24.1 Confidence Intervals and Margins of Error

Transcription

1 24.1 Cofidece Itervals ad Margis of Error Essetial Questio: How do you calculate a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Resource Locker Explore Idetifyig Likely Populatio Proportios I a previous lesso, you took samples from a populatio whose parameter of iterest is kow i order to see how well a sample statistic estimated that parameter. I this lesso, you will estimate a populatio parameter usig a statistic obtaied from a radom sample, ad you will quatify the accuracy of that estimate. Suppose you survey a radom sample of 50 studets at your high school ad fid that 40% of those surveyed atteded the football game last Saturday. Although you caot survey the etire populatio of studets, you would still like to kow what populatio proportios are reasoably likely i this situatio. Suppose the proportio p of the populatio that atteded last Saturday s game is 30%. Fid the reasoably likely values of the sample proportio p. Alamy I this case, p =? ad =?. μ p = p =? ad σ p = p (1- p)? (1 -? ) = _?? The reasoably likely values of p fall withi 2 stadard deviatios of μ p. μ p - 2 σ p =? - 2 (? ) =? μ p + 2 σ p =? + 2 (? ) =? Module Lesso 1

2 Draw the graph o grid paper, the draw a horizotal lie segmet at the level of 0.3 o the vertical axis to represet the iterval of likely values of p you foud i Step B. 0.9 Proportio of Successes i Populatio, p Proportio of Successes i Sample, pˆ Now repeat the process for p = 0.35, 0.4, 0.45, ad so o to complete the graph i Step C. You may wish to divide up the work with other studets ad pool your fidigs. Draw a vertical lie at 0.4 o the horizotal axis. This represets p = 0.4. The lie segmets that this vertical lie itersects are the populatio proportios for which a sample proportio of 0.4 is reasoably likely. Reflect 1. Discussio Is it possible that 30% of all studets at your school atteded last Saturday s football game? Is it likely? Explai. 2. Discussio Is it possible that 60% of all studets at your school atteded last Saturday s football game? Is it likely? Explai. 3. Discussio Based o your graph, which populatio proportios do you thik are reasoably likely? Why? Explai 1 Fidig a Cofidece Iterval for a Populatio Proportio A cofidece iterval is a approximate rage of values that is likely to iclude a ukow populatio parameter. The level of a cofidece iterval, such as 95%, gives the probability that the iterval icludes the true value of the parameter. Houghto Miffli Harcourt Publishig Compay Recall that whe data are ormally distributed, 95% of the values fall withi 2 stadard deviatios of the mea. Usig this idea i the Explore, you foud a 95% cofidece iterval for the proportio of all studets who atteded the football game last Saturday. Module Lesso 1

3 The graph that you completed i the Explore is show. You ca see from the graph that whe the horizotal lie segmet at p = 0.4 is rotated 90 about the poit (0.4, 0.4), it becomes a vertical lie segmet that captures all of the likely populatio proportios. Sice you already kow the iterval o the horizotal axis that defies the horizotal segmet, you ca fid the iterval o the vertical axis that defies the vertical segmet by usig the fact that μ p = p ad iterchagig the variables p o the horizotal axis) ad p (o the vertical axis). So, the horizotal axis iterval μ p - 2 σ p p μ p + 2 σ p becomes the vertical axis iterval as follows: Replace μ p with p. Iterchage p ad p. p - 2 σ p p p + 2 σ p p - 2 σ p p p + 2σ p I this case, the vertical axis iterval is the 95% cofidece iterval for p. This result ca be geeralized to a c% cofidece iterval for p. 0.9 Proportio of Successes i Populatio, p Proportio of Successes i Sample, pˆ Houghto Miffli Harcourt Publishig Compay A Cofidece Iterval for a Populatio Proportio A c% cofidece iterval for the proportio p of successes i a populatio is give by p - z c p (1 - p ) p p + z c p (1 - p ), where p is the sample proportio, is the sample size, ad z c depeds upo the desired degree of cofidece. I order for this iterval to describe the value of p reasoably accurately, three coditios must be met: 1. There are oly two possible outcomes associated with the parameter of iterest. The populatio proportio for oe outcome is p, ad the proportio for the other outcome is 1 - p. 2. p ad (1 - p ) must both be at least The size of the populatio must be at least 10 times the size of the sample, ad the sample must be radom. Module Lesso 1

