CHAPTER 8 CONFIDENCE INTERVALS

CHAPTER 8 CONFIDENCE INTERVALS Cofidece Itervals is our first topic i iferetial statistics. I this chapter, we use sample data to estimate a ukow populatio parameter: either populatio mea (µ) or populatio proportio (p). (See why it s called iferetial statitistics?) Populatio parameters are geerally ukow, because it is hard to have populatio data. We have talked about how we ca use x to estimate µ. I fact, we say that x is the poit estimate for µ; x is our sigle best guess at µ. I this Chapter, we estimate populatio parameters by providig a cofidece iterval of umbers that we are, say, 95% certai cotais the value of the parameter. The iterval is cetered at the poit estimate ad the exteded i both directios by the margi of error: cofidece iterval for parameter = poit estimate ± margi of error I this sceario, 95% is the cofidece level, CL. The margi of error depeds o the cofidece level. (If we wat to be more cofidet that the iterval cotais the populatio parameter, what must we do to the margi of error?) 1. A Sigle Populatio Mea (Cofidece Iterval) usig the Normal Distributio For the case of estimatig populatio mea µ, the poit estimate is x, ad the margi of error is kow as the EBM, or error boud for populatio mea. Thus, the cofidece iterval for populatio mea has the form cofidece iterval for µ = ( x EBM, x + EBM) Notice that the two equatios above demostrate two differet, but equally valid, ways to express cofidece itervals: usig ± otatio or iterval otatio. As oted above, margi of error (EBM) depeds o the cofidece level (CL). We choose the EBM so that we ca say somethig like: we are 95% certai that this cofidece iterval cotais the true populatio parameter. Actually, a more accurate statemet would be 95% of samples of this size from the populatio produce cofidece itervals that cotai the true value of the populatio parameter. (Remember that we use samples from the populatio to do our estimatig.) Coversely, we could say that 5% of samples of this size from the populatio produce cofidece itervals that DO NOT cotai the true value of the populatio. This value, 5% is kow as α (alpha). So, 1 CL = α is the probability that our cofidece iterval DOES NOT cotai the true value of the populatio parameter. α = 1 CL Note that CL ad α are both probabilities ad may be expressed as decimals or percets. 1

2 CONFIDENCE INTERVALS Example 1. Cofidece Iterval Form Suppose we have collected data from a sample. We kow the sample mea is seve, ad the error boud for the mea is 2.5 for a cofidece level of 95%. (1) x = (2) EBM = (3) The cofidece iterval is ± or (, ) (4) We estimate with cofidece that the true value of the popoulatio mea µ is betwee ad. Example 8.1 (edited). (Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013.) We tur our focus to the margi of error, EBM; where does this umber come from? Draw a ormal distributio with µ at the ceter. Suppose we wat a 90% cofidece iterval. Shade a area of 90% i the ceter. (What is α? Where does α show up i the drawig? ) This ormal distributio is NOT X, but is X (which we ca ofte assume is ormally distributed due to the ), because we wat the data values to be x s, sample meas. I our drawig, we ca see that 90% of samples (of the appropriate size) will have meas that lad o the umberlie iside the shaded area, ad 10% of samples will have meas that lie o the umberlie outside the shaded area (i the tails ). We eed for the samples whose meas fall withi the shaded area to result i cofidece itervals that capture the true value of the populatio mea, which is right i the ceter. So o the drawig, how wide must the EBM be?. More precisely, to capture the cetral 90% of data, we must go out 1.645 stadard deviatios from the mea o either side. So, for this example, EBM = 1.645 (stadarddeviatio). Due to the Cetral Limit Theorem, we kow that the stadard deviatio of X is. So we have EBM = 1.645 σ, where σ is the stadard deviatio of the populatio of iterest ad is the size of the sample we have collected. (Yes, it IS kid of reckless to assume we kow σ whe we do t kow µ, but we are goig to assume we have that iformatio for ow. We will hadle the dilemma of ukow σ i the ext sectio.) Example 2. Fid the percetile (P? ) of the z-score ad the z-score for calculatig the margi or error (EBM) for each cofidece level. Remember that z-scores come from a stadard ormal distributio: Z (1) 90%: z-score = P = (2) 95%: z-score = P = (3) 99%: z-score = P =

