Estimation4

Summary

In this lecture, Erin Heerey introduces the concept of interval estimates as a solution to the variability of point estimates. Interval estimates, unlike point estimates, provide a range where a true population parameter is likely to lie. Heerey compares them to fishing with a net instead of a line, explaining how confidence levels impact the width and precision of intervals. Throughout the lecture, Heerey emphasizes the importance of defining confidence levels and understanding the trade-offs between precision and informativeness. The use of bootstrap methods for calculating confidence intervals when normal distribution assumptions aren't met is also covered, alongside discussions on effect size and the importance of methodology.

Highlights

• Interval estimates solve the problem of variability in point estimates by offering a range where a true parameter is likely to fall. 🔄
• Confidence intervals resemble fishing with a net instead of a line, making them more reliable for catching the true parameter. 🎣
• Narrower confidence intervals suggest greater precision, while wider ones can be less informative. ⚠️
• Bootstrapping is a method used to calculate confidence intervals without relying on normal distribution assumptions. 🔄
• 95% confidence intervals are standard but varying them depends on the desired certainty and informativeness. 🤔

Key Takeaways

• Interval estimates provide a range of values within which the true population parameter is likely to fall, offering more precision than single point estimates. 🎣
• The width of a confidence interval reflects the level of certainty; narrower intervals imply greater precision. 🔍
• Bootstrap methods are useful when assumptions for normal distribution aren’t met, offering a flexible and universal technique. ⚖️
• Understanding and setting confidence levels beforehand ensures a more accurate estimation process. 🎯
• Confidence intervals can play a key role in assessing the effect size or magnitude of differences between groups in experimental research. 📊

Overview

In this educational lecture, Erin Heerey delves into the nuances of statistical estimation, focusing on the concept of interval estimates as a method to improve the reliability of research findings. Traditional point estimates, while useful, are prone to variability depending on sample composition. In contrast, interval estimates provide a probabilistic range, giving researchers a better chance of capturing the true population parameter.

Heerey illustrates the analogy of fishing with a net versus a line to explain the difference between point and interval estimates. The lecture touches on the definition of confidence intervals and their role in research, emphasizing how confidence levels influence the precision of these intervals. He further explains the use of bootstrap methods to derive confidence intervals, particularly when traditional assumptions of normal distribution are not met.

The session concludes with a discussion on the practical applications of confidence intervals in determining effect size and the differences between groups in experimental research. Heerey underscores the importance of solid methodology and empirical approaches in accurately interpreting data, ensuring that researchers can make informed and confident inferences in their scientific inquiries.

