken o We calculate the statistic take it and store it do the same process with another sample infinite number of samples theoretically Produces a plot distribution of possible means Plot of means will look normally distributed if done with a large sample size Central Limit Theorem variable o About sampling distribution says nothing about distribution of o Tells you that for means with a large enough sample size the sampling distribution of the mean will be normally distributed Standard Error of the Mean Standard dev of sample divided by square root on n sample size An estimate of the standard deviation of the sampling distribution Larger sample size smaller fraction smaller std smaller standard error narrow sampling distribution unlikely that we ll pull a sample which will lead us to a poor estimate of the mean AKA better estimates 95 Confidence Interval of the Mean o Xbar 1 96 std sqrt of n o Ex Xbar 7 Variance 25 Sample size n 100 Give answers for what numbers fill formula Answer 7 1 96 5 sqrt100 Actually calculating LB and UB LB 2 8 UB 3 2 We don t know the true mean just fairly certain that it falls somewhere between there 95 of the time 2 Tests The Different Distributions 03 24 2014 Standard Normal Distribution 95 conf interval o The sampling distribution of the mean is always a normal distribution o Mean 0 median 0 mode 0 o 68 95 99 Rule All values lie within three standard deviations of the mea in a normal distribution t Distribution diff of means two t tests o Essentially the normal distribution same shape mmm 0 o Deals with smaller sample sizes more likely to end up in tail ends of distribution Although used for various sample sizes much of the things we do have sample sizes large enough where t dist looks almost exactly like normal dist o Critical Values For large sample size critical value of t Pval 05 t 1 96 Pval 10 t 1 64 Ex If we calculate t 1 75 we can reject the null at 10 level because it is greater than that but can t reject 05 because it is less than 1 96 2 Distribution sampling distributions o 1 Calculate the marginal o 2 Calculate the expected table Row value times column value divided by total n o 3 Observed Expected 2 Expected For each cell o 4 Add up the values to get 2 o 5 Calculate degrees of freedom rows 1 columns 1 o 6 Is the calculated 2 larger than the critical value The Research Hypothesis The relationship we expect to see in our sample Relationship is expected or no relationship null none There is no relationship between the two variables The Null Hypothesis P values Ranges from 0 1 your result o The closer to 0 lower value the more confident you are in As a norm we want a p val smaller than 05 which means we are at least 95 confident in our result and no less than 01 o Systematic relationships no random chance Two mistakes that can be made Type I Error o Rejecting the null when the null is true o This treatment has an effect but it actually doesn t o Lower p val prevents this we want to be confident that the relationship were seeing is real Type II Error o Accepting the null when the null is false o 0 chance this drug cures disease when it actually does o Low p val increases chance of type II error Statistical Significance associated SS relationship means we are very confident that two variables are o Doesn t say anything about how large association is An actual calculated statistic T test Substantive Significance Given our sample we can rule out the probability that what we are observation is random chance how strong is the relationship between these two variables o About argument why this statistic matters Measures of Association o Tells you how strong the relationship between two variables is Tabular Analysis with the 2 test Need to know when do we go to 2 test o Categorical independent variable ordinal or non ordinal Ex Regions S SW E etc o Categorical dependent variable ordinal or non ordinal Ex Flavor of soda Degrees of freedom n 1 o rows 1 columns 1 o Two tables in back of test t table don t look at and 2 table Will ask what critical value is for DOF look up in tables Will provide table on test o Asks whats 2 How many DOF o Expected value if variables are random o Sum E 0 2 E 2 0 accept null no relationship between variables The correlation coefficient r Won t be
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