3 I geeral, z score = P (CL+ α 2 ) = ivorm(cl + α, 0, 1) 2 where α = 1 CL. To summarize, if we wat to use a radom sample of size to geerate a CL cofidece iterval estimate for the populatio mea µ of a populatio that has stadard deviatio σ, where cofidece iterval estimate = x ± EBM = ( x EBM, x + EBM EBM = z-score σ. Example 3. Calculatig a Cofidece Iterval Suppose scores o exams i statistics are ormally distributed with a ukow populatio mea ad a populatio stadard deviatio of three poits. A radom sample of 36 scores is take ad gives a sample mea (average score) of 68. Follow the steps to fid a 90% cofidece iterval estimate for the populatio mea exam score (the average score o all exams). (1) Idetify x, σ, CL, ad α. (2) Fid the appropriate z-score. (3) Calculate the margi of error, EBM. (4) Write the cofidece iterval i the ± form. (5) Write the cofidece iterval i the iterval otatio form. (6) Plot the cofidece iterval o a umber lie. Label the sample mea ad the EBM. (7) Iterpret the cofidece iterval i the cotext of this problem. Example 4. Calculatig a Cofidece Iterval from Data (Example 8.3 i textbook.) The Specific Absorptio Rate (SAR) for a cell phoe measures the amout of radio frequecy (RF) eergy absorbed by the user s body whe usig the hadset. Every cell phoe emits RF eergy. Differet phoe models have differet SAR measures. To receive certificatio from the Federal Commuicatios Commissio (FCC) for sale i the US, the SAR level for a cell phoe must be o more tha 1.6 watts per kilogram. Table 8.1 i the textbook shows the highest SAR level for a radom selectio of cell phoe models as measured by the FCC. (1) Fid the mea of the sample of SAR levels. (2) Fid a 98% cofidece iterval estimate for the true (populatio) mea of the SARs for all cell phoes. Assume that the populatio stadard deviatio of SARs is σ = 0.337. (3) Iterpret the cofidece iterval (ALWAYS i the cotext of the give problem).

4 CONFIDENCE INTERVALS Please do the followig to be tured i at the ext class meetig: (1) Fid the z-score that would be required to calculate the margi of error (EBM) for a 93% cofidece level. (2) Explore chagig the cofidece level: (a) Fid the EBM ad cofidece iterval for Example 3, but with a cofidece level of 95% (keepig everythig else the same). (b) Fid the EBM ad cofidece iterval for Example 3, but with a cofidece level of 99%. (c) How does icreasig the cofidece level affect the margi of error, EBM? The width of the cofidece iterval? (d) Explai i your ow words why the margi of error is affected i this way whe the cofidece level is icreased. (3) Explore chagig the sample size: (a) Fid the EBM ad cofidece iterval for Example 3, but with a sample size of 100. (b) Fid the EBM ad cofidece iterval for Example 3, but with a sample size of 500. (c) How does icreasig the sample size affect the margi of error, EBM? The width of the cofidece iterval? (d) Explai i your ow words why the margi of error is affected i this way whe the sample size is icreased.

