INTRODUCTION TO STATISTICS



Stat 1450 Course Notes – Chapters 14Confidence Intervals: The Basics Guided Notes associated with the Lecture Video for Section 14.1Connecting Chapter 14 to our Current Knowledge of Statistics Midterm I material, Probability Theory, Sampling Distributions (by way of the Law of Large Numbers, & the Central Limit Theorem) have all laid the foundation for us to use sample statistics to estimate population parameters. Statistical inference provides methods for drawing conclusions about a population from sample data.If we want to estimate the population mean, it is logical to use the sample mean as our basis.But, the sample mean might not be exactly equal to the population mean.So, we’ll need some “room for error.” Namely, give or take “room for error” “room for error” “margin of error”This “room for error/margin of error” is a function of the standard deviation (?).But, it must also reflect our knowledge of probability theory (let’s say a constant c), and the sample size (n).The “margin of error” for the population mean will contain “c”, ?, and n.14.1 Conditions (and cautions) for inference about a mean.Here are three simple conditions (and cautions) for inference about a mean:We have an SRS from the population of interest. The SRS may not be perfect.The variable we measure has an exact An exact Normal distribution Normal distribution N(μ, σ) in the population.is not always attainable. We don’t know the population mean μ. Slightly odd that we’d know σ, But we do know the population standard deviation σ.but not μ.The 3 simple conditions will often be “assumptions” even though there is room for some doubt. The Reasoning of Statistical EstimationEven though we find the simple conditions a bit questionable, they help us understand the reasoning of statistical estimation. Please read Example 14.1 on page 352 of the text, because it goes through this reasoning well. Example: Suppose IQ scores for adults follow a Normal distribution with a standard deviation of 15. A random sample of 56 adults yields an average score of 100. Recall that we do not know the true mean IQ score. To estimate the unknown population mean IQ score μ, we can start with the mean = 100 from the random sample. Based on what we learned in Chapter 11, we know the sampling distribution of the sample mean. The 68-95-99.7 Rule and the sampling distribution for the sample mean yield… (--68%--) (--- 95% ---) (------ 99.7% --------) If we focus uniquely on the 2nd interval, we would say:Guided Notes associated with the Lecture Video for Section 14.214.2 Margin of Error and Confidence LevelNote that the 95% confidence interval we calculated is centered at the sample mean (our estimate) and goes out about 2 standard deviations (margin of error) on either side of the sample mean.Similar to how you’d like to get the best grades possible – usually in the 90’s;in statistics, we aspire to be as confident as possible.Confidence levels of 95%, 90%, and 99% are quite common. A confidence level of 68% is not common. A level C ______________________________ for a parameter has two parts:1. An interval calculated from the data, usually of the form 2. A __________________________, which gives the probability that the interval will capture the true parameter in repeated samples. That is, the confidence level is the success rate for the method (not for an individual interval).The confidence level is the success rate of the method that produces the interval. We don’t know whether the 95% confidence interval from a particular sample is one of the 95% that capture μ or one of the unlucky 5% that miss.Note: Confidence is about the process, not about any one interval.Guided Notes associated with the Lecture Video for Section 14.314.3 Confidence Intervals for a Population MeanRemember from the 68-95-99.7 rule that the 95 we used above was approximate.We do not need to go a full 2 standard deviations away from the mean to be 95% confident. In fact, we need only go 1.96 standard deviations for 95% confidence. Let’s look at Table A for areas corresponding to +2.00 and -2.00.P(Z < 2.00) = .9772P(Z< - 2.00) = __________The difference is ________Let’s look at Table A for areas corresponding to +1.96 and -1.96.P(Z < 1.96) = .9750P(Z < -1.96) = __________The difference is ________So, we’ll adopt 1.96 vs. 2.00.In general, for a level C confidence interval, we need to find the value such that we have area C between and . These are the critical values of the standard Normal distribution.The following table gives the critical values for the most common confidence levels:Confidence level C90%95%99%Critical value 1.6451.9602.576To recap our confidence interval for ? has evolved…. “margin of error” Verbatim 100 (1, 2, or 3) Our example (1.96, or, another z-score) General CaseUsing the critical values, we can calculate a confidence interval for any confidence level:Draw an SRS of size n from a Normal population having unknown mean μ and known standard deviation σ. A level C confidence interval for μ isThe broader method of calculating confidence intervals, involves the Four-Step Process (page 358) “_______________”;_______: What is the practical question that requires estimating a parameter?_______: Identify the parameter, choose a level of confidence, and select the type of confidence interval that fits your situation._______: Carry out the work in two phases:Check the conditions for the interval you plan to use.Calculate the confidence interval._________: Return to the practical question to describe your results in this setting.1st Note: The four steps involve: 1) thinking about what we want to do; STATE / PLAN 2) checking to see if we can do it; SOLVE (a) 3) doing it; SOLVE (b)and 4) communicating our results in context.CONCLUDE2nd Note: The Four-Step Process outlines a general schema for the problems that are posed to us.We will continue using the “Steps for Success” model for the specific steps within each solution.Example: Recall that adult IQ scores follow a Normal distribution with a standard deviation of 15. A random sample of 56 adults is taken had an average IQ of 100. Give a 99% confidence interval for the average adult IQ score. Also provide its margin of error. We’ve focused on the confidence level. The standard deviation has remained constant.But, use your knowledge about the Law of Large Numbers and the CLT to answer this question with technology.Technology Tips – Computing Confidence Intervals (? known)TI-83/84 STAT TESTS ZInterval Enter. Select Stats. Enter ?,, n, and the confidence level. Select Calculate.(Note: Select Data when and n are not provided. Then enter the list where the data are stored.)JMP Enter the data. Analyze Distribution.“Click-and-Drag” (the appropriate variable) into the ‘Y, Columns’ box. Click on OK. Click on the red upside-down triangle next to the title of the variable from the ‘Y,Columns’ box. Proceed to ‘Confidence Interval’ -> Select the appropriate confidence level.Example: Let’s now use technology to compute 95% and 99% confidence intervals for the average adult IQ score. Recall that n = 56, ? = 15, with the sample average of 100. Also provide the margin of error for each confidence interval. Example: Let’s use technology to compute 99% confidence intervals for the average adult IQ scores for sample sizes of 49 and 225. Recall that ? = 15, with the sample average of 100. Guided Notes associated with the Lecture Video for Section 14.4 14.4 How Confidence Intervals BehaveThe confidence interval for the population mean, ?, with known standard deviation is of the form with margin of error equaling .A high confidence level indicates that our methods commonly yield correct answers. Smaller margins of error reflect increased precision in estimating the true value of the parameter. Smaller margins of error are quite desirable and can be attained mathematically through…Lower confidence levelsThese result in smaller values of z*.Smaller standard deviationsLess error in the data, with smaller values of ?. Increased sample sizesThis results in dividing ? by a larger number. Quadrupling the sample size cuts the margin of error in half. Example: Recall that earlier we computed a 95% confidence interval for the mean adult IQ score based upon a population standard deviation of 15 and a sample of 56. Suppose that the population standard deviation was 1/3 the size of our initial value of ?. Is this new margin of error 1/3 the size of earlier calculation? Five-Minute Summary:List at least 3 concepts that had the most impact on your knowledge of confidence intervals._________________________________________ Notes taken while Viewing the Demonstration for Chapter 14Notes taken while Completing the Recitation Activity for Chapter 14 (to be posted 10/27) Notes taken after Reviewing the Chapter 14 Material included in the Quizzes Notes taken while Completing the Homework Exercises from Chapter 14 ................
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