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# Example constructing a t interval for a mean

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.Q (LO)
,
UNC‑4.Q.1 (EK)
,
UNC‑4.Q.2 (EK)
,
UNC‑4.Q.3 (EK)
,
UNC‑4.R (LO)
,
UNC‑4.R.1 (EK)
,
UNC‑4.R.2 (EK)

## Video transcript

a nutritionist wants to estimate the average caloric content of the burritos at a popular restaurant they obtain a random sample of 14 burritos and measure their caloric content their sample data are roughly symmetric with a mean of seven art calories and a standard deviation of 50 calories based on this sample which of the following is a 95% confidence interval for the mean caloric content of these burritos so pause this video and see if you can figure it out alright what's going on here so there's a population of burritos here there is a mean caloric content that the nutritionist wants to figure out but Kent doesn't know the true population parameter here the population mean and so they take a sample of 14 burritos and they calculate some sample statistics they calculate the sample mean which is 700 they also calculate the sample standard deviation which is equal to 50 and they want to use this data to construct a 95% confidence interval and so our confidence interval is going to take the form and we've seen this before our sample mean plus or minus our critical value times the sample standard deviation divided by the square root of n the reason why we're using a t statistic is because we don't know the actual standard deviation for the population if we knew the standard deviation for the population we would use that instead of our sample standard deviation and if we use that if we used Sigma which is a population parameter then we could use a Z statistic right over here we would use a Z distribution but since we're using this sample standard deviation that's why we're using a t statistic but now let's do that so what is this going to be so our sample mean is 700 they tell us that so it's going to be 700 plus or minus plus or minus so what would be our critical value for a 95% confidence interval well we will just get out our T table and with a T table remember you have to care about degrees of freedom and if our sample size is 14 then that means you take 14 minus 1 so degrees of freedom is - one so that's going to be 14 minus one is equal to 13 so we have 13 degrees of freedom that we have to keep in mind when we look at our tea table so let's look at our tea table so 95% confidence interval and 13 degrees of freedom so degrees of freedom right over here so we have 13 degrees of freedom so that is this row right over here and if we want a 95% confidence level then that means our tail probability remember if our distribution let me see if I'll draw it really small a little small distribution right over here so if you want 95% of the area in the middle that means you have 5% not shaded in and that's evenly divided on each side so that means you have two and a half percent at the tails two and a half percent so what you want to look for is a tail probability of two and a half percent so that is this right over here point zero to five that's two and a half percent and so there you go that is our critical value two point one six zero so this so this part right over here so this is going to be - let me do that in a darker color this is going to be two point one six zero times what's our sample standard deviation it's 50 over the square root of n square root of 14 so all of our choices have the 700 there so we just need to figure out what our margin of error this part of it and we could use a calculator for that okay two point one six you write a zero there doesn't really matter times 50 divided by the square root of 14 square root of 14 we get a little bit of a drumroll here I think twenty eight point eight six so this part right over here is approximately twenty-eight point eight six that's our margin of error and we see out of all of these choices here if we were round to the nearest tenth that would be twenty eight point nine so this is approximately twenty-eight point nine which is this choice right over here this was an awfully close one I guess they're trying to make sure that we're looking at enough digits so there we have it we have established our 95% confidence interval now one thing that we should keep in mind is is this a valid confidence interval did we meet our conditions for a valid confidence interval and here we have to think well did we take a random sample and they tell us that they obtained a random sample of 14 burritos so we check that one is the sampling distribution roughly normal well either you take if you take over 30 samples than it would be but here we only took 14 but they do tell us that the sample data is roughly symmetric and so if it's roughly symmetric and it has no significant outliers then this is reasonable that you can assume that it is roughly normal and then the last condition is the independence condition and here if we aren't sampling with replacement and it doesn't look like we are if we're not sampling with replacement this has to be less than 10% of this has to be less than 10% of the population of burritos and we're assuming that there's going to be more than 140 burritos that the universe that the population that this popular restaurant makes so I think we can meet the independence condition as well so suing that you feel good about constructing a confidence interval this is the one that you would actually construct
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