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AP Statistics Test Prep Table of Contents Part 1: Test Overview Part 2: Test Content Part 3: 2013 Released Qs Part 4: Specific Concepts Part 5: Stats in a Nutshell Part 6: Example Questions Part 7: Formulas Part 8: Calculator Shortcuts Part 9: Study Guide by Chapter © Richard Newton 2013 Slide 2 Slide 4 Slide 11 Slide 21 Slide 26 Slide 31 Slide 47 Slide 52 Slide 59 AP Statistics Test Prep Part 1: Test Overview AP Test Prep – Part 1: Test Overview AP Stats Test is May 10th at Noon! AP Statistics Exam Study Materials! Barron’s “AP Statistics” Princeton Review’s “Cracking the AP” Kaplan’s “5 Steps to a 5” Part 1: Multiple Choice 50% of overall Grade 40 Questions 90 Minutes Part 2: Free Response 50% of overall Grade 6 Questions (5 and 1) 90 Minutes Breakdown: Each Multiple Choice Question is 1.25% of the overall score Each of the first 5 free response questions is 7.5% of the overall score The final question is 12.5% of the overall score There is no penalty for guessing. AP Statistics Test Prep Part 2: Test Content AP Test Prep – Part 2: Test Content The course can be divided into 4 major topics: 1. Exploratory Analysis 2. Planning a Study 3. Probability 4. Inference You will see 1 Free Response on each of these and 1 “Investigative Task” which is basically where you hit all of them in a single question. Being able to look at a multiple choice or free response question and put it into 1 of these categories is one of the best skills you can learn. AP Test Prep – Part 2: Test Content Exploratory Analysis Ch.1) Categorical vs. Quantitative Data Types of Graphs Describe the Distribution: SOCS Ch.2) Density Curves (Area = 1) Normal Curves z-scores (Percentiles) Testing Normal: 5 Number Summary Box Plot Use histogram to check normal 68-95-99.7 Test Normal Probability Plot Ch.3) LSRL (y = a + bx, r, r2, residual) Ch.4) Marginal Distributions Conditional Distributions Simpsons Paradox Exponential Regression Power Regression - Look back at the various types of graphs and be familiar with which can be used with categorical vs. quantitative. You can expect to see at least 2 questions on stem plots – possibly with a back to back stem plot. You will see 2+ histograms as well. It is possible that you will see an O-give but unlikely you would have to convert it backward into a histogram. - Make sure you can find the area under a density curve if it is blocky. They’ll likely ask you to check if a data set is normal. - Make sure you’re able to create and interpret a LSRL line – in particular, refresh your memory on r, r2, and residuals! - Marginal and Conditional Distributions just means you’re able to interpret 2 way tables and answer questions about the data. AP Test Prep – Part 2: Test Content Planning a Study Ch.5) Designing Observational Studies Stratified Random Sample Multistage Sampling Design Bias Voluntary Response Convenience Sampling Leading Question Non Response Bias Response Bias Designing Experiments Matched Pairs Block Design Control and Placebo Simulations - If you split your observational study up by strata, you are stratifying it. - Be sure that you understand the specific types of bias! “The design is biased if it systematically favors certain outcomes.” - If you have each participant experience both treatments (1 after the other), you’ve used a Matched Pairs design. You could also attempt to match each participant with a very similar participant so that for each treatment, you have similar experimental units. This would be an attempt at a Matched Pairs design but I don’t think they’ll ask about this. - Make sure you can diagram experiments, especially if they are blocked. - Using a placebo is an attempt to create a control group to which you can compare your results. - Use randomInt to conduct simulations. AP Test Prep – Part 2: Test Content Probability Ch.6) Probability And (*)vs. Or (+) Replacement without Replacement Disjoint (not overlapping) Non Disjoint (overlapping) Ch.7) Discrete (integer columns) Continuous z-scores Combining Means Combining Variance Ch.8) Binomial Distributions Geometric Distributions Permutations Combinations Counting Principal - Remember the Bridge and Moat! - Most sequential probability questions are done with replacement (unless it says otherwise). - Diagram – tree or chart – probability questions wherever possible. Most can be reasoned out with a diagram and fractions alone. - Remember that discrete involves integer columns which makes them easier to work with but that means > and ≥ are different. - Continuous is composed of infinitely small columns and unless it’s rectangular or normal, you don’t have the tools to calculate the areas. - BinomPDF is a single column while BinomCDF is a single column and all the columns to the left. AP Test Prep – Part 2: Test Content Inference Ch.9) Sampling Distributions Central Limit Theorem Ch.10) Inferences when σ is known (z) Type 1 Error Type 2 Error Ch.11) Inferences when σ is unknown (t) Ch.12) Proportions 1propZtest 2propZtest Ch.13) Chi-Squared GOF Chi-Squared Homogeneity Ch.14) LSRL Significance Test Confidence Intervals: They ONLY want to see: 1. The interval (either notation is fine) 2. What the interval means “We are 95% confident the true mean (or proportion) is between ____ and ____.” - The sampling distribution is normal because of the Central Limit theorem if the rules of thumb are met. This is the principal which allows us to calculate probabilities and estimate standard deviation. ALWAYS: Talk about the conditions!! 