Chapters

• 00:00 - 03:00: Introduction to Interval Estimates The chapter discusses the limitations of point estimates, which are single samples that can vary based on the sample composition. It introduces interval estimates as a solution, explaining that these are ranges within which the true population parameter is likely to fall, providing a more reliable estimate than point estimates.
• 03:00 - 10:00: Estimating Confidence Intervals This chapter discusses the concept of estimating confidence intervals, which are ranges of values likely to contain a population parameter (e.g., mu). The chapter emphasizes the importance of defining the level of confidence in the estimation process before collecting data, ensuring a higher likelihood of capturing the parameter. It likens using confidence intervals to fishing, where the goal is to capture the true parameter value within the defined range.
• 10:00 - 20:00: Distribution of Confidence Intervals This chapter explores the concept of confidence intervals, drawing an analogy to fishing techniques to help explain their nature. The transcript discusses how the size of a confidence interval depends on the desired level of confidence, which must be determined in advance. It analyzes the implications of the range size on precision, suggesting that narrower intervals indicate greater precision. The chapter further delves into interval estimation, which involves finding the upper and lower boundaries of an interval that may contain a population parameter, providing a valuable metric for description.
• 20:00 - 30:00: Bootstrapping for Confidence Intervals The chapter discusses the importance of confidence in the estimation process, emphasizing that point estimates of parameters are still significant. For example, the mean and standard deviation of a sample are calculated to make population parameter estimates. Rather than simply stating, for instance, that the mean is 26.78, the chapter explains a method that adds depth to this estimation process, likely involving bootstrapping techniques to enhance confidence intervals.
• 30:00 - 40:00: Theory and Calculation of Confidence Intervals The chapter focuses on the theory and calculation of confidence intervals. It introduces the concept of calculating an interval around a point estimate, aiming to ensure a 95% confidence level that the true population parameter is within this interval. Emphasis is placed on using the point estimator and its variance or standard deviation to compute the confidence interval. Examples are provided to illustrate 95% confidence intervals.
• 40:00 - 50:00: Assumptions and Bootstrapping This chapter discusses the concept of confidence intervals and their importance in statistical analysis. The focus is on understanding how the confidence interval provides a range within which the true population parameter likely lies. A key point is that a narrower confidence interval suggests greater confidence in the estimate. When the boundaries of the interval are tight around the point estimate, it indicates higher precision and reliability of the estimated parameter compared to a wider interval. The chapter highlights the impact of varying confidence levels, such as 95%, on the bandwidth of the interval, illustrating how they affect the certainty of the estimates.
• 50:00 - 60:00: Effect Sizes and Confidence Intervals This chapter discusses effect sizes and confidence intervals, focusing on the importance of confidence intervals in statistical analysis. Confidence intervals provide a range of values where the true population mean is likely to lie, offering a probabilistic framework for interpreting results. The accuracy of these intervals depends on the sampling plan, theory, and design of the research. Therefore, the more accurate these elements are in an experiment, the more likely the confidence interval will contain the true value.

