CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate whether or ot we got a plausible or implausible sample statistic. This uit, we have take that ad made it a bit more specific. Namely, we ve computed/costructed specific itervals that tell us what values of a statistic are plausible or ot. Cofidece itervals are costructed with the followig formula: statistic ± (critical value) * (stadard deviatio of statistic) Whether for proportios or meas, the statistic is always the value of the statistic from our particular sample, for example p =.32 or x = 64.1. Oce we have a statistic, we wat to subtract ad add a margi of error, or wiggle room. This margi of error (ME) has a simple formula: ME = (critical value)*(stadard deviatio of statistic) The stadard deviatio of a statistic tells us typically how off a sample statistic will be from the true parameter (either p or μ). The critical value is the umber of these stadard deviatios that we will subtract from our sample statistic: the larger the critical value, the more stadard deviatios we subtract/add; the more we subtract/add to our statistic, the more cofidet we ca be that our method will capture the true parameter a high percetage of times. This critical value will deped o the cofidece level. The width of these itervals depeds o what is called a cofidece level. A cofidece level tells us the hit rate of a certai method of samplig ad costructig itervals. NOTE: THE FOLLOWIN EXAMPLE CONTAINS A LOT OF SAMPLE QUESTIONS JUST LOOK AT HOW MUCH YOU CAN TELL FROM LITERALLY JUST KNOWIN THE INTERVAL ALONE. The Newto Couty school board selected a SRS of 100 couty residets ad costructed a 97% cofidece iterval for the proportio of its residets who were bor out of state. The resultig iterval was (0.17366, 0.36634). a. Iterpret the cofidece level i cotext. Solutio: If we took may, may simple radom samples of size 100 of couty residets, the 97% of the resultig cofidece itervals would capture the true proportio of Newto Couty residets who were bor out of state. What to ote here: (1) The sample size matters. (2) It is the itervals that capture the parameter, ot the samples. (3) CONTEXTCONTEXTCONTEXTCONTEXTCONTEXDTXTESADSFASDFASD b. Iterpret the cofidece iterval i cotext. Solutio: We are 97% cofidet that the iterval from 0.17366 to 0.36634 captures the true proportio of Newto Couty residets who were bor out of state. What to ote here: Do t say that the true proportio falls i the iterval. The true proportio does t move. It is what it is. The itervals will chage from sample to sample. What is importat is whether or ot they capture the o-movig parameter. c. What was the poit estimate used to create this iterval? Solutio: The poit estimate p is smack i the middle of the iterval, so p = @.ABCDDE@.CDDCF =.27. (I the sample take, 27 of the 100 residets selected were bor out of state.)
d. What was the margi of error for the iterval? Solutio: Method 1: The margi of error is what gets subtracted from ad added to p. So, just fid the distace from 0.27 to either ed of the iterval: ME = 0.36634 0.27 = 0.09634 Method 2: The legth of the iterval is just 2 times the margi of error (sice it gets added i oe directio ad subtracted i the other). Therefore, ME = 1 2 0.36634 0.17366 = 0.09634 e. TOUH: What if I HADN T give you the cofidece iterval? What if I just gave you = 100 ad the iterval (0.17366, 0.36634)? How would you calculate the cofidece level C? SOLUTION: We calculated that p = 0.27 ad ME = 0.09634. Let s use the followig equatio: ME = z p 1 p 0.09634 = z 0.27 0.73 100 0.09634 = z (.0443959458) 2.17 z Therefore, our critical value is z = 2.17. The, let s see what happes whe we go that may stadard deviatios away from (i both directios) the mea of the Stadard Normal distributio! ormalcdf( 2.17, 2.17,0,1) = 0.96999 The cofidece level C is 97%. PRACTICE PROBLEM LIKE THAT: A college professor coducted a simple radom sample of 50 of his Calculus II studets ad costructed a C% cofidece iterval about the proportio of Calculus II studets who are majorig i egieerig. The resultig cofidece iterval was (0.223, 0.457). What is the value of C? A. 86 B. 88 C. 90 D. 92 E. 94 The solutio is o the ext page.