1) Is the SRS from the population of interest? 2) How big is the sample size? Is that big enough to do the test you’re doing? You will get marked off big time if you don’t! - You will only see t tests (see chapter 11 middle board), prop tests (ch.12), chi-squared (ch.13), and LSRL tests (ch.14). - You can do this entire part with a calculator and get full credit! Just check the conditions, explain the hypothesis, show the t score and probability, and then clearly state if the Null or Alternative is supported! AP Test Prep – Part 2: Test Content Other topics that are not tested: Covariance (combining variance when they are not independent) Baye’s Theorem Pooled t-tests Chapter 14.2 – LSRL Confidence Intervals Read the Question Re-Read the Question Exploratory Analysis Categorical vs. Quantitative SOCS Center vs. Spread LSRL y a bx y ab x y 10 x a Designing Studies Significance Tests and Confidence Intervals Z-Tests vs. T-Tests Samples vs. Proportions Confidence Intervals Chi-Squared LSRL b Observational Studies vs. Experiments Simulations Probability Mutual Exclusivity Independence vs. Dependence And vs. Or Discrete vs. Continuous Binomial vs. Geometric AP Statistics Test Prep Part 3: 2013 Released Qs AP Test Prep – Part 3: 2013 Released Qs We are 95% confident that the true mean for lead levels in crows to be between 4.4157 and 5.3843 ppm At first I thought I had to do a 1propZinterval, but I realized since they gave me the mean mean and standard deviation of a single sample, I had to do a t-interval. AP Test Prep – Part 3: 2013 Released Qs Pop > 10n Football game attendees might be more school spirited which might cause bias in the results and overestimate the proportion which are satisfied with the appearance of the university buildings and grounds. Using the list of 70,000 names, assign each student a number from 1 to 70,000, then have the number generator generate 500 unique random numbers from 1-70,000. The students who match those numbers are selected for the survey. At first I thought this was saying the campuses were gender specific, but both campuses are mixed. If the campuses are different, such that the results of the survey when stratified by location is expected to give different results, then it would be important to survey each campus individually rather than by gender. AP Test Prep – Part 3: 2013 Released Qs Simple z-score question. i) z x 850 840 7.9 1.266 P % 100 89.8% 10.2% Combined mean = 840g 840g – 20g for the carton = 820g for the dozen eggs, divided by 12 = 68.33 grams per egg mean of X = 68.33g ii) Combined standard deviation = 7.9g, so Combined variance = 62.41g Container standard deviation = 1.7g, so variance = 2.89g 62.41 – 2.89 = 59.52g variance combined for 12 eggs 59.52 / 12 = 4.96g variance for a single egg 2.23g standard deviation for a single egg standard deviation of X = 2.23g AP Test Prep – Part 3: 2013 Released Qs H0: No association between age and fruit/vegi consumption. Ha: There is an association between age and fruit/vegi consumption. Because we’re given a 2-way table, I know that I need to do a Chi-Squared Test of Homogeneity. I need to first generate my Expected values. You can see those below. Then I need to do a Chi-Squared Homogeneity T-test. Yes, at a probability of 1.12% there is significant evidence at the 5% level to support the alternative hypothesis. There appears to be an association between age group and fruit/vegi consumption. AP Test Prep – Part 3: 2013 Released Qs This is an observational study, meaning that treatments were not applied. Thus, lurking variables could be at work so we would not want to make any conclusions about causation. If the situation were replicated and made into an experiment, control groups could be established to attempt to mitigate the impact of lurking variables. This would allow the researchers to make stronger conclusions about causation between meditation and blood pressure. AP Test Prep – Part 3: 2013 Released Qs b) There are a few issues – #1, the sample size of 11 is too small, it should be atleast 15 #2, To do a matched pairs design like this and set the u of the differences to zero for the Null, you want to have the same sample size Values over here indicate the proportion with high bp was higher for non meditators Values over here indicate the proportion with high bp was higher for meditators The simulated distribution