ࡱ> egd ӪbjbjQFQF 43,3,z5%((kkkkk8(I."kHmHmHmHmHmHmH$KJN|HQk999HkkH"""9kkkH"9kH""RCERxwL DWHH0(I)DNZN<EE8Nk/F(" YHH?(I9999N( 1: AP Statistics Chapter 8 Estimating with Confidence Confidence Intervals: The Basics Point Estimator and Point Estimate Confidence Interval, Margin of Error, Confidence Level Interpreting Confidence Levels and Confidence Intervals. AP Exam Tip Calculating a Confidence Interval Facts about CI Conditions for Constructing a CI 8.2 Confidence Intervals: Estimating a Population Proportion Standard Error Construct a Confidence Interval for p Confidence Intervals: Four-Step Process Choosing a Sample Size 8.3 Confidence Intervals: Estimating a Population Mean The z distribution Choosing a Sample Size The t distribution Facts about the t distribution: t Confidence Interval Robust procedures Objectives: -Interpret confidence levels and confidence intervals -Construct a confidence interval Our goal in many statistical settings is to use a sample statistic to estimate a population parameter. In Chapter 4, we learned if we randomly select the sample, we should be able to generalize our results to the population of interest. In Chapter 7, we learned that different samples yield different results for our estimate. Statistical inference uses the language of probability to express the strength of our conclusions by taking chance variation due to random selection or random assignment into account. In this chapter, well learn one method of statistical inference confidence intervals so we may estimate the value of a parameter from a sample statistic. As we do so, well learn not only how to construct a confidence interval, but also how to report probabilities that would describe what would happen if we used the inference method many times. A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called the point estimate. Ideally, a point estimate is our best guess at the value of an unknown parameter #1,3 A confidence interval for a parameter has two parts: An interval calculated from the data, which has the form estimate  EMBED Equation.3  margin of error or statistic  EMBED Equation.3  (critical value)* (standard deviation of statistic) The margin of error tells how close the estimate tends to be to the unknown parameter in repeated random sampling. Margin of error plus or minus 3 percentage points is the shorthand for this statement: If we took many samples using the same method we used to get this one sample, 95% of the samples would give a result within plus or minus 3 percentage points of the true population proportion (mean). #5 A confidence level C: The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. #7,9 Confidence level: To interpret a C% confidence level for an unknown parameter (mean/proportion), say Intervals produced with this method will capture the true ____mean/proportion in about ___% of all possible samples of this same size from this same population. Confidence interval: To interpret a C% confidence interval for an unknown parameter, say We are _____% confident that the interval from _____ to _____ captures the true ___[population parameter in context]____. The price we pay for greater confidence is a wider interval. If were satisfied with 80% confidence, then our interval of plausible values for the parameter will be much narrower than if we insist on 90%, 95%, or 99% confidence. #11,13 On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you understand the difference: the confidence level describes the long-run capture rate of the method, and the confidence interval gives a set of plausible values for the parameter. The confidence interval for estimating a population parameter has the form statistic  EMBED Equation.3  (critical value)* (standard deviation of statistic) where the statistic we use is the point estimator for the parameter. The confidence level decreases. There is a trade-off between the confidence level and the margin of error. To obtain a smaller margin of error from the same data, you must be willing to accept lower confidence. The sample size n increases. We like high confidence and small margin of error. A small margin of error says that we have pinned down the parameter almost precisely. (Remember that  EMBED Equation.3  and  EMBED Equation.3  . So as the sample size n increases, the standard deviation of the statistic decreases.) Random: The data come from a well-designed random sample or randomized experiment Normal: The sampling distribution of the statistic is approximately Normal For means: The sampling distribution is exactly Normal if the population distribution is Normal. When the population distribution is not Normal, then the central limit theorem tells us the sampling distribution will be approximately Normal if n is sufficiently large (n e" 30). For proportions: We can use the Normal approximation to the sampling distribution as long as np e" 10 and n(1  p) e" 10. Independent: Individual observations are independent. When sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use our formula for the standard deviation of the statistic. #17 Objective: -Carry out the steps in constructing a confidence interval for a population proportion: define the parameter; check conditions; perform calculations; interpret results in context. -Determine the sample size required to obtain a level C confidence interval for a population proportion with a specified margin of error. -Understand how the margin of error of a confidence interval changes with the sample size and the level of confidence C. - choose a sample size Assignment 8.2 page 496 to #27-47 odd, 49-54 all When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic. The sample proportion  EMBED Equation.3 is the statistic we use to estimate p. When the Independent condition is met, the standard