Week 5: Sampling Distributions and Confidence Intervals

Week 5: Sampling distributions and confidence intervals Ch. 9 Randomized, comparative experiments - > Observational study - > An experiment is a study where the experimenter manipulates attributes and observes the consequences. · The attributes factors: variables that will influence results ↳combination of factors = treatments · observed responses : outcomes - Experiment us Observational study ! Experimenters actively and deliberally manipulates the factors. 2. Experimenters assigns the subjects to those treatments at random. 4 principles of experimental designs I . Control - controlling external sources of variation - -havingeroup making all conditions as similar as possible for all treatment groups. 2. Randomize - assign treatments at random ① 3. Replicate - make more than one observation at each level ↳ # of replicates is the same for each treatment combination = experiment is balanced. ② - repeat experiment for a different group of subjects, under different circumstances. 4. Block-blocking subjects together, analyzing the experiment accounting for each block. ↳reduces variation by comparing subjects within these more homogeneous groups. Experimental designs · one factor designs-single factor experiment with two levels - > completely randomized design-treatment combinations are assigned to participants at vandom. · Randomized block design-randomize subjects to the treatments within each block. - > group population based on the uncontrollable factor (e . g. age,gender, etc). · Multifactor design = an experiment with more than one factor. Issues in experimental design ! Blinding and placebos - To avoid bias : blinding the participants to the treatments. - 2 classes who can affect the outcome of the experiment: I .who influences results subjects 2. who evaluate the results experimenters ↳ When all individuals in one class are blinded = Single-blind ↳ when everyone from both classes are blinded = double-blind - Have a "fake" treatment that looks like the real ones=placebo - > placebo effect-importance of effective blinding and importance of control group. 2. cofounding =when effects of two+ variables on a response cannot be distinguished from each other. - can arise because of lurking variables or poor design. ↳can be difficult to know which variable is responsible for the effect. Displaying data from designed experiments · When an observed difference is large enough in standard deviations to be convincing its: - > StatisticallySignificant = unlikely to have occurred only by chance Interaction plot ↳ explore whether an intersection between the two factors affects the response. Ch. 10 - > Sampling error - when a sample drawn from a population deviates from that true population. - > sampling variability-when results vary from sample toSample · can't control it but can predict how much different proportions will vary from sample to sample. Distribution of sample proportions Notations ↳The theoretical sampling distributions is the distribution of p = proportion in the population p = observed proportion in sample. all the sample proportions that would arise from all the possible samples 9 = l-p = proportion of failures of the same size with a constant probability of a "Success" · mean: Mp = =C · SD =p = 19 - should be D centered at the true proportion p · provided that the sample values are independent and the sample size is large enough, the sampling of p is modeled by a Normal model. · claim only approximately true, model becomes better as sample size gets bigger. · 68-95-99. 7 rule to describe the sampling distribution model. Assumptions and conditions · Two assumptions in the model for the distribution of sample proportion: 1. Independence assumptions : sampled values must be independent of each other. 2. Sample size assumptions: Sample size, n, must be large enough. · Randomization condition - subjects should have beenrandomly assigned to treatments · 10% condition - sample size, n, should be no larger than 10% of the population. · success/failure condition - up > 10 (success) - nq > 10 (failure) Confidence interval ↳refers to the probability that a population parameter will fall between a range of values. · An estimation ofthe SD of a sampling distribution=a Standard error - > what does "95% confidence" : SE = really mean? · we are 95% confident that the true proportion lies in our interval. -purple dots are the simulated proportions of adults in each sample who thought economy was improving. - OrangeSegments show the confidence intervals found for each simulated sample. - green line represents the true percentage of adults who thought the economy was improving. ↳most of the Simulated confidence intervals include the true value - except one. · Normal model assures us that 95% of the intervals in the pile are winners, covering the true value, and only 5 % on average miss the target. =>that why we 95 % confident that our interval is a winner. Independence assumption · Randomization condition: proper randomization can help ensure independence. · 10 % condition: sample is less than 10 % of the population sampled. Sample size assumption · success/failure condition: contain at least 10 successes and 10 failures . Margin of error ↳the extent of that interval on either side of p is the ME. -for 95% =2SE(p) its 2 SD-for 997% ↑ confident= margin its 3 SD of error. - >confidence intervals looks like this: estimate IMEX Critical values -the value of the test statistic (z-value) which defines the upperlower bounds of a confidence interval . · wider confidence interval = more confidence - >the number of SEs we must stretch out on either side of is called the critical value = 2* - E. g.for 95 % confidence interval, the critical value is 2*=1 . 96 . =>meaning 95% of a normal model is found within Fl . 96 SDs of the mean. choosing the sample size · larger sample size=less variability ME = z * · Finding the right sample size : 1 . Look at ME: 0. 03 = 1 . 96 · 2. To solve for n , we need a value for - we can guess a value, cautious approach is to use p = 0.5 0.03 = 1 . 96 · -0605n = 1 . 96 : 0.5 . 0 . 6 = 32. 67 0. 03 n= 1067. /obe sand -n = 1068 3. Company will need at least 1068 respondents to keep the margin of error as small as 3% with a confidence level of 95%.