favors the left side, where meditators have a lower proportion of individuals with high bp than non meditators. For this reason, it might be reasonable to conclude that there is a relationship between meditation and blood pressure. AP Test Prep – Part 3: 2013 Released Qs AP Test Prep – Part 3: 2013 Released Qs AP Test Prep – Part 3: 2013 Released Qs AP Statistics Test Prep Part 4: Specific Concepts AP Test Prep – Part 4: Specific Concepts Things that are important to remember: Type 1 Error: Rejecting the Null when it’s actually true - P(type 1 error) = alpha Type 2 Error: Accepting the Null when it’s actually false – P(type 2 error) is difficult to calculate (and not on the test, thankfully) Power = 1 – P (type 2 error) High power is good – it means you’re unlikely to commit a type 2 error Ways to increase power: increase sample size or move alternative further away Simulation: If you’re asked to explain how you would simulate something, make sure you clearly identify what different number values will mean, example: RandomInt(1,10) 1-3: Miss 4-10: make 10 trials, results are: 5, 3, 2, 8, 5, 10, 1, 1, 6, 7 4 misses, 6 makes Experimental Result = 40% miss rate Repeated this process 5 times and got: 40%, 60%, 10%, 30%, 40% Mean Miss Rate after 5 trials of experiment: 36% miss rate AP Test Prep – Part 4: Specific Concepts Any t-test will fall into one of these categories: 1 Sample t test – Use Calculator, set μ = whatever was given Matched Pairs – Use calculator (use 1 sample t test), set μ = zero 2 Sample T test – Use calculator, set μ1 = μ2 1 prop Z test – Use calculator, set p = whatever was given 2 prop Z test – Use calculator, set p1 = p2 Considerations: SRS comes from population of interest N ≥ 15 SRS comes from population of interest Pop ≥ 10n Np ≥ 10 or n(1-p) ≥10 X2 GOF – No shortcut in a TI83 - sum((L1-L2)2/L2) X2 Homogeneity – use calculator, Matrix A and B – df = (r-1)(c-1) SRS comes from population of interest LSRL – Use Calculator, df = n - 2 Observations are independent True relationship is linear S the same everywhere Residuals are normally distributed AP Test Prep – Part 4: Specific Concepts Confidence Intervals The formula is always the same: s C : x t* n You can just use your calculator to calculate most of them. Just remember that df = n – 1 unless it’s LSRL and then its df = n – 2 (but you shouldn’t see a LSRL confidence interval). Remember the definition of a confidence interval – it is very specific: “If you conducted the sampling process multiple times and created many additional confidence intervals, 95% of the created intervals should contain the true mean.” Or: “We are 95% confident that the true mean is between these two values.” AP Test Prep – Part 4: Specific Concepts Combining Means and Variances Let’s say that you and your friend are ordering 2 items at a restaurant – a burrito and a chicken salad. You know that the following about the time (in minutes) it takes to make each item: Burrito Chicken Salad mean 2 3 Combined mean is: Combined standard deviation is: standard deviation 1 2 AP Statistics Test Prep Part 5: Stats in a Nutshell AP Test Prep – Part 5: Stats in a Nutshell AP Test Prep – Part 5: Stats in a Nutshell AP Test Prep – Part 5: Stats in a Nutshell AP Test Prep – Part 5: Stats in a Nutshell AP Statistics Test Prep Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Test Prep – Part 6: Example Questions AP Statistics Test Prep Part 7: Formulas AP Test Prep – Part 7: Formulas AP Test Prep – Part 7: Formulas AP Test Prep – Part 7: Formulas AP Test Prep – Part 7: Formulas AP Statistics Test Prep Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Test Prep – Part 8: Calculator Shortcuts AP Statistics Test Prep Part 9: Study Guide by Chapter AP Test Prep – Part 9: Study Guide by Chapter Chapter 1 – Exploring Data AP Test Prep – Part 9: Study Guide by Chapter Chapter 2 – Normal Distributions – z scores AP Test Prep – Part 9: Study Guide by Chapter Chapter 3 – Examining Relationships - LSRL AP Test Prep – Part 9: Study Guide by Chapter Chapter 4 – 2 Variable Data AP Test Prep – Part 9: Study Guide by Chapter Chapter 5 – Producing Data AP Test Prep – Part 9: Study Guide by Chapter Chapter 5 continued – Producing Data AP Test Prep – Part 9: Study Guide by Chapter Chapter 6 - Probability AP Test Prep – Part 9: Study Guide by Chapter Chapter 7 – Random Variables AP Test Prep – Part 9: Study Guide by Chapter Chapter 8 – Binomial and Geometric Distributions AP Test Prep – Part 9: Study Guide by Chapter Chapter 9 – Sampling Distributions AP Test Prep – Part 9: Study Guide by Chapter Chapter 10 – Z Tests AP Test Prep – Part 9: Study Guide by Chapter Chapter 11 – T tests AP Test Prep – Part 9: Study Guide by Chapter Chapter 11 continued – T tests AP Test Prep – Part 9: Study Guide by Chapter Chapter 12 – Prop tests AP Test Prep – Part 9: Study Guide by Chapter Chapter 13 – Chi Squared AP Test Prep – Part 9: Study Guide by Chapter Chapter 14 – LSRL test