Confidence Intervals

The cartel intervals represent upper berth and raze bounds of variation around each reference forecast. Values may occur outside the corporate trust intervals payable to external shocks, such as extreme weather, structural changes to the economic system, geopolitical level offts, or technology development. The self-assurance intervals increase in width throughout the forecast period due to the increasing level of uncertainty in each subsequent year.The upper and lower bounds were based on one to two precedent deviations of the historic values, indicating at least a 68 percent chance that future values would be expected to fall indoors the confidence interval. The confidence interval for the first forecast year is based on one pattern deviation and grows linearly until it reaches two bar deviations, or a 95 percent probability. =====================================================================================For example, if we have polled a number of respondents from t he al-Qaeda owners lets say 3500 respondents, and from those lonesome(prenominal) 1190 are using electricity to hot pants their mobs, this means that 34. 0% are using electricity to heat their cornerstones, p? = 1190/3500 = 34. 0%. And we chi bay windowe that a second smack of 3500 property owners wouldnt have a sample remainder of exactly 34. 0%. If another group of theatre owners has taken and we found that they have a sample of comparison of 38. 0%, So the take semblance go forth be the key to our ability to generalize from our sample to the population.Now, we whop that the consume distri thoion model is centered at the square(a) proportion, p, of all floor owners who use electricity to heat their homes. But we simulatet get p. it isnt 34. 0%. Thats the p? from our sample. What we do know is that the sampling distribution model of p? centered at p, and we know that the standard deviation of the sampling distribution is SE(p? ) = v p? q? /n = v(. 34)(1-. 34)/3 500 = 0. 008 Because our sample (3500) is large, we know that the sampling distribution model for p? should look just about like the one shown below The sampling distribution model for p? s normal with a mean of p and a standard deviation we estimate to be v p? q? /n. because the distribution is normal, wed expect that about 68% of all samples of 3500 home owners taken in a specific time would have had sample proportion in spite of appearance 1 standard deviation of p. and about 95% of all these samples volition have proportions within p 2 SEs. But where is our sample proportion in this picture? And what value does p have? We still dont know We do know that for 95% of random samples, p? will be no more than 2 SEs away(predicate) from p. so lets reverse it and look at it from p? s point of view. If I am p? there is a 95% chance that p is no more than 2 SEs away from me. If I reach out 2 SEs, or 2 x 0. 008, away from me on both sides, I am 95% sure that p will be within my grasp. Of course, I wont know, and even if my interval does catch p, I still dont know its unbowed value. Now, We dont know exactly what proportion of home owners using electricity to heat their homes, but we know that its within the interval 34. 0% 2 x 0. 8%. That is, its amongst 32. 2% and 35. 6%. this is getting closer, but we still cant be certain. We cant know for sure that the true proportion is in this interval-or in any particular range.We dont know the exactly the proportion of home owners that use electricity to heat their homes, but the interval from 32. 4% to 35. 6% probably contains the true proportion. Weve now fudged in two waysfirst by giving an interval and second by admitting that we only think the interval probably contains the true value. That last statement is true we can tighten it up by quantifying what we mean by probably. We byword that 95% of the time when we reach out 2 SEs from p? , we fetch p, so we can be 95% confident that this is one of those times , after(prenominal) putting a number on the probability that this interval covers the true proportion,We are 95% confident that between 32. 4% and 35. 6% of home owners using electricity to heat their homes, this is now an appropriate interpretation of our confidence intervals. Its not perfect, but its about the best we can do. Level of Confidence = 95% =1- ? = 0. 05 Sample size (n) = 3500 Sample proportion (p? ) = 0. 34 P complement (q? ) = (1-0. 34) = 0. 66 Sample standard computer error of a proportion = SE(p? ) = v p? q? /n = v(. 34)(1-. 34)/3500 = 0. 008 Z Score(1 ? ) = Z Score95% = NORMSINV (1 ? /2) = NORMSINV (1 0. 025) = NORMSINV (0. 975) = 1. 96 Width of Half the Confidence Interval = Z Score(1-? *S(q? ) = 1. 96 *0. 008 = 0. 01568 Confidence Interval Boundaries = p? +/- Z Score(1-? )*S(p? )= 0. 34 (1. 96) * (0. 008) = 0. 34 (0. 01568) = 0. 32432 to 0. 35568 = 32. 43% to 35. 57% We can be 95% certain that the percentage of home owners using electricity to heat their h omes is between 32. 43% and 35. 57% 0. 01568 0. 01568 0. 316 0. 3243 0. 332 0. 34 0 . 348 0. 3557 0. 364 0. 316 0. 3243 0. 332 0. 34 0 . 348 0. 3557 0. 364