Solutio: p = @.CE@.FVB = 0.34, ME = @.FVBW@.C = 0.117 ME = z p 1 p 0.117 = z 0.34 0.66 50 ormalcdf( 1.746, 1.746,0,1) = 0.919 z = 1.746 The value of C is 0.92. The correct aswer is (D) there was a little bit of roudig error. It happes. CONDITIONS FOR CONSTRUCTIN A CONFIDENCE INTERVAL ABOUT A PROPORTION Below is a table similar to that from the last study guide, just abbreviated. CONSTRUCTIN A CONFIDENCE INTERVAL ABOUT A PROPORTION Calculator Commad: STAT -> TESTS -> A:1PropZIt Coditio What it gives us Helpful iformatio 1. Radom 2. 10% coditio: A A@ N 3. Large couts: p 10, 1 p 10 Ubiased estimator -> mea of samplig distributio of p is p idepedet of oe aother. The samplig distributio of p will be approximately Normal. Do t just say radom. Iclude cotext, ad make sure that you idetify WHETHER OR NOT THE CONDITION IS MET. Put a yes or a check mark. idepedet because the probability of success will stay about the same each time (sice you did t sample too much from the populatio). Always, always, always give the specific umber of successes i the problem. Do ot just say p 10". Use cotext/the umbers from the problem. If this coditio is t met, we would t be able to calculate critical values. CONFIDENCE INTERVALS ABOUT MEANS The t-distributio Whe we wat to costruct a cofidece iterval about a sample mea, we take a sample. This sample will have a stadard deviatio s _ that is probably a little bit smaller tha the populatio stadard deviatio σ _. Therefore, whe we are fidig our critical value the umber of stadard deviatios we wat to subtract/add to the mea we eed to use a slightly larger value tha the traditioal z. This value is called t. The t-distributio is symmetric, but has more value i the tails tha the stadard Normal curve. However, as the sample size icreases, s _ becomes closer ad closer to σ _ ; therefore, t will start to become very close to z. The shape of the t-distributio is therefore defied by how big the sample size is. We defie the t-distributio by the sample size mius 1, or the degrees of freedom.
EXAMPLE: What critical value t should be used for a 90% cofidece iterval whe the sample size is 15? Solutio: ivt(0.95,14) 1.761. EXAMPLE: Which of the followig is more likely? A. Radomly selectig a umber betwee -2 ad 2 from a stadard Normal distributio B. Radomly selectig a umber betwee -2 ad 2 from a t-distributio with 10 degrees of freedom C. Radomly selectig a umber betwee -2 ad 2 from a t-distributio with 100 degrees of freedom ANSWER: A. No matter how big t is, the t-distributio will always have more area uder the tails tha the stadard Normal curve. Therefore, the stadard Normal curve will cotai more area betwee the tails. CONSTRUCTIN A CONFIDENCE INTERVAL ABOUT A PROPORTION Calculator Commad: STAT -> TESTS -> 8: TIterval Coditio What it gives us Helpful iformatio 1. Radom 2. 10% coditio: A A@ N 3. Large couts/normality a. Populatio is Normal, or b. 30 (CLT), or c. Sample data is reasoably symmetric w/ o strog skew or outliers Ubiased estimator -> mea of samplig distributio of p is p idepedet of oe aother. The samplig distributio of p will be approximately Normal. Do t just say radom. Iclude cotext, ad make sure that you idetify WHETHER OR NOT THE CONDITION IS MET. Put a yes or a check mark. idepedet because the probability of success will stay about the same each time (sice you did t sample too much from the populatio). If the shape of the pop dist. is ukow ad < 30, you must look at a plot of the sample data. TIPS FROM HORNBECK 1. Whe costructig a iterval, always NAME the iterval ad give the FORMULA. Name Formula Oe-proportio z-iterval (Pla step) p ± z p 1 p Oe-sample t-iterval (Pla step) x ± t s _ 2. KNOW YOUR CONFIDENCE LEVELS. IT LL SAVE TIME. For proportios: 90% cofidece --> z = 1.645 95% cofidece --> z = 1.96 99% cofidece --> z = 2.576 For meas, the t value will always be greater tha the correspodig z value (if oly by a little bit).
3. Relatioships INCREASIN SAMPLE SIZE --> CONFIDENCE INTERVALS ARE NARROWER. DECREASIN SAMPLE SIZE --> CONFIDENCE INTERVALS ARE WIDER INCREASIN CONFIDENCE LEVEL --> CONFIDENCE INTERVALS ARE WIDER DECREASIN CONFIDENCE LEVEL --> CONFIDENCE INTERVALS ARE NARROWER 4. Stadard error v. stadard deviatio The stadard error is whe we use the sample stadard deviatio i place of the populatio stadard deviatio, i.e. s f = f AWf g as opposed to σ f = s _ = h i g as opposed to σ _ = j g f AWf g