5 2. A Sigle Populatio Mea (Cofidece Iterval) usig the Studet t Distributio I the last sectio, we used a radom sample from a populatio to create a cofidece iterval for the populatio mea µ of a populatio. The cofidece iterval had the form: x ± EBM = x ± (z-score) σ I this sectio we are still estimatig µ with a cofidece iterval. The differece is that i this sectio we DO NOT assume the stadard deviatio σ of the populatio is kow. (Do you thik it is more likely that σ is kow or ukow?) How does NOT KNOWING σ affect the costructio of the cofidece iterval? Well certaily that does ot affect the first part of the formula: x. However, ukow σ impacts the formula for the EBM i two ways: (1) Just like x is the best poit estimate for µ, the SAMPLE stadard deviatio s is the best poit estimate for the POPULATION stadard deviatio σ. So the best we ca do i the formula is to put s i the place of σ. (2) As you might guess, usig s i the place of σ itroduces some error ito the EBM, but we do t kow if it errs by makig the EBM too large or too small. If s < σ, a 95% cofidece iterval will be too small to capture µ 95% of the time. Statisticias couterbalace this possible error by usig a t-score istead of a z-score, which makes the EBM just a tad larger. Whe creatig a cofidece iterval estimate for populatio mea µ, if the populatio stadard deviatio σ is UNKNOWN, EBM = (t-score) s. Obviously we ow eed to kow more about what a t-score is. As stated above, the t-score for a CL is smidge larger tha a z-score for the same CL, i order to offset ay error itroduced by usig s i place of σ. How much larger? That depeds o the sample size: If the sample size is fairly large, s is expected to be very close to σ, the error itroduced by usig s will be small, ad the t-score will be oly slightly larger tha the z-score. O the other had, if the sample size is small, the error itroduced by usig s to estimate σ will be bigger, so the t-score has to be larger relative to the z-score to off-set that error. Just like z-scores arise from a Z distributio, t-scores arise from a T distributio. Here are a few importat facts about the T distributio: The T desity curve is bell-shaped ad has mea 0 ad stadard deviatio 1, just like the stadard ormal distributio, Z. The precise shape of the T desity curve depeds o the sample size, or more precisely, o the degrees of freedom, df: df = 1. The higher the degrees of freedom, the more closely the T distributio matches the stadard ormal distributio, Z.

6 CONFIDENCE INTERVALS William S. Goset (1876 1937) of the Guiess brewery i Dubli, Irelad ra ito [the problem of error itroduced by small samples]. His experimets with hops ad barley produced very few samples...this problem led him to discover what is called the Studet s t-distributio. The ame comes from the fact that Gosset wrote uder the pe ame Studet. Excerpt From: Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013. ibooks. How do we fid a t-score? There are two ways. (1) If you have a TI-84+, your calculator has a fuctio ivt, which works just like the ivorm fuctio, except that you have to eter the degrees of freedom: t-score = ivt(area to the left, df), where df = -1 (2) If you have a TI-83+, you have to use a table of t-scores. (You ca dowload this table o my website.) Sca across the top to fid the area to the left of your t-score. Sca dow to fid the degrees of freedom, df = 1. Note: You will ofte see a subscript o t ad z. The subscript is the tail probability associated with the score, which is α/2. For example, a cofidece level of 95% is associated with the z-score z 0.025 = ivorm(0.975, 0, 1). Example 5. Calculatig t-scores. Fid the t-score associated with each cofidece level ad sample size combiatio. Use ivt(area, df) if you ca ad the table otherwise. (1) CL = 90%, = 6 (2) CL = 90%, = 100 (3) CL = 95%, = 6 (4) CL = 95%, = 100 Summary: Cofidece Iterval for Populatio Mea µ Requiremets: The margi of error (EBM) calculatio requires that X is ormally distributed. By the Cetral Limit Theorem, we require that > 30 or X is ormally distributed, which we verify by checkig to see if the sample appears to be approximately ormally distributed. If σ is KNOWN: x ± (z-score) σ If σ is UNKNOWN: x ± (t-score) s Example 6. Cofidece Iterval for µ (σ ukow). You do a study of hypotherapy to determie how effective it is i icreasig the umber of hours of sleep subjects get each ight. You measure hours of sleep for 12 subjects with the followig results. Costruct a 95% cofidece iterval for the mea umber of hours slept for the populatio (assumed ormal) from which you took the data. 8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10. TryIt 8.8. (Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013.) There are calculator fuctios that calculate cofidece itervals for µ. If σ is KNOWN: STAT, TESTS, ZIterval If σ is UNKNOWN: STAT, TESTS, TIterval What you should kow about usig these calculator fuctios:

