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# Conditions for valid t intervals

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.P (LO)
,
UNC‑4.P.1 (EK)

## Video transcript

flavio wanted to estimate the mean age of the faculty members at her large university she took an SRS or simple random sample of 20 of the approximately 700 faculty members and each faculty member in the sample provided Flavio with their age the data were skewed to the right with a sample mean of thirty eight point seven five she's considering using her data to make a confidence interval to estimate the mean age of faculty members at her University which conditions for constructing a T interval have been met so pause this video and see if you can answer this on your own okay now let's try to answer this together so there's seven hundred faculty members over here she's trying to estimate the population mean the mean age she can't talk to all seven hundred so she takes a sample a simple random sample of 20 so the N is equal to 20 here from this 20 she calculates a sample mean of 38 point seven five now ideally she wants to construct a T interval a confidence interval using the T statistic and so that interval would look something like this it would be the sample mean plus or minus the critical value times the sample standard deviation divided by the square root of N and we use a T statistic like this and a T table and a t distribution when we are trying to create confidence intervals for means where we don't have access to the standard deviation of the sampling distribution but we can compute the sample standard deviation and now in order for this to hold true there's three conditions just like what we saw when we thought about Z intervals the first is is that our sample is random well they tell us that here that she took a simple random sample of 20 and so we know that we are meeting that constraint and that's actually choice a the data is a random sample from the population of interest so we can circle that in so the next condition is the normal condition now the normal condition when we're using when we're doing a T interval is a little bit more involved because we do need to assume that the sampling distribution of the sample means is roughly normal now there's a couple of ways that we can get there either our sample size is greater than or equal to 30 the central limit theorem tells us that then our sampling distribution regardless of what the distribution is in the population that the sampling distribution actually would then be approximately normal she didn't meet that constraint right over here here her sample size is only 20 so so far this isn't looking good now that's not the only way to meet the normal condition another way to meet the normal condition if we have a smaller sample size smaller than 30 is 1 if the original distribution of Ages is normal so original distribution normal or even if it's roughly symmetric around the mean so approximately symmetric but if you look at it this they tell us that it has a right skew they say the data were skewed to the right with the sample mean of 38.7 5 so that tells us that the data set that we're getting in our sample is not symmetric and the original distribution is unlikely to be normal think about it it's not going to be you're likely to have people who are you could have faculty members who are 30 years older than this 68 and 3/4 but you're very unlikely to have faculty members who are 30 years younger than this and that's actually what's causing that skew to the right so this one does not meet the normal condition we can't feel good that our sampling distribution of the sample means is going to be normal so I'm not going to fill that one in choice C individual observations can be considered independent so there's two ways to meet this constraint one is is if we sample with replacement every faculty member we look at after asking him their age we say hey go back into the pool and we might pick him again until we get our sample of 20 it does not look like she did that it doesn't look like she sampled with replacement and so even if you're sampling without replacement the 10% rule says that look as long as this is less than 10% or less than or equal to 10% of the population then we're good and the 10% of this population is 70 70 is 10% of 700 and so this is definitely less than or equal to 10% and so it can be considered independent and so we can actually meet that constraint as well so the main issue where our T interval might not be so good is that our sampling distribution we can't feel so confident that that is going to be normal
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