This is top 10 topic number three confidence in the balls to a confidence interval for the meter. And, in a suitable for the proportion and will start from the main bus trip to lap webpage code or exam review. Click on statistics to click on statistics or do and will download and poll up the PowerPoint presentation and this presentation starts on slides 34. But you know what rather than starting exactly and 34 Masaru on slide 49 shows for a moment and will go back to 34 so those who can talk a little about a confidence interval or point that the difference between point estimates an interval of stumbles so point estimate that's the first bullet here on slide 49 he is a statistic or a single number. A bull point number two examples and sample mean or sample abortion or some other six and others as well, number three in each sample gives a different point estimates a stop for a moment member you're pulling a sample from a population pulling a smaller sample from a much larger population each time you draw a sample and you compute for example, the sample mean the simple mean is not likely to be exactly the same number for each sample. It's going to be a little bit different probably right would be very far will the sample means be very far away from each other. Probably not the probably be fairly close to each other, but any case, what were asked to report the sample mean, what we'd really like to do is report the interval best of it rather than a point customer who take as each sample gives a different point estimates a bullet point member for an interval estimate is arrange a values rather than just listing all the numbers there's a specific formula for a four example population mean, is equal to the sample name plus the heir to a population mean is equal to the sample mean a single statistic plus or minus the air once we add the plus or minus the air we turned a single statistic into excuse me, we turn a point estimate into an interval estimate and that gives more information about our particular day that it's clearer to where the population and its and it's a clearer range as to where the population mean will actually fall okay, so the last bullet is to generalize that sentence. A parameter that is the population parameter is equal to a statistic that is the sample statistic plus or minus a mayor. And that helps turn a point estimate into an interval estimate, which gives more information. Okay up to and he gives us and a better confidence that the bet if we were to take a number of samples say 95 samples from store a hundred samples 95% of the time of the pop the true population mean, would be in that interval so let's go back to slide number 34 and will work her way forward so confidence intervals and slight number 35 years of formal definition of conference symbols a range of values within the within which the population parameters expected to occur. So it's a range it's a interval estimate rather than a point estimate what any factors for the confidence of a role of the factors that determine the width of the confidence of Lahr won the sample size also known as in the number of observations to the variability of the population usually estimated by the standard deviation and three to decide level of confidence a 95% or something like a pang slide number 37 was compute the confidence interval for the mean will do the proportion later on. The we use a normal distribution, which is to see table if the population if we know the population standard deviation which look lowercase ago, if that's known usually it's not known but if it's known in the problem it'll tell you the word improbable to audio and a very strong from a normal population of the words of the sample is strong from a population that's approximately normally distributed that usually what the sentence will be and the sample size is greater than or equal to 39 the sample sizes lower than 30 and we don't know the pop or we don't know the population standard deviation to use the tea table will use that later on the all it does is adjust the range of the inner fall, because it's changing the statistic that this excuse because it's changing of the error are about the point estimate about the the statistic in the middle of the range that a slight 38. So here is the formula there's a formula for the confidence interval for the mean if if if the samples pulled from a normal distribution in other words for use in is the table then the new which is the population Main is equal to the Psalm of acts of either buy and that would be there to be a sample statistic in this case a sample mean air of the air is disease value times and the out population standard deviation divided by the Square or to the and with that you would get our range with them the population mean in the middle that so that's the interval estimate the number is plus or minus here so you can you get something to the left of the middle something to the right of the middle and that all of those at all of that information together is the interval estimates rather than the damn sample is that rather than a point customer came next slide to number 39 day now the tale is one half of of one minus the confidence level. So if the Council Los Lobos 8.95 subtract one from app that would be .05 and we divided in half why a half, because were looking at both sides of the tale rule is is be looking at both sides of the normal distribution. We want both tales in the distribution nor bullet point number two for second Coldplay number three this review uses the tale of the Bell curve the normal distribution for example and 95% confidence the tale is .54 one half of one -.95 to one -.95 is .05 one half of .05 is .024. So those and then we what we would do is look up and is the table for the see of .0 to five, and that's equal to 1.96's store are no tables on the exam if you have to look up something you'll be given in the problem of paying to this means disease value in the table for P. .0 to five minutes 