7 (1) The calculator prompts you o what to eter, but you have to kow what the symbols mea. (2) You have a choice of Data or Stats. Choose the Data optio if you have the data stored i a list o your calculator. (If you have the data i a sigle list (ot as a frequecy table i two lists), eter Freq: 1.) (3) The cofidece iterval you get back is i iterval otatio. If you eed to recover x ad EBM from the cofidece iterval, remember that x is the midpoit of the iterval ad EBM is half the width of the iterval. For a cofidece iterval (a, b), x EBM = a+b 2 = b a 2 Example 7. Cofidece Iterval for µ usig a Calculator Fuctio. Redo Example 6 usig the appropriate calculator fuctio. Example 8. Cofidece Iterval for µ usig a Calculator Fuctio. A sample of 20 heads of lettuce was selected. Assume that the populatio distributio of head weight is ormal. The weight of each head of lettuce was the recorded. The mea weight was 2.2 pouds with a stadard deviatio of 0.1 pouds. The populatio stadard deviatio is kow to be 0.2 pouds. (1) Use appropriate symbols to label all of the iformatio, icludig X ad X ad their distributios. (2) Which distributio should you use: T or Z? (3) Use a calculator fuctio to make a 95% cofidece iterval for the mea weight of all heads of lettuce. (a) iterval otatio: (b) ± otatio: (4) Iterpret the cofidece iterval you foud. Ch 8 Exercises. (Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013.) 3. A Populatio Proportio (Cofidece Iterval) I this sectio we switch from estimatig µ to estimatig p, populatio proportio. Whe estimatig populatio mea µ, the data is quatitative ad ofte cotiuous. The uits o the cofidece iterval bouds match the uits of the data. For example: With 95% cofidece, a perso usig hypotherapy sleeps is betwee 8.17 ad 8.46 hours per ight o average. Whe estimatig populatio proportio p, the data is categorical (with two categories of iterest). The uits o the cofidece iterval bouds are percetages. For example: With 95% cofidece, betwee 8.7% ad 9.5% of kidergarteers kow how to read whe they start school. The sceario where the data is categorical with two categories should soud familiar, like a biomial radom variable. Recall that if X B(, p), the X = the umber of successes out o trials, µ X = p, σ X = pq, ad the probability distributio of X is bell-shaped, like the ormal distributio.

8 CONFIDENCE INTERVALS However, what we are actually iterested i here is ot the umber of successes, x, but the proportio of successes p = x. Now, if you divide all the x data values by, you have ew radom variable X = the proportio of success i trials. I other words, X represets the populatio of sample proportios, p (much like X represets the populatio of sample meas). Notatio: p = p = populatio proportio sample proportio The radom variable X p has mea = p ad stadard deviatio pq = pq. The histogram of X matches that of X: it is bell-shaped. I fact, the histogram has such a ice bell-shape that as log as the sample size is fairly large ad p is ot very close to 0 or 1, we ca approximate the populatio of sample proportios by a ormal distributio: ( ) X pq N p,. Recall our geeral formula for cofidece itervals: poit estimate ± margi of error. I this sceario, the best poit estimate for populatio proportio p is sample proportio p. The margi of error is called the error boud for the proportio, EBP ad follows the same geeral formula as the EBM: EBP = (z-score)(stadard deviatio) = (z-score) ( ) pq Notice that the formula requires ukow p. The best we ca do is to replace p by it s best poit estimate, p, ad q by q = 1 p. We calculate EBP as follows: ( ) p EBP = (z-score) q. Now, our cofidece iterval for populatio proportio p is where ad ( ) p p q ± (z-score), p = x = umber of successes sample size q = 1 p. Example 9. Cofidece Iterval for Populatio Proportio. Suppose 250 radomly selected people are surveyed to determie if they ow a tablet. Of the 250 surveyed, 98 reported owig a tablet. Usig a 95% cofidece level, compute a cofidece iterval estimate for the true proportio of people who ow tablets. TryIt 8.10. (Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013.)