4 Use the values i the table for z c. Note that you should use 1.96 rather tha 2 for z 95% for greater accuracy. Desired degree of cofidece 90% 95% 99% Value of z c Example 1 I a radom sample of 100 four-year-old childre i the Uited States, 76 were able to write their ame. Fid the specified cofidece iterval for the proportio p of four-year-olds i the Uited States who ca write their ame. Fid a 95% cofidece iterval. Idetify the sample size, the proportio p of four-year-olds i the sample who ca write their ame, ad the value of z c for a 95% cofidece iterval. = 100 p = 0.76 z c = 1.96 Substitute the values of, p, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to two decimal places. p - z c p (1 - p ) = (1-0.76) p + z c p (1 - p ) = (1-0.76) So, you ca state with 95% cofidece that the proportio of all four-year-olds i the Uited States who ca write their ame lies betwee 68% ad 84%. Fid a 99% cofidece iterval. Idetify the sample size, the proportio p of four-year-olds i the sample who ca write their ame, ad the value of z c for a 99% cofidece iterval. = 100 p = 0.76 z c = Substitute the values of, p, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to two decimal places. p - z c p (1 - p ) p + z c p (1 - p ) = = ( ) ( ) So, you ca state with 99% cofidece that the proportio of all four-year-olds i the Uited States who ca write their ame lies betwee 65% ad 87%. Reflect Houghto Miffli Harcourt Publishig Compay 4. Discussio Do the data from the sample of four-year-old childre satisfy the three coditios for usig the cofidece iterval formula? Explai. 5. Discussio Does icreasig the value of c icrease or decrease the rage of values for a cofidece iterval of a populatio proportio? Explai why it happes mathematically ad why it makes sese. Module Lesso 1

5 Your Tur 6. Isabelle surveys a radom sample of 80 voters i her large tow ad fids that 46 support raisig property taxes i order to build a ew library. Fid a 95% cofidece iterval for the proportio p of all voters i Isabelle s tow who support raisig property taxes i order to build a ew library. Explai 2 Fidig a Cofidece Iterval for a Populatio Mea You ca use reasoig similar to the argumet i the Explore to develop a formula for a cofidece iterval for a populatio mea. A Cofidece Iterval for a Populatio Mea A c% cofidece iterval for the mea μ i a ormally distributed populatio is give by x _ σ - z c μ x _ + z σ c, where x _ is the sample mea, is the sample size, σ is the populatio stadard deviatio, ad z c depeds upo the desired degree of cofidece. Note that it is assumed that the populatio is ormally distributed ad that you kow the populatio stadard deviatio σ. I a more advaced statistics course, you ca develop a cofidece iterval that does ot deped upo a ormally distributed populatio or kowig the populatio stadard deviatio. Example 2 For the give situatio, fid the specified cofidece iterval for the populatio mea. I a radom sample of 20 studets at a large high school, the mea score o a stadardized test is 610. Give that the stadard deviatio of all scores at the school is 120, fid a 99% cofidece iterval for the mea score amog all studets at the school. Houghto Miffli Harcourt Publishig Compay Idetify the sample size, the sample mea _ x, the populatio stadard deviatio σ, ad the value of z c for a 99% cofidece iterval. _ = 20 x = 610 σ = 120 z c = Substitute the values of, x _, σ, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to the earest whole umber. _ σ x - z c = _ 541 x + z c σ 20 = So, you ca state with 99% cofidece that the mea score amog all studets lies betwee 541 ad 679. I a radom sample of 30 studets at a large high school, the mea score o a stadardized test is Give that the stadard deviatio of all scores at the school is 141, fid a 95% cofidece iterval for the mea score amog all studets at the school. Idetify the sample size, the sample mea _ x, the populatio stadard deviatio σ, ad the value of z c for a 95% cofidece iterval. = _ 30 x = 1514 σ = 141 z c = 1.96 Module Lesso 1