Estimation4 Transcription

• 00:00 - 00:30 so we've talked about the problem with Point estimates being that they are single samples That Vary depending on who we've included in the sample and there's a solution to this problem which we call interval estimates and we'll be talking about those next so interval estimates are just exactly what you think they are they are intervals within which a true population parameter is likely to fall so given that point estimators are likely to vary a bit if we consider a
• 00:30 - 01:00 range of values that are likely to contain an estimator for example mu we can be much more confident in our estimation process and I will note that the estimation process begins before we collect the data we Define our level of certainty our level of confidence the interval before we collect the data now because we use an interval within which the population parameter is likely to occur we can be more certain of capturing it it's a bit like fishing
• 01:00 - 01:30 with a Net versus fishing with a pole in line the size of the interval depends on how confident we want to be and we Define that level of confidence in advance intervals that have a narrower range for the same level of confidence suggest greater Precision than those with a wider range so let's look at what that looks like so interval estimation in which we find the upper and lower boundaries of an interval that might contain the population parameter um gives us a metric for describing our
• 01:30 - 02:00 confidence in the estimation process so the thing I want to point out here is that we're not doing away with the point estimate of the parameter we're still calculating that we're still calculating for example the mean of our sample and the standard deviation of our sample and we're using those to make estimates of the population parameter but what we're doing instead of saying we think the mean of the sample is I don't know 26.78 [Music] um instead of saying that what we are doing
• 02:00 - 02:30 is we are going to calculate an interval around which that point estimate Falls um so we'll calculate an interval around the point estimate that gives us a 95 chance of being certain that the true population parameter lies within that interval so we're going to use that point estimator and it's for example it's variance or standard deviation to calculate a confidence interval and what you see here are two examples of that so if both of these are 95
• 02:30 - 03:00 confidence intervals here's our population parameter our Point estimate and this confidence interval has a has a it has a has a narrower range it's lower and upper boundaries are more tightly clustered around that point estimator in that case we are more confident in our estimate than we are in an estimate that has a 95 confidence interval with a bandwidth here the range of the
• 03:00 - 03:30 interval is greater so confidence intervals give us a range of values within which we have some confidence that the true population mean lies so it's a probabilistic statement about the interval within which mu happens to be the more accurate your sampling plan your theory your research design the more accurate those elements of your of your experiment are the more likely it is that your confidence interval will include the true
• 03:30 - 04:00 population mean so I've put this word sum in bold here so we have some confidence that mu lies within this interval so what does sum mean it's an estimate of how certain we are that this specified range of values contains the mean so I said this was a probabilistic statement so it's probably going to include a probability and that depends on the percentage or probability of confidence that we're looking for so we often calculate what we call a 95 confidence interval where we're looking
• 04:00 - 04:30 for the 95 percent boundary within which the population interval is likely to lie often this is reported in psychology paper papers as 95 Ci or the the abbreviation CI with a subscript of 95 and what that means is we are 95 percent confident that mu is within the calculated interval with a five percent margin of error
• 04:30 - 05:00 so let's consider that had the mean of the estimator plus or minus the variation in the estimate that so we're not as I said we're not doing away with these point estimators at all we're using them to calculate the interval estimate um so if we decide that we want a confidence level of some amount that's what we're going to calculate so often the confidence interval we use and this is sort of a convention and psychology is 95 percent this is based on the idea that we want a
• 05:00 - 05:30 five percent margin of error in our decision making we don't want more than a five percent margin of error so we typically calculate a 95 confidence interval because we are interested in a 0.05 percent margin of error so we often describe our confidence interval as one minus Alpha um so if our margin of error is Alpha then our confidence at level would be 0.95 or 95 percent but it doesn't have to be 0.95 it could be another number as well we could
• 05:30 - 06:00 choose a 50 confidence interval we could choose a 99 confidence interval it just depends on how certain we want to be and the confidence interval itself relies upon the theoretical distribution of the parameter for example if we're interested in estimating mu the population mean then we need to consider its theoretical distribution in order to calculate that confidence interval although there's another way to do it which we'll talk about later so I want to call your attention to the top
• 06:00 - 06:30 picture that's over here what you're seeing is a series of 50 confidence intervals so these are not 95 they're 50 and I wanted to show you the ones that are 50 confidence intervals because they're more variable um rather than less variable so this is a population from which the each of these samples is being drawn these are each samples of 10 scores and as you can see each of these lines here includes a
