In this video, we are going to look at how to find confidence intervals using sample statistics to estimate the population mean new. So here, it says from a population that is normally distributed, a random sample of size and was drawn and it was found that in that sample, the sample mean is 50 and the sample standard deviation is eight. So one of the things also from this type of question is it could have said in words the sample mean is 50 and you would have had to recall that the notation for the sample mean is X with a bar over it. So that you had that expire is equal to 50. And then also recalled that the sample standard deviation is symbolized with the S So that you would have said S is equal to eight when they would report in words that the sample standard deviation was eight. So be sure that you're getting a good good connection between your vocabulary words and your symbols. So you're able to utilize the formulas correctly. So, this first one says find in 98% confidence interval for μ the population mean if N is equal to 20. Well, the setup for finding confidence intervals for the population means using sample statistics. When we pull our sample from a population that is normally distributed when our sample size is small or if we have a large sample size is expe r minus tisa alpha over to times S over the square root of N. Yeah, and X. Bar plus t sub alpha over two times S over the square root of N. Mhm. Now, where that tisa Alpha over two is the critical value that you can get either from your table or from the student's t distribution table or from your calculator so far tisa alpha over to work. Remember alpha Is the amount of area in the tales. So if I have a 98% confidence interval, I take one And i subtract the 0.98 and that gives me an alpha value of 0.02. Mhm. Now alpha over two then is just divide that 0.2 by two and I get 0.1. And then with the student's t distribution, remember there's a degree of freedom and your degrees of freedom is n minus one Here. In part a my N is 20, so my degrees of freedom is going to be 19, so 20 subtract one is 19. So looking that up on the table, I'm going to go underneath the .01 column and across from the 19 row and we see that that value is 2.539. Yeah. And now I have everything from my previous information to set up my bounds on my confidence interval. So X bar is the sample mean that's 50, then minus my critical value of tisa, alpha over two is the 2.539. And that's time. My s is my sample standard deviation of eight And divide that by the square root of my sample size and my sample size is 20. And then when you do the upper bound of your confidence interval, you just going to the same thing just with a plus in between so X bar is 50 then plus the critical value of 2.539 and times my s which is a sample standard deviation of eight divided by the square root of 20. Um my 20 is my sample size. And when you go through in key that through your calculator you will have a 98% confidence interval from you is between 45.5 and 54.5. And to write it out in a sentence that that's required, you would say a 98 confidence interval, Mhm, Mhm. The other one for the population mean, Mhm mhm is between 45.5 and 54.5. Now, if you're allowed to use your graphing calculator, then what you want to do here is to push the stat button and the step button is right under the delete cursor, Right to tests, cursor down until you cross from t interval, we're doing intervals instead of tests and its teeth because we again have that critical value of t stable for over two. And so then when we look at pushing enter to make that choice data is if you had the list of numbers, the sample um gathered um stats as if it you've pre done the sample mean and the sample standard deviation and that's how they gave it to us. We're going to stay on stats. Our X bar is 50 Our s is eight and our end here is 20, Then go to our confidence level of .98 and calculate It'll take just a second, but then you'll see rounded off I have 45.5 is my lower value and that's what we got. And then comma so that's the going to the other end of your confidence interval, the 54.5. So that's how you could use your calculator. Now, you can also use a calculator to get the Teesta about over two critical value. If we go to second and the virus key for the distribution and we do in verse t. The format for the inverse T. Let's clear out the stuff that I had here before. The format for the university is area, that's to the left of your critical value. And if my point no one is the alpha over to which is to the right, then one minus that, one minus the point 01 is my .99. And that's the number you want to put in for your area. So we're gonna go .99, our degrees of freedom is 19. And then when we go through and paste that and enter, We see that 2.539 number that we got for a critical value. So you can find that either on the table or from your calculator. So let's go to