12 12 2010 Chapter 12 Comparing Two Groups Two Sample z Statistic Two Independent Simple Random Samples o Possibly coming from two distinct populations with M1 sd1 M2 sd2 o Use xbar1 and xbar2 to estimate unknown M1 and M2 Has standard normal sampling distribution Unpooled Two Sample t Statistic Two independent Simple Random Samples o Possibly coming from two distinct populations with M1 sd1 and M2 sd2 o Use xbar1 s1 and xbar2 s2 to estimate M1 sd1 and M2 sd2 Both populations should be normally distributed TO COMPARE MEANS Follows approximately the t distribution with a standard error spread reflecting variation from both samples Degrees of freedom is equal to smallest of n1 1 n2 1 Null hypothesis is that both population means M1 and M2 are equal o Thus their difference is equal to zero o With either a one sided or two sided alternative hypothesis We find how many SE away from M1 M2 is xbar1 xbar2 by standardizing with t Assumptions o Main populations must be normally distributed Use if sd1 is different than sd2 Because we have two independent samples we use the difference between both sample averages xbar1 xbar2 to estimate M1 M2 o C is the area between t and t Common mistake o Calculate a one sample confidence interval for M1 and then check whether M2 falls within that confidence interval or vice versa This is WRONG because variability in the sampling distribution for two independent samples is more complex and must take into account variability coming from both samples Hense more complex formula for standard error Two sample t procedures are more robust than one sample t procedures o Most robust when both sample sizes are equal and both sample distributions are similar o Choose equal sample sizes if you can o A combined sample size n1 n2 of 40 or more will allow you to work with even the most skewed distributions Pooled Two Sample t Statistic Assumes equal variance o Will be told that the population variances are the same Was often used before computers because it has exactly the t distribution for degrees of freedom n1 n2 2 o Assumption of equal variance is hard to check o Unequal variance test is safer When both populations have the same standard deviation the pooled estimator of Sampling distribution for x1 x2 has exactly the t distribution with n1 n2 2 sd sq is sp sq degrees of freedom Assumptions o Same variances o 2 populations are normally distributed Paired Samples tests means of two related populations o paired or matched samples ex Two different appraisals on the same home o repeated measures before after ex Weight before and after a diet program o use difference between paired values eliminates variation among subjects Assumptions o Both populations are normally distributed o Or the population of the differences is normally distributed The point estimate for the population mean paired difference is dbar n is the number of pairs in the paired sample Chapter 13 Inference for Counts Chi Square Tests only for categorical data overall technique for comparing any number of population proportions testing for evidence of a relationship between two categorical variables Testing a Probability Model o Data count data for a categorical variable with r categories o Model a probability model assigns a probability to each category o Aim want to know whether this model is appropriate for the data i e is the model a good fit Notation o Pi the probability that a given individual falls in the ith category o R v Ni represent number of individuals falling in the ith category when N1 N2 Nr n individuals are observed Takes on only positive values Chi square statistic against the null o The larger the value of the chi sq statistic the more evidence that we have o If null is correct chi square statistic follows a xsq distribution with r 1 df o r is the number of categories Characteristics of Chi Square distribution o It is not symmetric o The shape of chi square distribution depends on degrees of freedom just like Student s t distribution o As number of df increases chi square distribution becomes more symmetric o Values are non negative P value o Always the area to the right o Interpreted and compared with a significance level a in the usual way Xsq statistic can be unreliable if some of the categories have expected counts i e n pi I ei smaller than 5 o Categories with small expected counts should be combined to overcome the problem Goodness of fit Test is known Assumptions o Main null is true Test for Independence Inferential procedure conclusion reached on evidence and reasoning o Used to determine whether a frequency distribution follows a claimed distribution Null hypothesis states that the model is correct o Pi1 pi2 pir category probabilities according to model o Alternative hypothesis is that the null is not correct two way table expected count is computed differently Take one simple random sample and classify the individuals in the sample according to two categorical variables Use x sq test to test hypothesis of no relationship Have a single sample from a single population o for each individual in simple random sample of size n we measure two categorical variables o results are summarized in a two way table when testing null hypothesis we compare actual counts from the sample data with expected counts given the null relationship o expected count is mean over many repetitions of the study assuming no doesnt need to be a whole number chi square statistic o a measure of how much the observed cell counts in a two way table diverge from the expected cell counts o tip calculate x sq components first for each cell of the table and then sum them up to arrive at x sq statistic if observed and expected counts are very different o x2 will be large indicating evidence against the null o p value is always based on right hand tail of distribution the values summed to make up x sq are called the x sq components o when test is statistically significant the largest components point to the conditions most different from the expectations based on the null Test for Homogeneity n2 nc Comparing several populations Randomly select several simple random samples each of c populations of sizes n1 o Classify each individual in a sample according to a categorical response variable with r possible values o There are c different probability distributions one for each population Use x sq test to test hypothesis of no relationship Chapter 14 Inference for Regression Simple Linear Regression Data is given in
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