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Unit 8: Estimating with Confidence 8.1 Confidence Intervals (The Basics)

Understand confidence interval Interpret confidence level Objectives: Understand confidence interval Interpret confidence level and confidence interval Construct a Conf Int Use Conf Int under the appropriate conditions Emphasize the appearance of “and confidence interval” b\c there is a huge difference between the two parts of this objective. Let students know that the conditions required to use a ConfInt should be familiar.

A single value produced by your estimator Point Estimator A statistic that provides an estimate of some population parameter using a sample Point Estimate A single value produced by your estimator Remind students of German Tanks simulation and how each of them invented an estimator. When def for Pt. Est. comes up add in the idea of “best guess” but clearly a long-shot to be absolutely on target!

What makes a good estimator? Unbiased 𝒑 and 𝒙 (so far) Low Variability Recall: What makes a good estimator? Unbiased 𝒑 and 𝒙 (so far) Low Variability Sample size! Write on the board how p-hat and x-bar are calculated, just to remind students. Recall that increasing the sample size always decreases variability of the sampling distribution, but we also want a particular shape…

Recall Penny Ages Applet: Population mean: 12.26 yrs Population stdev: 9.61 yrs Grab the poster of the sampl. distr. of x-bar when n=49 and remind them that this sampl. distr. had low variability (spread) and is approx norm (shape) b/c n > 30. See if you can elicit a recall of when you get 68-95-99.7 and emphasize that these are approx % for 1sigma-2sigma-3sigma Then go out 2sigma (of the sampl distr NOT the pop) and count the number of dots within that 2sigma

Confidence Interval for a population parameter: Interval Itself: pt.est. ± margin of error Confidence Level C: Success rate of the method in repeated sampling Link plausible-reasonable-likely

www.whfreeman.com/tps4e Go to website and run 95% confidence with a few samples, one at a time, so that stu see ConfInt created. Show them the point and the ME. Then do 50 at a time…note the running % of those CIs which capture the true pop parameter

This is a screen shot, just in case the website doesn’t work.

Observations about applet: The center of each interval is the pt.est. The dist. from pt.est. to either end of the interval is the margin of error IF we created EVERY interval poss. then ______% of the intervals would contain pop. parameter Give examples of intervals and have students identify the pt. est. and the ME. The blank in bullet 3 is predetermined when we declare how confident we wish to be….

A set of plausible values for the pop. parameter Interpreting!! Confidence Level How likely is our method to produce an interval that captures a parameter if we use it many times? Confidence Interval A set of plausible values for the pop. parameter Give examples of intervals and have students appropriately interpret both the interval and the confidence level.

www.whfreeman.com/tps4e Back to website & show how intervals change when conf level changes. What would it take to be 100% confident?

Our confidence level and the type of statistic determines this value Constructing a Con Int pt. est. ± margin of error statistic ± (crit value)(stdev of stat) Our confidence level and the type of statistic determines this value

Normal (b/c sample size is large enough) Conditions for constructing a confidence interval: Random Independent Normal (b/c sample size is large enough) Random: SRS Indep: 10n < N Normal: p-hat vs. x-bar

Example: In a survey, researchers asked adults aged 35 to 50 years if they used social media. A 90% confidence interval was calculated as being 73%, plus/minus 2.35%.