6 Substitute the values of, _ x, σ, ad z c ito the formulas for the edpoits of the cofidece iterval. Simplify ad roud to the earest whole umber. _ x - z c _ x + z c σ = σ = So, you ca state with 95% cofidece that the mea score amog all studets at the school lies betwee 1464 ad Reflect 7. What must you assume about the test scores of all studets to use the formula for the cofidece iterval? Your Tur 8. I a radom sample of 42 employees i a large compay, the mea weekly umber of miutes spet exercisig is 86. Give that the stadard deviatio of all employees is 22.4, fid a 99% cofidece iterval for the mea weekly umber of miutes spet exercisig amog all employees i the compay. Explai 3 Choosig a Sample Size I Part B of Example 2, you foud the 95% cofidece iterval 1464 _ x 1564, which is a rage of values cetered at _ x = You ca write the cofidece iterval as 1514 ± 50, where 50 is called the margi of error. The margi of error is half the legth of a cofidece iterval. Margi of Error for a Populatio Proportio The margi of error E for the proportio of successes i a populatio with sample proportio p ad sample size is give by E = z c p (1 - p ) degree of the cofidece iterval. Margi of Error for a Populatio Mea, where z c depeds o the The margi of error E for the mea i a ormally distributed populatio with stadard deviatio σ, sample mea x _ σ, ad sample size is give by E = z c, where z c depeds o the degree of the cofidece iterval. From the formulas you ca see that the margi of error decreases as the sample size icreases. This suggests usig a sample that is as large as possible. However, it is ofte more practical to determie a margi of error that is acceptable ad the calculate the required sample size. Houghto Miffli Harcourt Publishig Compay Module Lesso 1

7 Example 3 Fid the appropriate sample size for the give situatio. A researcher wats to kow the percet of teeagers i the Uited States who have social etworkig profiles. She is aimig for a 90% cofidece iterval ad a margi of error of 4%. What sample size should she use? Step 1 Rewrite the margi-of-error formula for a populatio proportio by solvig for. E = z c p (1 - p ) E 2 = z 2 c p (1 - p ) Write the formula. Square both sides. E 2 = z 2 c p (1 - p ) Multiply both sides by. = z 2 c p (1 - p ) E 2 Step 2 Estimate the value of p. Divide both sides by E 2. The researcher has ot coducted the survey ad is tryig to fid p. So, she must estimate p as 0.5, which is the value of p that makes the expressio p (1 - p ) as large as possible. Step 3 Idetify the values of E ad z c. E is the margi of error writte as a decimal ad z c is the z-score that correspods to a 90% cofidece iterval. So, E = 0.04 ad z c = Step 4 Substitute the values of p, E, ad z c i the rewritte margi-of-error formula from Step 1. = z 2 c p (1 - p ) 2 = (1.645) (1-0.5) E (0.04) So, the researcher should survey a radom sample of 423 teeagers. Caleb is a restaurat maager ad wats to kow the mea umber of secods it takes to complete a customer s order. He is aimig for a 95% cofidece iterval ad a margi of error of 6 secods. Based o past experiece, Caleb estimates the populatio stadard deviatio to be 21 secods. What sample size should he use? Step 1 Rewrite the margi-of-error formula for a populatio mea by solvig for. σ E = z c E 2 = z 2 c _ σ 2 Write the formula. Square both sides. E 2 = z 2 c σ 2 Multiply both sides by. = z c 2 σ 2 E 2 Divide both sides by E 2. Module Lesso 1