9 The calculator fuctio that fids cofidece itervals for populatio proportio directly is: STAT, CALC, 1-PropZIt. It is very simple to use. Ca you figure out what all the parts of the ame 1- PropZIt mea? Example 10. Cofidece Iterval for Populatio Proportio Calculator. Use 1-PropZIt to redo Example 9. Iterpretig Cofidece Iterval Overlap. We say that there is statistical evidece that the meas (or proportios) are differet as log as either cofidece iterval captures the poit estimate of the other cofidece iterval. Example 11. Iterpretig cofidece iterval overlap. Suppose we have cofidece itervals for the populatio mea scores o a statistics test by geder. What ca we coclude about the populatio mea scores of boys ad girls i each case: is there statistical evidece that the mea scores are differet? (1) Boys: 73 ± 2, Girls: 78 ± 2 (2) Boys: 73 ± 3, Girls: 71 ± 2 (3) Boys: 73 ± 3, Girls: 69 ± 2 Sample Size Determiatio. A importat questio whe coductig statistical aalysis is, How large must the sample be? Ofte gettig larger samples is expesive, labor itesive, ad/or time cosumig. Fortuately, it is possible to aalyze how large your sample must be to achieve the desired level of cofidece ad margi or error. The strategy for determiig sample size is to use the margi or error formula, EBM ad EBP, ad solve for. Oe thig to recogize is that icreasig sample size decreases margi of error. We will wat at most some amout of error. If our sample size falls short, we will have too much error. Therefore, with sample size, we always err o the side of too may samples. This reasoig accouts for the followig two covetios: Set p = q = 0.5 i the formula. (Naturally, sample size calculatio occurs before data collectio, so we do t yet have p. The choice of p = 0.5 maximizes (p )(q ), prevetig a uderestimate of the required sample size.) Roud UP sample size calculatios to the ext whole umber. (Sample size is a whole umber. If we roud dow, we uderestimate the required sample size.)

10 CONFIDENCE INTERVALS Example 12. Required Sample Size for a Cofidece Iterval for p. Suppose a mobile phoe compay wats to determie the curret percetage of customers aged 50+ who use text messagig o their cell phoes. How may customers aged 50+ should the compay survey i order to be 90% cofidet that the estimated (sample) proportio is withi three percetage poits of the true populatio proportio of customers aged 50+ who use text messagig o their cell phoes. Example 8.14. (Barbara Illowsky & Susa Dea. Itroductory Statistics. OpeStax College, 2013.) ( ) p (1) EBP = (z-score) q. Our strategy is to fill i everythig except, the to solve for. (a) Fid the z-score for a 90% cofidece iterval. (b) What is the desired EBP? (c) That leaves p, q, ad. For sample size calculatio, set p = q = 0.5. is ukow, so leave it as. (2) Use the EBP formula with the values you have foud to solve for. (3) Roud = sample size UP to the ext whole umber. (4) Write a statemet explaiig what you have foud. We ca use a similar strategy to discover the required sample size whe estimatig populatio mea µ. Of course we use the appropriate margi of error formula: EBM = z-score σ. Example 13. Required Sample Size for a Cofidece Iterval for µ. Suppose a medical researcher eeds to estimate the mea white blood cell cout (i cells per microliter) for adults i the US. A aalysis of past studies reveals a approximate populatio stadard deviatio of 1.6 cells per ml. What sample size is required for a 99% cofidece iterval of the mea white blood cell cout to withi 0.2 cells per ml?

11 Please COMPLETE THE FOLLOWING. Due: Next class meetig. (NOTE: Sample size determiatio for cofidece itervals for populatio mea µ are ot i WebAssig, but you are expected to kow it.) Example 14. How may cars would we eed to sample to estimate the average speed of cars passig by Augusta Uiversity o Walto Way? Assume we require 95% cofidece that the estimate is withi 2 miles per hour (mph) of the average speed. A previous study of speeds o a similar road leads us to approximate the stadard deviatio of speeds at 6 mph. Example 15. A ecoomist wats to kow if gas prices are affectig the proportio of people i the US who commute to work via carpoolig. How may people must he sample to be 90% cofidet that estimate is withi 2% of the the true proportio of carpoolers i the US?