1.96%. So if we go back to the formula that we have before, if and is equal to four this is on slide for a if and is equal to 49 of the sum of ex is equal to 490 of the population standard deviation is two and were working on 95% confidence level that we have the population mean, is equal to 49 he divided by 49, which is 10 plus or minus a 1.96 because that is the value that we just found looking up times to the standard in the population standard deviation divided by the square root of an 49 is equal to 490 divider by 49, which is 10 plus this of these all look the 1.96 times to divide them by swearword event, which is .56 okay so if we take 10 and we add .56 we get 10.56 sets the upper bound of the general and with we take 10 and we subtract .56. That's 9.44, and that's the lower bound of up the confidence of noble came we think the population mean will be between these two. That's all that an interval estimate is a better estimate than a point estimate a thing here's another example slide 41 to one of the SOM professes a sea son wants to estimate the mean number of hours worked per week by students a sample a 49 students showed a mean of 24 hours. It is assumed that the population standard deviation is four hours. What is the population mean, and paying me in an easy answer to this question is the population mean is 20 for the same as the sample status to be mean but that you don't know that for sure. There is no way to know that even if we took a whole lot of samples were strong not absolutely sure about that. So what we really want is a one that won the question says, what is the population mean the best answer is not a single statistic, but rather a single assistance to him and not a single statistic error that changes it from a point estimate to an interval estimate, which provides more information to this so one professor once this information of paying up to right slide 42 more working in a 95% confidence interval for the population mean in and ex-bar, which is the sample means of plus or -1.96 divided by Sigma divided by square root of and that's the formula so we plug in our numbers for this particular form of 24 plus or my this 1.96, which is the value it .025 times for which the population standard deviation all given the problems divided by the square root of 49 all given the problem that results in an and a form of an answer that's 24 plus or my's 1.12 so if we take 24 and we bad. 1.12 we end up with an upper bound of 25.12. If we take 24 and we subtract 1.12 we end up with 22.88 and that would be the lower bound of the confidence level of excuse me a cup of the confidence of both and I'll just read the bottom part of 42 the confidence limits range from 22.88 225.12 we estimate with 90% confidence that the average number of hours worked per week by students lies between these two values actually use if you took a hundred samples 95% of the time. The true population mean, would be within the range of paying to 43 conference or a bowl for the mean, but we use his tea distribution, and we can we use a tea distribution as if it's for if the sample was drawn from a normal population, but the population standard deviation lowercase sigma is not known and will point number two is here given the sample standard deviation S. use the tea table and just assume those in normal population of a pain. And if it's one population remember with a tea table, you're stuck to have to calculate degrees of freedom and degrees of freedom is equal to an minus one and is the number of observations and let's go to slide 44, here's the tears the form of the algebraic formula for the conference in a role for the means using the tea distribution and you see it's a very similar to the formula for the see the first parts exactly the same. The only thing that changes we use is the rather than tea leaves team rather than as a it's basically the only difference and we use the sample standard deviation of the population standard aviation. Okay, so a former list of population mean new sequel to the sum of ex-divided by and that's the sample mean plus or my the team value for an minus one (that's the tea for the degrees of freedom keepers of freedom is a cold and minus one number of observations plus one times to pass, which is the sample standard deviation divided by the spirit of the tank slide 45. Develop the confidence interval for proportion you use this bullet point number the first bullet point here is a use if if if if use proportions of his success or failure wide because those are the only two options of the total for exec other examples are defective and not defective satisfactory or unsatisfactory. Here are a few more male female. For example, those are two dichotomous outcomes of okay in the the normal lock up you can use the normal approximation to the binomial that's okay if the number of items times the proportion is greater than five and end times one minus five years pies, used his portion is greater than five and is the sample sizes and pie is the proportion is a population proportion of and you never use the tea table, if you're working with proportion should only use the tea table with means. Not with proportionate use the binomial table if it's a proportion of paying and let the excuse reuses the table of its proportion sorry about that gave the formula is slightly different for proportion to let me show it to you here on slide 46 or go to the.