• 06:30 - 07:00 different sample from one through twenty the mean of this population is marked by this dashed line here and what you can see is you can see the distribution of scores that were randomly sampled from this population this was done with Monte Carlo sampling and sometimes these are in red sometimes they're in blue but what you'll see in each line in each row in this um in this distribution here as you'll see a little diamond shaped
• 07:00 - 07:30 box which is the point estimator the mean surrounded by some bars which is our 50 confidence interval in this case and if the colored bar is red that means the true population mean is not included in the confidence interval if it's blue it's included so you can see that the bars in this blue in row four here the the error bars on the confidence
• 07:30 - 08:00 interval overlap the mean of the population um and so it's been color-coded blue and in a 95 conference info we don't get very many of these that are deviants this is why I'm showing you the 50 confidential confidence interval and you can see that about half of them are around 10 out of 20 are we are are not capturing the true population mean they're missing it whereas 50 of them are actually getting that true population mean so you can see that
• 08:00 - 08:30 this is a confidence interval of about 50 percent now the confidence interval itself relies as I said on this theoretical distribution of the parameter which we have to calculate when we when we make our own confidence intervals so we have to figure out what that theoretical distribution looks like and then we have to find its lower and upper boundaries where 95 of the time that population mean will fall within
• 08:30 - 09:00 those boundaries we need to figure out what those boundaries are such that 90 95 of the time are estimate of the parameter mu here will be inside of that of that region so only 2.5 percent of the time will the mean be above the high the upper bound and only 2.5 percent of the time will the mean be below the lower bound
• 09:00 - 09:30 so the width of a confidence interval is important now to be more certain that we capture the population parameter we can increase the width of the interval say from 95 to 99 percent and the problem with doing that is if we increase that interval too much it becomes so wide as to be uninformative so that's the downside of just increasing the confidence interval we say well why don't we ever have 100 confidence interval and the problem is is that that interval becomes so wide
• 09:30 - 10:00 and it therefore becomes uninformative as this cartoon suggests this is Garfield here he's taking a look at at the weather forecast on television good old-fashioned television but by the looks of this thing over here taking a look at tomorrow's weather the high temperature will be between 40 below zero and 200 above and Garfield's happy because this guy's never wrong that does not tell you a lot about the what the temperature tomorrow is going to be that's you know sort of the average forecast for most of what we get in Southern Ontario
• 10:00 - 10:30 it could they could have gone a bit lower here on this one I think there were a few weeks in the winter where maybe we would be beyond that those boundaries so if we want to make inferences about the population we usually do that with a single experiment so we've collected just one sample and we want to use that one sample to make an inference about the population so what we need to do is we need to we obtain our estimate of the terror Point
• 10:30 - 11:00 estimate of our population parameter from a single sample and we can then use that population parameter and its measure of dispersion that corresponds to it so its standard deviation or variance to calculate or make an inference about how wide where our upper and lower boundaries of our confidence interval should fall there were two methods we can use to do that we can use a calculated formula and we can also do bootstrap resamplings we talked a little bit about bootstrapping
• 11:00 - 11:30 in last lecture and now here we're going to put that into practice today and in this week's lab so let's start with the theoretical version of this the confidence interval calculated by a formula which really does rely on a couple of on a theoretical distribution this normal distribution so I'm going to walk you through the formula we're going to talk about the elements so we have the confidence interval
• 11:30 - 12:00 which is the thing that we're trying to calculate we have our population parameter let's say the population mean here mu plus or minus this quantity over here which is z star the critical value of Z and I'll tell you what that is in a minute multiplied by the standard deviation of the population Sigma divided by the square root of n which is the population size now of course we're estimating remember n from the sample and we're estimating Sigma based on the
• 12:00 - 12:30 samples standard deviation now Z Star this one we have to unpack so you learned about Ze scores or Z scores I think is what they're called in Canada you learned about said scores last year in data science and a z-score is a score that has a mean of zero and its value is associated with a number of standard deviations above and below the mean that that score happens to fall so
• 12:30 - 13:00 if I have a z score of 1.1 it's going to be a score that's right here at about the one point the the one standard deviation plus about a tenth of a standard deviation so it's going to be plus 1.1 I can have a standard I can have a z score of minus two or minus 2.5 and that'll fall way down here on this distribution so what we're looking for with a critical value means is this critical value is the point in the distribution where 95 of the values fall within a
• 13:00 - 13:30 specified range so if we want to calculate a 95 confidence interval where we are 95 certain that the true population parameter lies within that con within that confidence interval what we're looking for is a zed score that gives us these exact population parameters so plus 1.96 and minus 1.96 because that's the point in this distribution if we if you think about
