Confidence Intervals: The Basics Section 8.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Lesson 6.1.1

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

So… Statisticians are never 100% confident in their results. To 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!

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.

Can We Use 𝑥 ? We can use the value of the statistic 𝑥 because the value 𝑥 is an unbiased estimator of the population mean µ.

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, would you say somewhere around ____? Between what numbers, low and high?” Question of the day is: How confident are you with that interval you gave me? Do you want to be 95% confident?

Recall Our Bell Curve

Confidence Interval 𝒙 ± 𝟐∗𝝈 𝒙 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 two parts: An interval calculated from the data, which has the form Estimate ± margin of error The margin of error tells how close the estimate tends to be to the unknown parameter in repeating random sampling Margin or error can be elaborated on for 95% CI: Statistic ± (critical value)* (standard deviation of statistic) 𝒙 ± 𝟐∗𝝈 𝒙 **don’t worry about calculating critical value for now**

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 Example: If many samples are taken and 95% confidence intervals are constructed based on these samples, then about 95% of the intervals will capture the true parameter being estimated. “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.”

Simulating Confidence Intervals http://www.rossmanchance.com/applets/ConfSim.html

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.

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.

Calculating a Confidence Interval The confidence interval for estimating a population parameter has the form: statistic ± (critical value) · (standard deviation of statistic) 𝒙 ± 𝟐∗𝝈 𝒙

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 dist. That is at least approximately normal. For 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 populations: 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

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