the next one. Find a 98% confidence interval from you if N is 15. So again recall that are mean expert is 50 and our sample standard deviation is eight And here our end is 15. So when we go to calculate artists of alpha over to We still have that alpha is 0.02. We have alpha over two is 0.01. Our degrees of freedom is N -1 here. My Ennis 15. So my degrees of freedom Is going to be 14. Yeah. And from the table or from your calculator you'll get that your critical value. Tisa Balfe over two is 2.624. And I'll verify that over here. So again, second and far is key for distribution. Go to inverse T. We're gonna put in the area of the .99. And again that was 1- the alpha over to value. And that's how we got the .99 entry. But my degrees of freedom this time is 14. And when we push enter on paste and then enter to have it done We see again the 2.6- four. So finding my confidence interval exposure minus tisa alpha over two times as over the score to end for the low end and then X bar plus T symbol for over two times s over the square to end for the upper? So I have my 50 minus the two point 6-4 times my S. Which is the eight Divided by the square root of and again this time our end is 15. Mhm. And expires 50 Then plus the 2.6-4 times eight divided by the square root of 15. And when you run that through you'll get our 44.6 And 55.4. So what happened with this is it was a 90% confidence interval in both part a and part b. Part a. Had a sample size of 20, part B had a sample size of 15. And when we look at what happened with the margin of air between them, the confidence interval got wider. I started with a lower number and ended with a higher number in part B than I did in part A. So that means my margin affair is bigger. So decreasing the sample size actually increases the margin of air. Yeah. Yeah. Mhm. Yeah. Mhm. Yeah. Mhm. Yeah. Yeah. Okay. And then over on the calculator again, if we push stat, go to tests, go down to t interval because I'm doing a confidence interval from you which enter we have stats Experts still 50 SS eight and this time we change to 15, our confidence level is still .98. And when we calculate this And just compare we see it surrounded to the 44.6 for my low and the 55.4 for number one. Okay, how about finding a 95% confidence interval from you with an equal 20. Yeah. So again we have our expire minus Teesta alpha over two times as over the score to end and X bar plus Teesta alpha over two times as over the score to end. The reason we put the and in between is because when you go to the full sentence you say a 95% confidence interval for the population mean is between the low number and the high number. So that's why it's just an abbreviation with the word and in there now with my teeth of alpha over to in this case I have 95% confidence interval. So one minus the 10.95 is 0.5. That's your alpha Alpha over two then is divide that by two, My degrees of freedom Is N -1 and is 20. So my degrees of freedom is going to be 19. So when you look up or do your university on the calculator, your teeth of alpha over to for that case is 2.093. And so I have my 50. Yeah -2.093 times The s. S. eight Divided by the square root of N is 20. And for the Top number of our interval we have 50 Then plus 2.093 times eight Divided by the square of 20. And so calculating that through, we get 46.3 and 53.7. So again wording it out, a 95% confidence interval for the population mean is between 46.3 and 53.7. And here it's with my sample size of an April 20. Now, what if it's asks me, how does the margin of air change if the confidence level is decreased? Well, if the confidence level is decreased, here's an equal 20, I'm going to compare that with part A And part A. had a low end of 45.5 and a high end of 54.5, Part C has a low end of 46.3 and a high end of 53.7. So my lower number is bigger than the start into in part A. And my higher number is smaller, so it squeezed in a bit, so my margin of air decreased when my confidence level decreased. Okay. Mhm. And then lastly, could we have used this process to construct confidence intervals in part A through C. If the population wasn't known to be normal. Well, remember with the central limit theorem, if I have a population that's normally distributed, then the sampling distribution of the sample means will also be normal. Yeah, If I don't know if my population is normally distributed, but I have sample sizes that are greater than or equal to 30. Then the the central limit theorem says that the distribution of your sample means will be approximately normal as you have that larger sample size. Well, these sample sizes were 20 And 15 smaller than 30. So I had to have the population be normally distributed in order for me to do this process to find confidence intervals. So could we have that answer is no. Well, I hope you found this video to be beneficial and keep an eye out. We will be sharing more as time goes on.