• 13:30 - 14:00 drawing a little line down from where this Arrow hits the edge right down and down here that's minus 1.96 and up here that's plus 1.96 and so our confidence interval is going to be within that range so that's what we mean here so we're calculating the critical value which from 95 confidence interval in a z distribution is 1.96 standard deviations above or below the mean
• 14:00 - 14:30 um multiplied by our standard deviation divided by the square root of n if we wanted to calculate a 99 confidence interval our critical value of Z would be 2.58 plus or minus so that's what we're looking for when we calculate the confidence interval from a formula we're looking for the critical value of Z the place at which only 2.5 percent of the scores if we're looking for 95 confidence interval only 2.5 percent of the scores are greater than that
• 14:30 - 15:00 um than that value and only 2.5 percent of the scores are more extreme in the lower lower end of that distribution and so that's where we're going to draw our 95 confidence boundary when we're thinking about a 95 Ci or a 95 confidence interval in a standard normal distribution but of course this requires us to have a standard normal distribution so there are some conditions that must be true in order to use the account or
• 15:00 - 15:30 to calculate the confidence interval using this formula and these are called assumptions and we're going to be talking more and more about statistical assumptions moving forward so in the sample we drew in order for that confidence interval to be meaningful the sample needs to be truly a random sample from the population so remember in Psychology we tend to use convenience samples rather than random samples and even if we take random samples they're often not random in one way or another right so I sometimes
• 15:30 - 16:00 conduct research online in which case I have access to a much wider range of of individual participants but still it's not a truly random sample they have to have internet access they have to have an account with a place that allows me to recruit them and so on and so forth so it's not really a truly random sample from the population even though it might be better than the number of people who are living like directly in London Ontario and willing to come into my lab
• 16:00 - 16:30 that sample must be normally distributed because its formula relies on these this theoretical normal distribution in which 1.5 or in which 90 1.96 of in which the 95 confidence interval ranges from minus 1.96 to plus 1.96 um and often in in real life our samples are not perfectly normally distributed like that and finally the scores need to
• 16:30 - 17:00 be independent so we can't have a relationship between the scores and these are the conditions that we rely on when we are considering calculating a 95 confidence interval from a formula so by and large these are theoretical um and they're things that we don't really we very rarely meet these conditions fully and properly and when we don't meet these conditions actually we can conduct a bootstrap analysis and the idea here is that if we have
• 17:00 - 17:30 data from the full population we could just calculate the mean right we wouldn't need to if we if we had a sample with every single person's score in the population we wouldn't need to estimate we could just calculate that would give us a perfectly accurate estimate of the true population mean unfortunately we don't right we don't collect those every every individual within a population it costs too much it's too time consuming to do that and instead we can build a model of the
• 17:30 - 18:00 population and sample from that so remember we talked about using the sample as a model of the population so under specific conditions the distribution of the bootstrapped statistic approximates the distribution of the sample statistic so remember when we take a we have a population that we've defined and we take a sample from that population and we calculate a statistic from our sample that we're going to use to estimate the true population parameter
• 18:00 - 18:30 now we know that the population has a very large n our our sample let's pretend it has an N of 100 right this number is going to vary from sample to sample but let's say it's got an N of 100 now the important thing here is that we've sampled from our population without replacement so we've sampled a set of scores from this population of individuals without replacing them now our model of the population is going
• 18:30 - 19:00 to be based on this sample here so we have an N of 100 in our model of the population and what we're going to do is we're going to repeatedly sample with replacement from that population model which is actually our sample and we're going to take samples of 100 with replacement so that because we're using our sample which is only 100 people to approximate a huge population of people we have to
• 19:00 - 19:30 sample with replacement which allows us to make those draws independent of one another and by calculating the distribution of the statistic from our bootstrap sample in fact we're going to take many bootstrap samples which means we take our sample we calculate for example it's mean or its standard deviation and then we but each time we take a score we note it down and throw it back so that each single sample from this model of the population each single score that we draw is independent
• 19:30 - 20:00 we can create a distribution of our test statistic that's interesting to US based on many bootstrap samples so if we're thinking about a bootstrap distribution what we're doing is we're using the bootstrap method to resample our data and build a distribution of sample means so last week we used Monte Carlo methods to build a distribution of sample means this week we'll use bootstrapping so we take the sample we have and over off many many iterations we are going to resample the sample by selecting samples of size n with
• 20:00 - 20:30 replacement so this is a little bootstrap distribution that was created in this way where there was a sample of um actually people who have run the Boston Marathon um and their ages and the sample mean is uh actually the population mean of these folks which we never really know in real life the population mean of these folks is um somewhere between 35 and 36 30 let's say
