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AP Statistics- Final Review.pdf

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New York University
James P.Stanley

AP Statistics: Final Test 7 I. Forms of Studies/ Sampling A. Simple Random Sample (SRS): each subject is equally likely to be selected; each sample of size is equally likely to be selected B. Stratified Random Sample: divide the population into strata (plural); from each stratum (singular), select an SRS C. Proportionate Allocation: the sample size is partitioned among the strata proportional to the size of the strata D. Stratified Random Sampling: population is divided into homogeneous (same) groups. An SRS is them selected from each group. (Note: Groups are heterogeneous with respect to each other.) E. Systematic Random Sampling: a random sample is sampled, then every kth subject after him/her is sampled F. Cluster Sampling: separate the population into heterogeneous groups. With respect to each other, the groups are homogeneous G. Convenience Sampling: Conducted in 1 location to obtain the responses of a convenient group. Problem: Sample does not always represent the population H. Multistage Sampling: combines different types of sampling II. Bias A. Selection Bias: the sample is not representative of the population B. Voluntary Response Bias: people volunteer to respond causing the sample to be non- random C. Measurement Bias: the instruments used for recording data are not accurate D. Response Bias: anything in a survey that influences responses E. Nonresponse Bias: people refuse to respond or can’t be reached III. Studies A. Observational Study: we observe people who are selecting their own habits and we collect data on the variable(s) of interest B. Experimental Study: treatments are imposed on subjects. The goal is to attribute cause/ effect C. Retrospective Study: an observational study in which subjects are selected and then their previous conditions or behaviors are determined D. Prospective Study: an observational study in which subjects are followed to observe future outcomes IV. Experimental Design A. Experimental Units: that which we are applying a treatment to (when experimental units are people, we call them subjects) B. Explanatory Variable (Factor): whatever we expect to affect outcomes in the study (there can be more than 1 in a study) C. Response Variable: the variable whose values are compared after treatment D. Levels of the Factor: values that factor(s) can take on E. Treatments: if 1 factor, treatment is same as factor; if more than 1 factor, all combinations of levels of the factors F. Diagram of Experimental Studies (in order) SUBJECT --> (RANDOM ALLOCATION) --> GROUPS --> TREATMENT --> MEASURE ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ RESULTS G. Principles of Experimental Design 1. Randomization a) Goal is to equalize the groups with respect to variables that may impact the response 2. Control a) Control sources of variation 3. Blocking a) When groups of subjects are similar, gather them into blocks (reduces variability in results) 4. Replication a) Repeat the experiment using different subjects OR use multiple subjects in each treatment group H. Matched Pairs Design: participants in different conditions are matched to certain characteristics V. Mean and Standard Deviation A. Use x for sample mean which is a part of a population B. Use µ for population mean which is the mean of all sample means x C. Use S xfor the sample standard deviation D. Use σ xfor the population standard deviation VI. Sampling Distribution of Sample Means A. We select an SRS of size n and record mean ( x ) for the variable of interest B. Continue selecting SRS’s of size n and recording x . There are N C n possible SRS’s ( N < population size) C. The distribution of all means is the sampling distribution of sample means VII. Properties of the Sampling Distribution of Sample Means A. The mean of the sampling distribution of sample means is equal to the population mean B. The standard deviation of the sampling distribution of sample means is found by σ = σ x where σ is the population standard deviation and n is the sample size x n x C. In most cases, the sampling distribution of sample means is roughly Normal. VIII. Sample Size and Binomial Model A. When a parent population is Normal, the sampling distribution of sample means is Normal B. If the sample size ≥ 30, the sampling distribution of sampling means is Normal C. The binomial model is approximately Normal when np ≥ 10 and nq ≥ 10 Test 8 IX.Confidence A. How to Find the Confidence Interval * * 1. The confidence interval is found by p±margin of error or p±(z ×se(p))ˆ where z is the z score for the confidence level and se(p)is the standard error 2. The z score is found by marking the percentage of the confidence level as the center of a Normal distribution with half of the confidence level to the left of the center and the other half to the right of the center; then, find the area left or right of the marked off point; then find the z score of that and use the absolute value of it pq 3. The standard error is , found by se(p) = n 4. The confidence interval is reported in parenthesis, where the lower value is followed by a comma and the higher value B. Margin of Error 1. The margin of error decreases when the sample size increases 2. If we want to keep our confidence level but reduce the size of our confidence interval, we need to make the margin of error smaller 3. Smaller samples will produce larger margin of errors C. Standard error is the same as standard deviation but it takes into account of the sample size. D. Standard error decreases as sample size increases E. Reasonable Variability: the amount of variability we expect to see between the different samples of the same size X. One- Proportion Z-Test A. Assumptions and Conditions 1. Was the selection done randomly? 2. Is each selection independent of each other? 3. np ≥10;nq ≥10 o o 4. Is the population more than 10 times the sample? B. Alpha Value 1. Lower alpha value --> rejection region is lower 2. Higher alpha value --> rejection region is higher 3. If p-value < or = to alpha value, reject the null hypothesis 4. If p-value > alpha value, fail to reject the null hypothesis C. How to do a 1- Proportion Z-Test 1. Set up null and alternate hypotheses 2. Verify conditions and assumptions 3. Find p 4. Calculate zˆp 5. Find the p-value and interpret it D. A p-value is statistically significant if the null hypothesis is rejected E. In a two- tailed hypothesis test, multiple the p-value by 2 because both extremes are being accounted for Test 9 I. Confidence Intervals A. Confidence Level = 100(1-α)% B. If 0 is not captured by a 100(1-α)% confidence level (p is no0 in the range of the interval), reject 0 C. If p0is captured by a 100(1-α)% confidence level (p is in0the range of the interval), fail to reject H0 II. Errors and Power A. Type I Error: made when H was0true but was rejected B. Type II Error: made when H was0false but was failed to be rejected C. Power: made when H is f0lse and was rejected D. The correct decision occurs when H was 0rue and failed to be rejected E. The probability of a Type I error is α F. The probability of a Type II error is β G. α and β do not add up to 1! H. The power is 1- β I. Effect Size: the distance between P an0 P (null and true) J. All other things being held equal, as effect size increases, the probability of a type II error decreases, the probability of a type I error increases and the power increases K. If we want to decrease the probability of a type I error, we can decrease α L. If we want to decrease the probability of a type II error, we can increase effect size or sample size III. 2-Proportion A. Assumptions and Conditions for Inference about a Proportion 1. Both data are from an SRS from the population of interest 2. Both populations are at least 10 times as large as the samples 3. n is so large that n1p1, n1(1− p1), n2p2, n2(1− p 2 are 5 or more IV. Hypothesis Testing for Means A. Assumptions and Conditions 1. Randomization Condition OR The sample is representative of the population 2. Independent Condition 3. 10% Condition 4. Normal or nearly Normal Condition B. Because this is concerned with means and not proportions, the mean for testing is µ x, σ x the mean of the sample is x and the standard deviation of the sample is
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