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Confidence Intervals: The Basics Statistic ± (critical value)* (standard deviation of statistic) Section 8.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore From Chapter 7 to Chapter 8 We are making a transition. In chapter 7 we assumed that we knew the true value of a parameter and then asked questions about the distribution of the statistic used to estimate that parameter. Now, in chapter 8 we no longer pretend to know the true value of a parameter. We start with the more realistic situation where we know only the value of a sample statistic. Then we use that to estimate the parameter. Objectives • Point Estimator/ Point Estimate • Idea Of Confidence Intervals and Levels – Confidence Interval – Confidence Level C • Applet! Lets explore! • Calculating Confidence Intervals • Conditions of Constructing Confidence Intervals So… • Statisticians are never 100% confident in their results. So taking into account of any variability and eliminate any possibility to be proven wrong with one example that’s outside of our results, we (Statisticians) use confidence intervals. These intervals are used to describe a specific range of low and high numbers with a certain percent of certainty depending on how large of a gap we are leaving between numbers! Can We Use 𝑥? WHY? We can use the value of the statistic 𝑥 because the value 𝑥 is an unbiased estimator of the population mean µ. Point Estimator / Point estimate • If we had to give a single number to estimate the value of the statistic 𝑥 what would it be? (such a value is known as a point estimate). Example: “Jenny, if you had to guess what the sample mean grade of your brother’s Algebra 1 class would be, what would you guess?” • Point estimator is a statistic that provides an estimate of a population parameter. Point estimate is the number. Ideally, a point estimate is our “best guess” at the value of an unknown parameter. Intro To Confidence Intervals • Example: Jenny, if you had to guess what the sample mean of your brother’s Algebra 1 class would be, what would you guess? • She says:______ • I say, “now you don’t imagine that’s exactly the score, could you give us a low score and a high score that you think the class mean would fall between?” Question of the day is: How confident are you with that interval you gave me? Do you want to be 95% confident? Making connections… • In Chapter 2 we learned about the Empirical rule: – 68% of values lie within (+1) σ of the mean – 95% of values lie within (+2) σ of the mean – 99.7% of values lie within (+3) σ of the mean. • 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 if we take multiple samples each may vary by a certain amount, but if we take all the possible combinations of samples, then, we can construct a sampling distribution. Recall Our Bell Curve Confidence Intervals and Levels If I want to be 95% confident, I would need to create an interval that is +2σ and -2σ A confidence interval for a parameter has 2 parts: Confidence Interval Confidence level “interpret the confidence interval…” “interpret the confidence level…” Confidence Interval 1) An interval calculated from the data, which has the form for 95% CI: Statistic ± (critical value)* (standard deviation of statistic) “We are 95% confident that the interval from ____ to ____ captures the true [parameter in context]” **don’t worry about calculating critical value for now** Estimate ± margin of error The margin of error tells how close the estimate tends to be to the unknown parameter in repeating random sampling Confidence Level 2) A confidence level C, which gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter “If we take many samples of the same size, (n), from this population, about 95% of them will result in an interval that captures the true [parameter value]” Example: Do You Use Twitter? In late 2009, the Pew Internet and American Life Project asked a random sample of 2253 U.S. adults, “Do you ever…use Twitter or another service to share updates about yourself or to see updates about others?” Of the sample, 19% said “Yes.” According to Pew, the resulting 95% confidence interval is (0.167, 0.213).2 PROBLEM: Interpret the confidence interval and the confidence level. Interval: We are 95% confident that the interval from 0.167 to 0.213 captures the true population proportion of all US adults who use Twitter or another service for updates Example: Do You Use Twitter? In late 2009, the Pew Internet and American Life Project asked a random sample of 2253 U.S. adults, “Do you ever…use Twitter or another service to share updates about yourself or to see updates about others?” Of the sample, 19% said “Yes.” According to Pew, the resulting 95% confidence interval is (0.167, 0.213).2 PROBLEM: Interpret the confidence interval and the confidence level. Level: If we take many samples of 2253 US adults, 95% of confidence intervals will result in an interval that captures the true population proportion of US adults who use twitter or other services to share updates about themselves or others. CHECK YOUR UNDERSTANDING How much does the fat content of Brand X hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard deviation σ is 2.84 to 7.55. 1. Interpret the confidence interval. 2. Interpret the confidence level. 3. True or False: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation σ. Justify your answer. Simulating Confidence Intervals http://www.rossmanchance.com/applets/ConfSi m.html Calculating a Confidence Interval statistic ± (critical value) · (standard deviation of statistic) Lets just get some straight up Algebra number crunching down for now, we will most definitely elaborate further with more context! Construct a 95% confidence interval. 𝒙 = 𝟐𝟒𝟎. 𝟕𝟗 𝝈𝒙 = 𝟓 Critical value = 1.96 statistic ± (critical value) · (standard deviation of statistic) Conditions for Constructing A Confidence Interval Random: The data should come from a well-designed random sample or randomized experiment. Normal: Constructing confidence intervals should come from a sampling distribution that is at least approximately normal. – For Pop.Means: if pop is normal, sample is normal. If pop is not normal the CLT of a sample greater than or equal to 30. – For Pop.Proportions: normal conditions checked Independent: The procedure of calculating confidence intervals assume that individual observations are independent. Objectives • Point Estimator/ Point Estimate • Idea Of Confidence Intervals – Confidence Interval – Margin Of Error – Confidence Level C • Applet! Lets explore! • Calculating Confidence Intervals • Conditions of Constructing Confidence Intervals Homework 8.1 Homework Worksheet Start Chapter 8 Reading Guide