So in this question, it says a sample of size n equals 100 produce the sample mean of 16, assuming the population, population deviation is three, compute a 95% confidence interval for the population mean. Alright, so we're talking about populations, so we're going well above our normal number of measurements within a given population. And remember we said that if we're going above 30 that usually means that we're not using the T test, but instead the Z test Now, because I don't give you a z value here, go to your students tea table, look at your student's T table and we're dealing with a 95% confidence interval. So look at the com column that's dealing with 95%. Since we're dealing with populations that are incredibly large, we're gonna look at the degrees of freedom as being equal to infinity. So, if you line up infinity with your 95% confidence interval, you'll see that your Z score then would be 1.960. Okay, so that's the logic we use when we're dealing with incredibly large populations like we are in this question. So here we're gonna say that it is our mean or average plus or minus R. Z score here, times now we're dealing with our population deviation. So that's our our population standard deviation. Remember your standard deviation is s and when we transition to a population standard deviation, it becomes sigma. So it'll be times sigma over the number of measurements. Look at this, look at the similarities that this has with a typical confidence interval. A typical confidence interval would just be the mean plus or minus your T score times your standard deviation divided by n the square root of n. Again, we've transitioned our standard deviation to the population standard deviation. And because we're dealing with so many numbers that are much greater than normal, because we're dealing with a population with a larger data set, T has transitioned into Z. All right. Other than that we plug in the values and we'll have our answer. So our mean here is 16 plus or minus 1.960 times your deviation, which is three divided by the square root of 100. So here, when we plug all this into our calculator gives us .588. So this is 16 plus or -188. So what does that mean? That means we have 16 minus 160.588 and then we have 16 plus 160.588. So that means that we're 95% confident That our value alive in between 15 412 - 16.588. So that would be our level of confidence within this particular question. So, remember when we're going beyond 30, we transition for more of AT score to a Z score here, we're dealing with infinity in terms of degrees of freedom and therefore because we're dealing with 95% confidence interval when we line it up on the T table, that gives us a score of 1.960. Now that you've seen this one, attempt to do the practice question left here on the bottom once you do come back and see how I approach that same exact practice question.