Population proportion and that's by his equal to the sample proportion that's the year of plus or minus zero we look that up in the table times the square root of Pete Times one minus Pete divided by and so this part of the equation the first part of the equation is similar in some ways, but they be ending part of equation is used as a little different. We look up a value in saying, but what we multiply against in other words, the part of the formula related to the air is different for proportion than it is for a mean. So for example the middle of page 40s of slight 46, a defective set of a hunter to the probability is a over a hundred witches point away and is equal to 195% confidence. So what would what we deem you do is need to plug these values into the form or .08 said that that is the number of that is the sample portion of the defect is out of a hundred is point away plus or minus remember it's always closer minus because we want to build both sides of the inner role we want to change a point to an interval so we want to go all we want to put a whole bunch of points within that interval on each side. So plus or minus will look up as the value of your 95 for her confidence would be one half a one -95% of a one -95 per 195% is .951 -.95 is .0 5 1/2 of .05 is .0 to five and a free look up as the table for .025 we find at that value is 1.96, and we multiply that times the rest of the formula, which is P.28 times one, minus the one -.08 is .92 so .08 times .92 divided by a hand in SQL 2.08 plus or -.05. We do the same kinds of calculations to get an upper bound and lower down just a formalist differ for a proportion that it is for me. A slight 47 here's another work example in the more examples you see in you that you work the better off you are conferencing over for a proportion of sample of 500 people who own their homes revealed that 175 of them planned to sell the home within the next five years to develop a 98% confidence are also sold different than 95% but develop and 98% conference a mole for the proportion of people who plan to sell the house within five years in a case of the first thing you compute it is the problem to is the sample proportion the simple proportion is the number of people wish to sell the home of hundred and 75 divided by the sample 500, and that would be .35 or 35% though many many problems would just leave it there but not in statistics we want to know more about the data okay and that's a function of a college graduate is to know more about the data than a typical person walking down the street so we want an interval rather than a point asked him a okay so the interval here would be .35 sample proportion of plus or minus one plus and minus cozy intervals on both sides to see value in this particular case is one half minus one -.98 three, which is a different number instead of 1.96 if he look it up in the table. It's 2.33 times the square root of teen times one minus peace of its .35 minus one times one -.35 richest .65 all divided by 500 tickets where it are that and you get .0497 so the confidence interval for this particular portion proportion schismatic is up .35 plus or -.0497 to pay the interpretation of bout is on slide 48 of what you're doing that contain bullet point number one if you're working at the 95% confidence than 95% of all the confidence intervals that you compute will come will include the true population parameter when they restate that if you draw sample from a population you draw another sample from another sample drawn the other sample than 90 excuse me and I apologize then 95% of all the confidence intervals that you bet you develop will include the true population parameter. That's what 95% means and the interesting at the second bullet is an interesting note on the terminology note never used the term probability when estimating a parameter to pay this is actually a really nice little paragraph or for example do not say the probability that the population of the probability that the population to mean is between 23 and 32 is .95 or 95% to cut why because a parameter or a population parameter is not a random variable and the variables are for samples. Not for populations population is fixed but a now something we can see the sample is what we say. Okay, in fact, the population mean is fixed but an unknown quantity. That's just restating what I should set so don't use the word probability when talking about population parameters with a slight 49 is just the same things that we talked about her that the but in the very first part of those lecture, which is instead of using point estimates we want a given interval at some of us call the conference are all go through that again and though they're a less live in is us are open, not of the necks of the flashlight is the width of the interval you might ask how wide is the interval going to be okay here's a couple of a full point about that for examples sample mean of 23 in the air is three to the point of point estimate would be 23 you could just answer a question by giving a point estimate 23 will play number three is the end of a lesser men is really 23 or 23 plus or minus three. So 23 plus three is 26. That's the upper bound 23 minus three is to one lane, which is lower brown than paying and the width of the interval is 26 -20 46 six is twice the big air of their year to three times two in the last bullet is a wide interval you have a very wide and her mobile point estimate is unreliable because you don't know exactly where it's going to be in there, which one is narrow and are false on the talk a little more about the deaths on slide 51 and you end up with a wide conference honorable. If one you have a relatively small sample size, increasing the sample size is usually a good idea and most care in almost all cases. But if he ever relatively small sample size can end up with a wide confidence and role, which means that your results are less reliable than you would like to you or you have a large state standard deviation of the research large variation among the data so it's difficult to be able to so say were all points or are three of a very high confidence level for example 99% confidence and that for example a 99% confidence interval is whiter than 95% confidence level to the last bullet if you want to narrow confidence interval you need a large sample size, not small sample slut is or is small or relatively small standard deviation, if possible or choose a lower confidence level like 95% as opposed to 99% okay. And that's confidence intervals and paying