now we're going to look at a type of confidence interval problem where Sigma is unknown uh which we've seen that kind of problem before but what you're given is raw data let's take a look at this question so the grade point averages GPA for 12 randomly selected college students are shown to the right here so complete these questions so uh first we're going to find the sample mean notice that in previous problems the sample mean and then also Part B find the sample standard deviation in previous problems those were given what we have right now is raw data we actually have to figure out what that is so in the in the software if you can it says little rectangle click on that and if you click on that the data will pop up in your software um you might be using Excel I'm using LibreOffice they look about the same so from this we can calculate the sample mean just type equals average open parenthesis and then highlight your data to get the cell names right close parenthesis enter and your mean has been calculated for you next equals uh and that's what I should say we're going to calculate the standard the sample standard deviation so equals stdev that's the command for that open parenthesis highlight your data close parenthesis hit enter and there is your sample standard deviation so we're going to take that information and we're just going to jot it down so sample mean oops okay sample mean X bar was 2.3 and the sample standard edition s was let me check that again 1.1901 [Music] four decimal places should be good enough there okay and now we're ready to do the problem the way that we normally do um we're ready to Constructor confidence interval uh by our usual method so we know that a confidence interval I should say comments interval for Mu um and again we know that because we don't have Sigma we'll be using S which also means we're using the T distribution so the formula for the current interval is going to be X bar plus or minus t times s for X bar so we already know X bar we just calculated it that's 2.3 and s x bar we don't know but we can figure it out pretty quick that is 1.1901 or let me write down the whole formula s over rad n which is 1.1901 divided by the square root of quickly count up I see that it's 12. so let me actually just jot that so I remember that n equals 12. and now I'm just going to punch that into my calculator so I'm going to use this for my calculator 1901 divided by the square root of 12. okay so 0.3436 if I'm rounding so there's my S for X bar and that's another thing that I needed for the formula and then finally we have to find T so to find T uh we go into status there's other technology that you can use um my favorite is statis so click analysis distribution student to distribution and now we need to fill this in so since we know that uh n is 12 degrees of freedom let me actually just jot this down over here degrees of freedom is n minus one so that's going to be 11. so degrees of freedom is 11. and we're trying to find the T value so that means we have to know what is the area to the right of the t-score so it is a confidence interval which means our picture has two tails and it is a 95 percent confidence interval so 0.95 is in the middle and that means 0.025 is in each tail so the area to the right of my t-score is 0.025 click evaluate and there's a t-score it's 2.2 I guess if I go to three decimal places it's 2.201 if I round it or using status there's the t-score and then the rest is just a quick arithmetic calculator so let's figure this out so 2.201 times 0.3436 so that gives me 0.7563 if I'm rounding um and actually let's see I think typically it says round to three decimal places so we're going to round the answer to three decimal places there you go uh and now we just do the plus minus so 2.3 minus 0.756 and then 2.3 Plus 0.756 and that gives you your confidence interval 1.544 and 3.056 so here I rounded to three decimal places I think that's typical you of course you can round to however many places you want or however many uh is requested for the particular problem