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1、Point and Interval Estimation with Confidence,Another Spectacular Power Point Presentation,Point Estimates,Definition: A single number that is based on sample data, and represents a reasonable value of the characteristic of interest. Things Like: x s M - The simplest approach to estimating a populat
2、ion value.,Lets Look at an Example,A recent survey asked high school seniors how many hours they spent on line each day. Find the point estimate of with the data below. 4 7.25 5 5.25 6.25 6.25 4.5 3 .5 4.25,What could we use to estimate ? M or x Find both: x = 46.25 = 4.625 10 M = 4.75 Which one is
3、better? Recall when it is best to use each one,Confidence Intervals (estimating ),Things to Remember About the Distribution of x: 1) Has a normal distribution (by CLT) 2) Is an unbiased estimator of (x = ) 3) If permission is granted: s = /n Issues with Point Estimates: if we re do the sample, will
4、we get same stat?,Confidence Intervals,Def: the interval of reasonable values of the characteristic of interest Pattern for the formula: (whats on AP test) statistic critical value(standard deviation of stat) For Estimating : x z*(/n) Margin of Error,Confidence Levels,Def: the success rate of the me
5、thod used to construct the interval. Ex: A 95% confidence interval = if we took lots of samples and found the mean over and over again, in the long run 95% of the resulting intervals would capture the true value of the parameter. If we go through the same process with the same sample size, , and z*,
6、 but different x, 95% of those intervals will contain . (CI applet),Calculating Confidence Intervals,What is z*? A location on the normal curve that results in the needed confidence level. Ex: Use Table C! z* = 1.96,-z*,z*,95%,Calculating Confidence Intervals,Lets try another z*: What is z*? z* = 2.
7、326,-z*,z*,98%,Critical Values,z* is also called a critical value. Any ideas why? The area outside of the interval are called critical areas,PANIC (not at the disco),When calculating a confidence interval, remember to PANIC. P = Parameter of interest (what you are estimating). A = Assumptions (like
8、10n) N = Name the interval (x z*/n) I = Interval: low to high C = Conclusion in context,Conclusion in context: Mantra,I am 95% confident that the true population mean of context of problem is contained in this interval , (confidence interval). If this same procedure were repeated many times, then ap
9、proximately 95% of all intervals created would capture the true mean of context of problem (confidence level).,Calculating Confidence Intervals,Lets do one already! Gosh Suppose we need to verify the amount of active ingredient in a new drug. is known to be .0068 grams per liter. Here are the result
10、s of 3 measurements. Find a 99% confidence interval for the true concentration. .8403 .8363 .8447,Calculating Confidence Intervals,Lets PANIC: Parameter of interest is the mean . Assumptions: 10(3) = 30, should be OK. Name the interval: x z*(/n) What we need: x z* n .8404 2.576 .0068 3,Calculating C
11、onfidence Intervals,Fill in the formula: x z*(/n) .8404 2.576(.0068/3) .8404 .0101 Interval: .8303 to .8505 Conclusion: I am 99% confident that the true population mean amount of active ingredient is contained in the interval .8303 to .8505 g/L.,Calculating Confidence Intervals,Use the same problem,
12、 but do a 90% C.I. .8404 1.645 (.0068/3) .8404 .0065 .8339 to .8469,Compare the 90% to 99%,Compare the 90% to 99%,What happens to the range as the confidence level increases?,In order to be more confident you must have a wider range.,Compare the 90% to 99%,What happens to the margin of error as the
13、confidence level increases?,In order to be more confident you allow yourself more room to mess up!,How Intervals Behave,Higher confidence gives up “precision.” Want to have a small margin of error: - smaller z*: again calls for lower confidence - smaller : smaller variation among individuals - large
14、r n: better picture of population AKA: Law of Large Numbers Need 4x sample size to cut m of e in half,Choosing a Sample Size,Usually decided by desired margin of error (m) Formula: m z*(/n) Make sure to round appropriately! Population still needs to be 10X as big as sample!,Choosing a Sample Size,Ex
15、ample: From previous problem, what sample size is required to produce results .005 with a C.I. of 95%? .005 1.96 (.0068/n) Do some algebra! n(1.96.0068)/.005 n 7.1 so n = 8 because 7 is not enough!,Cautions!,1) Data must come from a SRS! We dont get into stuff for stratified, or blocked. 2) Bias is still bad! Fancy formulas cannot rescue badly produced data! 3) Outliers can have a great effect on intervals (think about why). 4) Need sample sizes of 15 or more. 5) For now we are assuming we know .,
