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Analytical Chemistry
Learn the toughest concepts covered in your Analytical Chemistry class with step-by-step video tutorials and practice problems.
Statistics, Quality Assurance and Calibration Methods
Confidence Intervals
A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample.
Confidence Intervals
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concept
Confidence Interval
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So here it states that a confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample. So what does that really say? Well, a confidence interval is a way of knowing for certain based on a percentage that a certain value will lie within a certain range. So when we're talking about confidence intervals, um we tend to look at percentages from 50% all the way to 99.9%. So by using our confidence interval, let's say 50%, I can say that I am 50% confident that my value lies within this given range. So that's the whole premise behind a confidence interval, it gives us a level of certainty that will find a particular value that we're looking for within a given per parameter or range of values. Now we're gonna say here for example, 95% confidence interval means we are 95% confidence that the mean lies within a given interval. So basically what we've been saying, so our value or average value will fall within a given range based on using our confidence interval formula. Now the confidence interval formula is we're going to say equals X plus t. S over square root of N. Here, we're going to say that the students T is obviously the t value, our standard deviation is S. R number of measurements that we're looking at is N. And then our average or our mean is just X here, with this line above. Now we're gonna say related to the confidence interval is a student's t statistical table. Now here we take a look at it, realise here that we have our degrees of freedom, that's just number of measurements -1. And we can see that our degrees of freedom go from one all the way to infinity infinity just means that it's a number beyond the 120 individual measurements that were customers seeing for large sampling. And we're gonna say here we have confidence interval percentages. So we go from 50% all the way to 99.9% confidence intervals. So we use this table once we determine our degrees of freedom to determine what our T value is. So for example, if we had 10 measurements, we'd say that is 10 minus one. So that's a degree of freedom of nine. And let's say we're looking for 95% confidence interval. So we just move all the way to 95% which would get us here. This tells us that the T value is 2.262. We take that T value, we plug it in here. Once we've determined our average or mean from an earlier calculation, we know our standard deviation. We know that the number of measurements we have originally is 10 From this, we'd be able to figure out what our confidence interval is, we'd figure out that our value is some mean value. Let's say 50.3 plus or minus some value. So it'd be 95% confident that our mean lies within that given range. Now we've seen the fundamentals in terms of confidence interval, we'll take a look at the example left on the bottom of the page, so make sure you click on the next video and see how I approach the same exact example in determining the 95% confidence interval based on the information provided.
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example
Confidence Interval
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So here it says constructed 95% confidence interval for an experiment that found the main temperature for a given city in july as one oh 3.5 degrees Celsius with a standard deviation of 1.8 from 10 measurements. Now obviously this temperature is much too high for a normal city. So this is a fictitious city. Um with the temperature this high. Now they're asking for the 95% confidence interval. What we have to do here is we have to determine what our T value is, which we kind of already did When talking about our tea table here we have 10 measurements. And remember your degrees of freedom is N -1. We have 10 measurements -1 gives me a degree of freedom of nine. So here's our degree of freedom of nine. They're telling us that we're dealing with a confidence interval of 95%. And then we just traced down and we saw that this was R. T. Value 2.262. So that represents our T. Value here. So now we're gonna say that our confidence interval which I'll abbreviate is ci equals your mean or average plus or minus t times your standard deviation divided by the number of measurements. All right, so we have one oh 3.5 degrees Celsius plus or minus R. T value is 2.26 to our standard deviation is 1.8, Divide by number of measurements which is 10. So then we're gonna get here 103.5°C plus or -1.28°C. So that means that our range here Would be 103.5°C -1.28°C two one oh 3.5 plus one point To 8°C. So that means here that on this day in July the temperature would range between 102.2 2°C to 104.78°C. So we are 95% confident that our range on this particular day would between would be between 102.22 degrees Celsius to one oh 4.78 degrees Celsius. Now that's how we approach questions like this. When we have to deal with confidence intervals, remember we're never going to be 100% confident because the highest we can be is 90 9.9% confidence with any um range when given a mean standard deviation and number of measurements. So just remember when it comes to using the tea table, we have to first determine our number of measurements and from there are degrees of freedom. Then match up the T value with the percentage that's given to you within the question in this one, it was 95
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example
Confidence Intervals Calculations 1
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continuing our discussion on confidence intervals. Let's take a look at the following example. Here it says the barium content of a metal ore was analyzed several times by a percent composition process here are given four measurements of 40.10 point 11.4 and 0.11. Here we need to calculate the mean median and mode now realize here that we have two types of means that we've dealt with up to this point. We have our traditional mean which is just X. With a lineup of it. And then we have mu which is termed our population mean remember to get the population mean? We would need to do several measurements. So many in fact that are typical average mean with transition into the population mean since here we only have four measurements. We know we're dealing with our standard mean. So we're gonna say here are mean equals all the measurements added up. So we're gonna take all of these measurements, add them together And then divided by the total number of measurements. So the total number of measurements is four Doing that. We're gonna have .036 as the total for the top portion divided by four. So that gives me .009 As my standard. Mean next we need to figure out our median for median. We're gonna list these measurements from smallest to largest. So we have .004.010.011 and .011 for the median. We look for the middle of all these measurements. If it's just one number, if it were five digits there would be one number in the middle and we would choose that number as our median. In this example though, two numbers are technically in the middle. So what we do here is we take those two numbers and we add them together And we divide by two. Doing that would give me .0-1, divided by two, which is .0105 as my median. Finally, we need to determine our mode. Our mode is just the measurement that appears the most. The measurement that appears the most is .011, so that would represent our mode. So these are just some basic statistical definitions and calculations that you should be familiar with. Now that we've done this example, continue on to the next one where you're asked to calculate the standard deviation attempted on your own but if you get stuck, just come back and see how I approach that same question
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example
Confidence Intervals Calculations 1
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So from example one they asked us to calculate the standard deviation. So remember your standard deviation which is s equals square root, you have the summation of each measurement minus the average or mean? And that will be squared divided by n. Which is the number of measurements minus one. Now remember we're dealing with only four measurements here, so we're just dealing with our standard deviation. If we're dealing with numerous measurements then it would change into the population standard deviation and in that case would be called sigma. But here we're just dealing with standard again because there's only four measurements, so this is going to equal each measurement. So remember X I stands for each measurement, so each one of these minus your mean or average which was this and that will be squared. Alright, so plugging this in we're gonna say we have 0.104. 1st measurement minus the average and that'll be squared plus the next measurement which was 0.11 minus the average. And that's squared plus 0.4 minus the average. That squared plus 0.11 minus the average. And that's squared Divided by the number of measurements which is 4 -1. Remember N -1 is also coined the degrees of freedom. So when we subtract and then square all of those and add them together, that's gonna give me square root of 0.34 divided by three Which is square root of .000011 Which gives me .003367. At the end. For my standard deviation. Remember the smaller standard deviation? The more precise your numbers are, the more closely there they are to one another. But a small standard deviation does not mean that your measurements are accurate. We can only tell if they're accurate if we compare them to the true value of the barium content within the metal or. Alright, so again, all we know for sure is that this standard deviation is pretty small. Um So it's gonna be pretty precise in terms of how close the measurements are to one another. Now that we've seen this, attempt to figure out the 95-90% confidence interval that's left as the practice question. Once you're done with that, come back and take a look and see how I approach that same question.
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Problem
From the examples given above, find the 90% confidence interval.
A
0.009 ± 0.002758
B
0.009 ± 0.003589
C
0.009 ± 0.003961
D
0.009 ± 0.002581
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