• 20:30 - 21:00 35 and a half is the average age of the Boston Marathon of a Boston marathon runner I didn't say winner I said Runner and there's a range and then what we can do is we can take the we can take samples let's pretend we have samples of size 50 from that population we can sample a size of 50. we can and we can do we can resample from our popular from our from our original sample many times with replacement and record our sample statistic into an array this gives us a
• 21:00 - 21:30 distribution of sample means built by bootstrapping now and not by Monte Carlo analysis so what we've done is we've simply resampled our same scores that we had and we've produced an array that can then be plotted as a histogram that contains sample means so it's the average age for each one of the samples that was taken from our original sample and that allows us to determine the confidence interval based on the distribution of sample
• 21:30 - 22:00 means that we created and allows us to compare the calculated confidence interval and we can compare that to our estimator of the true population mean here we actually know the true population mean because the folks who run the Boston Marathon have been keeping these records for ages and ages and ages so we actually know what the true population mean is there and what its range is we know what the youngest person and the oldest person who have ever run the Boston Marathon so but what we can do is we can sample from for example this year's sample of Runners so
• 22:00 - 22:30 the people who ran the or this past year the people who ran the Boston Marathon this past year and we could use their ages as an estimate and so that sample mean might be slightly older so here in this in this version in this sample um the average age of the sample was 37 approximately give or take and so that's slightly overestimating the true population mean because maybe this
• 22:30 - 23:00 year sample is a little bit older than usual but in the grand scheme of things if we calculate this confidence interval which is a 95 confidence interval around the mean of our sample using this bootstrap distribution here what we can come up with is we can come up with this interval here that ranges from this lower bound of a little bit older than 33 to this upper bound of you know slightly older than 40 but definitely not quite 41.
• 23:00 - 23:30 um and we can be 95 certain that the true population mean which we never know it's unknown in the real world um we can be 95 certain that that true population mean Falls within that um within that parameter and that allows us to make an estimate so we never actually get to compare our estimator to the true population mean but it allows us to make an act because we don't know the true population mean but it does allow us to make an estimate
• 23:30 - 24:00 of what that mean is likely to be so in Python terms here's what a bootstrap function might look like it's got our definition statement we have a function name my bootstrap it has some data a sample size and a number of iterations and then we have our distribution of sample means that we're going to be returning we Define it by using the NP array with a length of however many samples we're going to take and these number of iterations these are
• 24:00 - 24:30 pretty large quantities they might be a thousand they might be five thousand they might be ten thousand it depends on your level of precision that you're interested in um but it's going to be a big number and again your computer can do this much faster than you can so this is why we can do bootstrapping um just like the for Loops that we've defined in the past for I and range number of iterations we will take a sample from our
• 24:30 - 25:00 from our original sample and we will take a sample with replacement this keyword here replace equals True by default this data.sample function replace equals false but we have we'll set it to true so that they so that it's resampling meaning that every time it takes a number it looks at what it is and then it throws the number back in the pot so this is sampling with replacement just like you did in the m m lab
• 25:00 - 25:30 through your m m back instead of eating it and then what we're doing is we're taking the mean of that sample and we're placing it in our dosm at the if position and that's what we're returning here and we'll also call your attention a couple people have problems with this in the lab where they didn't indent properly so when you are writing a function or anything else so this for statement right here notice
• 25:30 - 26:00 that there's a four spaces in between the beginning here and where this lives that tells python that the that these two elements in fact are part of this Loop and that this one right here is not if for example you indented the return statement it would iterate this one time so without being mistakes you to pay attention to it weak this in Python convention this stuff is called the white space so the space on the page and
• 26:00 - 26:30 your statements need to be indented properly in Jupiter these statements will turn red if they're not indented properly so it's a good idea to make sure that you have them intended but it doesn't always know because this is one of these coding things that the user has to know rather than the program itself um and then what we do is we have this function the function is totally flexible it can take any data we give it it can take any sample size we give it
• 26:30 - 27:00 and it could take the number of iterations and then we make a call to that function and we're going to allow this return variable to be set into a new variable mydosm mostly because we're going to make a histogram of it later so we want to make sure that we save it and then we have the call to my bootstrap so we have our call to my bootstrap we're going to pass in some data we are going to pass in a sample size which is going to be the same length as the vector of data that we're