8 Step 2 Idetify the values of E, σ, ad z c. E is the margi of error, σ is the populatio stadard deviatio, ad z c is the z score that correspods to a 95% cofidece iterval. So, E =6, σ = 21 ad z c = Step 3 Substitute the values of E, σ, ad z c i the margi of error for a populatio mea formula that was solved for. 2 ( ) ( ) ( 6 ) = So, Caleb should survey a radom sample of 47 orders. Reflect 9. Discussio I Part A, do you expect the sample size to icrease or decrease if the researcher decides she wats a smaller margi of error? Explai usig the margi-of-error formula for a populatio proportio. 10. Discussio I Part B, do you expect the sample size to icrease or decrease if Caleb decides he wats a 99% cofidece iterval istead of a 95% cofidece iterval? Explai usig the margi-of-error formula for a populatio mea. Your Tur 11. Zoe is a editor of a ewspaper i a state capital ad wats to kow the percet of residets i her state who are i favor of baig the use of hadheld cell phoes while drivig, a bill that is beig cosidered i the state legislature. After researchig similar polls coducted i other states, she estimates that p = She is aimig for a 95% cofidece iterval ad a margi of error of 5%. What sample size should Zoe use? Elaborate 12. How ca a iterval that captures 95% of the sample proportios i a samplig distributio be used to fid a 95% cofidece iterval for a populatio proportio? 13. Describe how icreasig the sample size affects the cofidece iterval of a populatio mea. 14. Essetial Questio Check-i What is the relatioship betwee a cofidece iterval ad a margi of error for a populatio proportio or populatio mea? Houghto Miffli Harcourt Publishig Compay Module Lesso 1

9 E Evaluate: Homework ad Practice Idetify the values of the sample proportio p that fall withi 2 stadard deviatios of the give populatio proportio p for each situatio. 1. Suppose that 44% of all employees at a large compay atteded a recet compay fuctio. Alaah plas to survey a radom sample of 32 employees to estimate the populatio proportio. What are the values of p that she is likely to obtai? 2. Suppose the proportio p of a school s studets who oppose a chage to the school s dress code is 73%. Nicole surveys a radom sample of 56 studets to fid the percet of studets who oppose the chage. What are the values of p that she is likely to obtai? For the give situatio, fid the specified cofidece iterval for the populatio proportio. 3. Huter surveys a radom sample of 64 studets at his commuity college ad fids that 37.5% of the studets saw a film at the local movie theater i the last 30 days. Fid a 90% cofidece iterval for the proportio p of all studets at the commuity college who saw a film at the movie theater i the last 30 days. 4. I a radom sample of 300 U.S. households, 111 households have a pet dog. Fid a 99% cofidece iterval for the proportio p of all U.S. households that have a pet dog. 5. A quality cotrol team at a compay that maufactures digital utility meters radomly selects 320 meters ad fids 12 to be defective. Fid a 95% cofidece iterval for the proportio p of all digital utility meters that the compay maufactures ad are defective. 6. I a radom sample of 495 four-year-olds i a state, 54.7% ca provide the first ad last ame of at least oe paret or guardia. Fid a 99% cofidece iterval for the proportio p of all four-year-olds i the state who ca provide the first ad last ame of at least oe paret or guardia. For the give situatio, fid the specified cofidece iterval for the populatio mea. 7. A olie website that tracks gas prices surveys a radom sample of 73 gas statios i a state ad fids that the mea price of 1 gallo of regular gasolie is $ If the website estimates from past surveys that the populatio stadard deviatio is $0.117, fid a 95% cofidece iterval for the mea price of regular gasolie i the state. 8. Caide maages the security team at a large airport ad surveys a radom sample of 149 travelers. He fids that the mea amout of time that it takes passegers to clear security is 28.3 miutes. From past experiece, Caide estimates that the populatio stadard deviatio is 6.4 miutes. Fid a 90% cofidece iterval for the mea amout of time that it takes passegers to clear security. Module Lesso 1

