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# Introduction to t statistics

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.O (LO)
,
UNC‑4.O.1 (EK)
,
VAR‑7 (EU)
,
VAR‑7.A (LO)
,
VAR‑7.A.1 (EK)

## Video transcript

we have already seen a situation multiple times where there is some parameter associated with a population maybe it's the proportion of a population that supports a candidate maybe it's the mean of a population the mean height of all the people in the city and we've determined that it's unpractical or we just there's no way for us to know the true population parameter but we could try to estimate it by taking a sample size so we take n samples and then we calculate a statistic based on that we've also seen that not only can we calculate the statistic which is trying to estimate this parameter but we can construct a confidence interval about that statistic based on some confidence level and so that confidence that confidence interval would look something like this it would be the value of the statistic that we have just calculated plus or minus some margin of error and so we'll often say this critical value Z and this will be based on the number of standard deviations we want to go above and below that statistic and so then we'll multiply that times the standard deviation of the sampling distribution for that statistic now what we'll see is we often don't know this to know this you oftentimes even need to know this parameter for example in the situation where the parameter that we're trying to estimate and construct confidence intervals for is say the population proportion what percentage of the population supports a certain candidate well in that world the statistic is the sample proportion so we would have the sample proportion plus or minus Z star times well we can't calculate this unless we know the population proportion so instead we estimate this with the standard error of the statistic which in this case is P hat times 1 minus P hat the sample proportion times 1 minus the sample proportion over our sample size if the parameter we're trying to estimate is the population mean then our statistic is going to be the sample mean so in that scenario we are going to be looking at our statistic is our sample mean plus or minus Z star now if we knew the standard deviation of this population we would know what the standard deviation of the sample distribution of our statistic is it would be equal to the standard deviation of our population times the square root of our sample size but we often will not know this in fact it's very unusual to know this and so sometimes you will say okay if we don't know this let's just figure out the sample standard deviation of our sample here so instead we'll say okay let's take our sample mean plus or minus Z 2 star times the sample standard deviation of our sample which we can calculate divided by the square root of n now this might seem pretty good if we're trying to construct a confidence interval for our sample for our mean but it turns out that this is not not so good because it turns out that this right over here is going to actually underestimate the actual interval the true margin of error you need for your confidence level and so that's why statisticians have invented another statistic instead of using Z they call it T instead of using AZ table they use a T table and we're going to see this in future videos and so if you are actually trying to construct a confidence interval for a sample mean and you don't know the true standard deviation of a population which is normally the case instead of doing this what we're going to do is we're gonna take our sample mean plus or minus and our critical value we'll call that T star times our sample standard deviation which we can calculate divided by the square root of N and so the real functional difference is that this actually is going to give us the confidence interval that actually has the level of confidence that we want if we have one a 95% level of confidence if we keep compute if we keep computing this over and over again for multiple samples that roughly 95% of the time this interval will contain our true population mean and to functionally do it and we'll do it in future videos you really just have to look up a T table instead of a Z table