• 27:00 - 27:30 passing in and then we have a number of iterations here I've put in a thousand but it could be more than that as well so that's remember that when we're making functions these things need to be flexible enough to deal with all different kinds of data types and everything else so here we're trying to make our our sample as flexible as we can so the magic here is that if we meet all the conditions that we've selected and we have a representative sample we
• 27:30 - 28:00 have independent sampling this process allows us to approximate the population value by determining the interval within which that value is likely to fall based on our specific sample so this isn't a theoretical distribution anymore this is an empirically derived distribution um and so that's a key word that you should make sure you understand this idea that this distribution is derived based on empirical methods based on
• 28:00 - 28:30 resampling the data and not a theoretical perfectly normally distributed population which is a theoretical distribution uh this is a universal technique so it's really nice it can be applied to any statistic we're interested in so not just the mean we can calculate the bootstrap sample of a test statistic called T we can calculate a bootstrap distribution of a test statistic called chi-square you can bootstrap any test statistic we want we can calculate the standard deviation
• 28:30 - 29:00 or variance that way as well so any test statistic we want we can calculate using bootstrap using bootstrapping so it's really a universal thing it does not require assumptions about the underlying distributions we don't have to have this normally distributed distribution as long as our sample is representative and independent we're all good disadvantages it requires programming skill but you now have the skill to do this it requires Computing time but the computer you have that you're watching
• 29:00 - 29:30 this lecture on is perfectly sufficient in terms of computing power to calculate this kind of variable now the tricky bit is if the sample is too small these can be noisy to the extent that they they won't give you a good estimator of the population but once your sample size gets above somewhere in the neighborhood of 30 scores these bootstrap distributions actually look pretty good and pretty similar to more um it's pretty similar to population
• 29:30 - 30:00 distributions so that's the magic of bootstrapping it can help us figure out where our confidence intervals lie based on an empirical rather than a theoretical distribution and it's especially important if the data are not perfectly normally distributed I will also note in terms of confidence intervals that if we calculate the 95 percent confidence interval for Amine we can be 95 confident that the interval
• 30:00 - 30:30 contains that population perimeter so the confidence interval tells us what range of values we can expect to find if we redo the experiment in exactly the same way what we cannot show we cannot be 95 certain that the parameter is within that is inside that interval so it's the boundaries of the interval that variate not the parameter itself the population parameter is what it is and it's something that we never know about unless we're doing you know these sort
• 30:30 - 31:00 of silly things that we do in class but in the real world we never know what that perimeter is and so what we're doing is we're calculating the intervals within which that parameter estimate that parameter might lie and not we're not 95 certain that the parameter is in the interval we are we are calculating the boundaries that's what we're working on and we can be approximately certain that the parameter
• 31:00 - 31:30 lies within the boundaries but we can't be 95 certain that the perimeter is actually in those boundaries so the confidence interval cannot tell us how likely it is that we found the true value of the population parameter because it's based on sampling and not 100 of the population so you need to be really careful when you're interpreting confidence intervals for that reason confidence intervals allow us to estimate the interval within which our population parameter of Interest lies
• 31:30 - 32:00 they depend just like any experiment on having a solid methodology with good sampling techniques appropriately calibrated measurement and so forth but if we have those things we can actually be more confident or more certain that we have captured some element and can make inferences about how that pop what that population parameter is how our Theory Works within that population but of course all of these things depend on the methodology
• 32:00 - 32:30 which we select before we start the experiment now the other thing we need to consider is an idea of effect size or magnitude now often we want to know whether one group differs from another whether our treatment was effective whether there's a relationship between the two variables so we talked about last lecture
• 32:30 - 33:00 the fact size being a quantitative measurement of the strength or an associate of an association or an experimental effect and confidence intervals can help us to answer these questions does the confidence interval around the mean of Group 1 include the mean of group two so now you can start to see that these confidence intervals might be useful in the context of decision making about our estimators if the mean of group one if the 95
• 33:00 - 33:30 confidence interval around the mean of Group 1 does not include the mean of group two it's highly likely that group 1 differs from group two so we can use these confidence intervals to tell us something about the effect size or magnitude of the effects that we're studying so how strong is the relationship or association between variable X and variable Y and the effect size will tell us about that so the effect size is really about the true difference between two groups if we're
• 33:30 - 34:00 interested in the difference between two groups or the true size of of a parameter of Interest out in the real world and we'll leave it there for today