10 9. A quality cotrol team at a compay that maufactures smartphoes measures the battery life of 24 radomly selected smartphoes ad fids that the mea amout of cotiuous video playback from a full charge is hours. Give that the populatio stadard deviatio is 0.83 hour, fid a 99% cofidece iterval for the mea amout of cotiuous video playback from a full charge. 10. Stephe surveys 53 radomly selected studets at his large high school ad fids that the mea amout of time spet o homework per ight is miutes. Give that the populatio stadard deviatio is 22.7 miutes, fid a 90% cofidece iterval for the mea amout of time studets at the school sped o homework per ight. Fid the appropriate sample size for the give situatio. 11. Executives at a health isurace compay wat to kow the percet of residets i a state who have health isurace. Based o data from other states, they estimate that p = 0.8. They are aimig for a 99% cofidece iterval ad a margi of error of 1.5%. What sample size should they use? 12. Tyler is a maager at a utility compay ad wats to kow the mea amout of electricity that residetial customers cosume per moth. He is aimig for a 90% cofidece iterval ad a margi of error of 10 kilowatt-hours (kwh). From past experiece, Tyler estimates that the populatio stadard deviatio is 91.1 kwh. What sample size should he use? 13. Teeka is a restaurat ower ad wats to kow the mea amout of reveue per day. She is aimig for a 95% cofidece iterval ad a margi of error of $200. Give that the populatio stadard deviatio is $870, what sample size should Teeka use? 14. Biology Mirada is a biologist who is measurig the legths of radomly selected frogs of the same species from two locatios. a. Mirada measures the legths of 25 frogs from locatio A ad fids that the mea legth is 7.35 cm. Give that the populatio stadard deviatio is 0.71 cm, fid a 95% cofidece iterval for the mea legth of frogs from locatio A. b. Mirada measures the legths of 20 frogs from locatio B ad fids that the mea legth is 7.17 cm. Give that the populatio stadard deviatio is 0.69 cm, fid a 95% cofidece iterval for the mea legth of frogs from locatio B. c. Is it clear that the mea legth of frogs from oe locatio is greater tha the mea legth of frogs from the other locatio? Explai your reasoig. Module Lesso 1

11 15. Which of the followig sets of desired margi of error E, desired cofidece level c, ad give sample proportio p or populatio stadard deviatio σ require a sample size of at least 100? Select all that apply. A. E = 0.1, c = 95%, p = 0.5 B. E = 1000, c = 99%, σ = 4000 C. E = 0.04, c = 90%, p = 0.9 D. E = 4, c = 95%, σ = 18.6 E. E = 0.06, c = 99%, p = 0.3 F. E = 50, c = 90%, σ = 309 H.O.T. Focus o Higher Order Thikig 16. Multiple Represetatios The margi of error E for the proportio of successes i a populatio may be estimated by 1, where is the sample size. Explai where this estimate comes from. Assume a 95% cofidece iterval. 17. Draw Coclusios I the Explore, you wrote a iterval of the form μ p - 2 σ p < p < μ p + 2 σ p that captures 95% of the sample proportios. You kow that μ p = p, so you ca rewrite the iterval as p - 2 σ p < p < p + 2 σ p. Solve this compoud iequality for p. What does this result tell you? 18. Explai the Error A quality assurace team for a LED bulb maufacturer tested 400 radomly selected LED bulbs ad foud that 6 are defective. A member of the team performed the followig calculatios to obtai a 95% cofidece iterval for the proportio p of LED bulbs that are defective. Explai the error. = 400 p = p - z c p (1 - p ) p + z c p (1 - p ) = z c = ( ) = ( ) = Houghto Miffli Harcourt Publishig Compay With 95% cofidece, the proportio of LED bulbs that are defective lies betwee 0.3% ad 2.7%. Module Lesso 1

12 Lesso Performace Task Betwee 2010 ad 2011, a research group coducted a survey of youg workig wome ad me, ages 18 to 34. Of the 610 wome surveyed, 66% idicated that beig successful i a highpayig career is either oe of the most importat thigs or very importat i their lives. Of the 703 me surveyed, oly 59% replied that they attach such importace to career. a. Fid a 95% cofidece iterval for each sample proportio. b. Fid the margi of error for each result at the 95% cofidece level. c. Is it possible that the populatio proportios could actually be equal? Explai. Houghto Miffli Harcourt Publishig Compay Module Lesso 1