DOC PREVIEW
UA MATH 167 - Lecture Notes

This preview shows page 1-2-3-4 out of 13 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 13 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Estimating a population proportionIntroductionPoint estimatesConfidence intervalsConfidence interval for pUseComputationDetermining sample size for desired EConfidence Interval Belt GraphsSummaryAdditional examplesIntroductory Statistics LecturesEstimating a population proportionConfidence intervals for proportionsAnthony TanbakuchiDepartment of MathematicsPima Community CollegeRedistribution of this material is prohibitedwithout written permission of the author© 2009(Compile date: Tue May 19 14:50:07 2009)Contents1 Estimating a populationproportion 11.1 Introduction . . . . . . . 1Point estimates . . . . . 21.2 Confidence intervals . . 31.3 Confidence interval for p 3Use . . . . . . . . . . . . 3Computation . . . . . . 3Determining samplesize for desired E 7Confidence IntervalBelt Graphs . . . 81.4 Summary . . . . . . . . 121.5 Additional examples . . 121 Estimating a population proportion1.1 IntroductionExample 1. We want to estimate the proportion of people in the US who wearcorrective lenses. Assuming our class data represents an unbiased sample ofthe US population, (1) what would our estimate be and (2) how precise is it?R: summary ( c o r r e c t i v e l e n s e s )NO YES8 1012 of 13 1.1 IntroductionPOINT ESTIMATESNotationp population proportion.Note: proportion, percentage, and probability can all be consid-ered as p.ˆp estimate of sample proportion with x successes in n trials.ˆp =xn, ˆq = 1 − ˆp (1)point estimate.Definition 1.1A single value (or point) used to approximate a population parameter.The sample proportion ˆp is the best point estimate of the popula-tion proportion p.Importance of proper sampling.If a sample is not representative of the population, ˆp will not be a usefulestimate of p. Use proper sampling techniques!Example 2. Point estimate of proportion of people who wear corrective lensesin the US using class data:R: x = sum( c o r r e c t i v e l e n s e s == ”YES”)R: x[ 1 ] 10R: n = l eng t h ( c o r r e c t i v e l e n s e s )R: n[ 1 ] 18R: p . hat = x/nR: p . hat[ 1 ] 0.5 5 5 5 6Question 1. How good is the estimate of p? How precise is the estimate?Question 2. What do we need to know about ˆp to determine the precision ofthe estimate?Anthony Tanbakuchi MAT167Estimating a population proportion 3 of 131.2 Confidence intervalsConfidence interval. Definition 1.2is a range of values — an interval — used to estimate the true valueof a population parameter. It provides information about the inherentsampling error of the estimate. (In contrast to point estimate.)Just as we used the empirical rule to estimate an interval 95%of the data would fall within if the data’s distribution was normal,we can construct a similar interval for a statistic given it’s samplingdistribution.“We are 95% confident that the interval (ˆpL, ˆpU) actually contains the truevalue of p.”Confidence level. Definition 1.3is the probability that the confidence interval contains the true popu-lation parameter that is being estimated, if the estimation process isrepeated a large number of times.confidence level = 1 − α (2)where α is the probability that the confidence interval will not containthe true parameter value.Typical confidence levelsCL α99% 0.0195% 0.0590% 0.10Most commonly used is 95%.1.3 Confidence interval for pUSEOften used to answer:1. What is a reasonable estimate for the population proportion?2. How much variability is there in the estimate for the population propor-tion?3. Does a given target value fall within the confidence interval?COMPUTATIONSampling distribution of ˆpIf np and nq ≥ 5 then p will have a normal distribution1and the CLT tells usthat ˆp is approximately normally distribution where:µˆp= p (3)1Normal approximation of binomial.Anthony Tanbakuchi MAT1674 of 13 1.3 Confidence interval for pσˆp=rpqn≈rˆpˆqn(4)Confidence interval for p.Definition 1.4The confidence interval for p at the (1 − α) confidence level is:ˆpL< p < ˆpU(5)F−1norm(α/2) < p < F−1norm(1 − α/2) (6)0 5 10 150.00 0.05 0.10 0.15Binom dist of x assuming p=0.5xPbinom((x))0.0 0.2 0.4 0.6 0.8 1.00.0 1.0 2.0 3.0Sampling dist. of p.hatp^p^Lp^USampling distribution for ˆp: If the requirements are met it will have anormal distribution with µˆp≈ ˆp = 0.556, σˆp≈qˆpˆqn= 0.117. Total shadedarea is α = 0.05, each tail has an area of α/2 = 0.025. Thus, 95% confidenceinterval for p is (ˆpL, ˆpU) = (0.326, 0.785).Variation in CI of p from sample to sampleSimulate study of corrective lens use 50 times with random sample size of 18assuming true p = 0.5.Anthony Tanbakuchi MAT167Estimating a population proportion 5 of 130.2 0.4 0.6 0.8 1.00 10 20 30 40 50pRandom Sample NumberConfidence intervals for p||||||||||||||||||||||||||||||||||||||||||||||||||95% CI’s, tick marks represent each point estimate ˆp.In general, 95% of the confidence intervals will contain p.Confidence intervals for p in RTo construct a CI (ˆpL, ˆpU) at (1 − α) confidence level:ˆpL= ˆp − EˆpU= ˆp + Ewhere E is the margin of error.With the following requirements:1. Simple random sample.2. Satisfies binomial distribution.3. Satisfies normal approximation to binomial.Margin of error E. Definition 1.5The confidence interval can be expressed in terms of the margin oferror E:CI: ˆp ± E (7)where the margin of error for ˆp is:E = zα/2· σˆp(8)or if the upper and lower values are known:E =upper − lower2=ˆpU− ˆpL2(9)Anthony Tanbakuchi MAT1676 of 13 1.3 Confidence interval for p−4 −2 0 2 40.0 0.1 0.2 0.3 0.4Standard normal distribution, shaded area = alpha.zf(z)zαα 2Critical value zα/2.Definition 1.6The critical value zα/2is the value of z on the standard normal distri-bution with α/2 area to the RIGHT.Example 3. Find the critical value zα/2for the 95% confidence interval.R: alpha = 1 − 0 . 9 5R: z . c r i t i c a l = qnorm ( 1 − alph a / 2 )R: z . c r i t i c a l[ 1 ] 1. 9 6 0 0zα/2for 95% CLzα/2= 1.96 for α = 0.05 (10)Question 3. How does this differ from the Empirical Rule?Example 4. Using our class data to estimate the 95% confidence interval forthe proportion of people in the US who wear corrective lenses.What’s known:R: alpha = 1 − 0 . 9 5R: n[ 1 ] 18R: x[ 1 ] 10R: p . hat = x/nAnthony Tanbakuchi MAT167Estimating a population proportion 7 of 13R: q . hat = 1 − p . hatR: sigma . p . hat = s q r t (p . hat ∗ q . hat /n)Finding the 95% CIR: z . c r i t i c a l = qnorm ( 1 − alph a / 2 )R: E = z . c r i t i c a l ∗ sigma . p . hatR: p


View Full Document

UA MATH 167 - Lecture Notes

Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?