deviation of the sampling distribution of  EMBED Equation.3  is  EMBED Equation.3  If we dont know the value of p, we replace it with the sample proportion  EMBED Equation.3   EMBED Equation.3  This quantity is called the standard error (SE) of the sample proportion EMBED Equation.3 . It describes how close the sample proportion  EMBED Equation.3 will be, on average, to the population proportion p in repeated SRSs of size n. One-Sample z Interval for a Population Proportion: Choose an SRS of size n from a large population with unknown p of successes. A level C confidence interval for p is Formula: Estimate ( margin of error Statistic ( (critical value)*(Standard deviation of statistic)  EMBED Equation.3  Calculator: STAT ( TESTS (1-PropZ Interval ( Data or Stats C.I. Tail area z*(Critical Value) 80% 0.1 1.282 90% 0.05 1.645 95% 0.025 1.960 99% 0.005 2.576 Conditions for constructing a CI for p: 1. Data must come from a random sample or randomized experiment 2. At least 10 successes and failures; that is  EMBED Equation.3 and  EMBED Equation.3  3. Observations are independent; 10% condition is met State: What parameter do you want to estimate, and at what confidence level? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform the calculations. Conclusion: Interpret your interval in the context of the problem Choosing a Sample Size for Desired Margin of Error To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n:  EMBED Equation.3  where EMBED Equation.3 is a guessed value for the sample proportion. The margin of error will always be less than or equal to ME if you take the guess EMBED Equation.3 to be 0.5 Objective: - When  EMBED Equation.3 is known, construct and interpret a one-sample z interval for a population mean - When  EMBED Equation.3 is unknown, construct and interpret a one-sample t interval for a population mean - -Determine the sample size required to obtain a level C confidence interval for a population mean with a specified margin of error. Assignment 8.3 page 518 #55-73 odd, 75-80 One-Sample z Interval for a Population Mean: Choose an SRS of size n from a population with unknown mean  EMBED Equation.3 and known standard deviation  EMBED Equation.3 of successes. A level C confidence interval for  EMBED Equation.3  is Formula:  EMBED Equation.3  Calculator: STAT ( TESTS (Z Interval ( Data or Stats Conditions for constructing a CI for EMBED Equation.3  : 1. Random: Data must come from a random sample or randomized experiment 2. Normal: Population distribution is normal or a large sample ( EMBED Equation.3 ) 3. Independent: Observations are independent; 10% condition is met Choosing a Sample Size for Desired Margin of Error When Estimating EMBED Equation.3  To determine the sample size n that will yield a level C confidence interval for a population mean  EMBED Equation.3  with a maximum margin of error ME, solve the following inequality for n:  EMBED Equation.3  (Always roundup when finding n) When we do not know , we substitute the standard error (SE)  EMBED Equation.3  of x for its s.d.  EMBED Equation.3 . The statistic that results does not have a normal dist. It has a distr. called the t distribution. The t distribution has a different shape than the standard Normal curve: still symmetric with a single peak at 0, but with much more area in the tails.  INCLUDEPICTURE "http://jpkc.njmu.edu.cn/course/tongjixue/file/jxzy/ybzzSD/images/fig14-2_0.jpg" \* MERGEFORMATINET  The statistic t has the same interpretation as any standardized statistic: it says how far  EMBED Equation.3 is from its EMBED Equation.3  in standard deviation units. There is a different t distribution for each sample size. We specify a particular t distribution by giving its degrees of freedom (df). The t Distribution; Degrees of Freedom Choose an SRS of size from a large population with mean  EMBED Equation.3 and standard deviation  EMBED Equation.3   EMBED Equation.3  with degrees of freedom df = n-1. The density curves of the t distr. are similar in shape to the standard normal curve. They are symmetric about zero, single peaked, and bell shaped. The spread of the t distribution is a bit greater than the standard normal distr. The t distribution has more probability in the tails and less in the center than the normal. As the degrees of freedom increase, the t density curve approaches the N(0,1) curve. This happens because  EMBED Equation.3 for the fixed parameter  EMBED Equation.3  causes little extra variation when the sample is large. One-Sample t Interval for a Population Mean: Choose an SRS of size n from a population with unknown mean. A level C confidence interval for  EMBED Equation.3  is Formula:  EMBED Equation.3  with df = n 1 Where t* is the critical value for  EMBED Equation.3 distribution. Table B gives critical values for the t distribution. By looking down any column, you can check that the t critical values approach the normal values as the degrees of freedom increase. When the actual df does not appear in the table, use the greatest df available that is less than the greatest df available. Calculator: STAT ( TESTS (T Interval ( Data or Stats Conditions: 1. Random: Data must come from a random sample or randomized experiment 2. 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Using One sample t Procedures Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more important than the assumption that the population distribution is normal. Sample size less than 15. Use t procedure if the data are close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t. Sample size at least 15. The t procedures can be used except in the presence of outliers or strong skewness. Large samples. The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ( 30.     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