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## AP®︎/College Statistics

### Unit 11: Lesson 1

Constructing a confidence interval for a population mean- Introduction to t statistics
- Simulation showing value of t statistic
- Conditions for valid t intervals
- Reference: Conditions for inference on a mean
- Conditions for a t interval for a mean
- Example finding critical t value
- Finding the critical value t* for a desired confidence level
- Example constructing a t interval for a mean
- Calculating a t interval for a mean
- Confidence interval for a mean with paired data
- Making a t interval for paired data
- Interpreting a confidence interval for a mean

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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)

An introduction to why we use t statistics.

## Video transcript

- [Instructor] 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's it's unpractical or there's no way for us to know
the true population parameter. But, we can 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
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 one minus p hat. The sample proportion times
one 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're
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 sampling 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'll 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 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 and instead of using a z-table, they use a t-table. Now, 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 your population, which is normally the case, instead of doing this, what
we're going to do is we're going to 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 want a 95% level of
confidence, 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.