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### Course: Statistics and probability > Unit 11

Lesson 3: Estimating 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
- Sample size for a given margin of error for a mean
- Sample size and margin of error in a confidence interval for a mean

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# Introduction to t statistics

An introduction to why we use t statistics.

## Want to join the conversation?

- What is the difference between a statistic and a parameter? Please explain like you would to someone who barely knew anything about statistics.(8 votes)
- We use 'statistic' in order to approximately estimate 'parameter'.

Let's say we want to know what percent of all male population of USA (or another random country) do some jogging in the morning. This percent is called 'parameter'.

Can we really survey and analyze every male in the USA ?

Well, maybe we can, but it would be to costly to do so in terms of the time, money or human rights infringements of those who don't want to share what the do in the morning.

So, in practice we just randomly select some men from all over the country and count what percent of them run in the morning. This percent is called 'statistic', which approximately estimates 'parameter'.(26 votes)

- This explanation of the distinction seems really confusing. If the population is Bernoulli distributed then the population proportion and population mean are the same thing! And yet we can estimate one with a Z-stat but the other needs a T-stat?

Also, when Sal calculates confidence intervals for the sample mean he uses the sample variance, which is presumably Bessel corrected and therefore less biased. But when he calculates the intervals for the sample proportion there's no Bessel correction!

Again, the population proportion and population mean are the same for a Bernoulli distribution. And the sample proportion and sample mean are also the same. Yet, when calculating confidence intervals, why do we use Z-stats for one and T-stats for the other? Why do we use Bessel's correction for one, and not for the other?

Finally, why is there no mention of the sample size? I thought that small n is the determining factor for when to use T-stats instead of Z-stats.(13 votes) - why when calculating p hat (sample proportion), we dont use t score?(11 votes)
- When calculating phat, we know sigma. However, now we don't, as mentioned in3:12, so we use a thing called a t score.

EDIT:

Sorry for my original unclear answer. Looking at Edexcel S3 and S4 manuals I am pleased to confirm that JW and chris are correct. When n is large(>30 for IAL) the Student-t tends toward a normal. Also remember that the t- and z-statistics are basically the same thing (s is unbiased estimate of \sigma) and the difference is that in one case s (sample variance) is also an r.v. and in the other it's not because of extra data given. So which on to use ultimately depends on whether you want to make the approximation that s==\sigma (which is accurate when n>30).

PS this vid is an intro to t-score so presumably he wants to connect the z- and t-scores first.(3 votes)

- Why is the expression at3:02'not so good'? Where can I get to read the math behind calculating z and t?(8 votes)
- What is the difference between z and t that fixes the problem?(7 votes)
- At3:10, Sal claims that using z* as part of making the confidence interval for a sample mean actually leads to an underestimate for the confidence interval. Why is that?(4 votes)
- The actual sampling distribution of means doesn't really follow a normal distribution (which is what z is based on). The sampling distribution of means has more "extreme" values than does the normal distribution, particularly when you use small samples to estimate the mean. This means more of samples will have means further from the population mean than they would if the sampling distribution was normal. So the confidence interval is narrower than should be, and the intervals don't contain the parameter the "correct" proportion of time. The t-distribution accounts for these "fatter tails".(2 votes)

- here I'm a bit stuck...

p is the proportion of something in the population.

p_hat is the proportion of the same parameter in the sample we take. So to speak our statistic.

So isn't p "just" the population_mean (of the something) / N?

And isn't p_hat the sample proportion: p_hat = sample_mean / n ?

All this by definition?

What am I missing?

Is the X_mean we are searching for the mean of the sampling distribution of the sample means? But wouldn't that be mu = p.... so back to the beginning of my question...

And if it is the real mean value, so not the proportion wouldn't it be just p_hat * n ?

So if we have a mean, but not the proportion, then why can't we just do mean / n to get the p_hat. And from here go the old way with p*(1-p)... ?(4 votes)- the x_ mean is the sample mean from some random independent sample of a population, which Sal discusses in the beginning of the video(1 vote)

- Why using sample standard deviation leads to underestimate?!

It's square root of sample variance right? And sample variance is divided by "n-1" rather than "n", so it seems to have larger value. why doesn't it lead to overestimate?(2 votes)- It's not about the sample standard deviation (the standard error), it's about the shape of the sampling distribution (all the possible means for a particular sample size). This distribution is not a normal distribution, particularly if you have small samples. It actually follows what's called student's t-distribution. This distribution has "fatter tails" (ie, more values that are far from the mean) than the normal distribution, and this is what causes the underestimation.(3 votes)

- Where does sigma over square root of n come from? Why and how did we put it there?(2 votes)
- Interesting question! In this discussion, we use theoretical (or population) standard deviation and variance.

To derive this, we use the following properties:

1) The variance of a sum of**independent**random variables is the sum of their variances.

2) When a random variable is multiplied by a factor that doesn't depend on the random variable, the variance is multiplied by the**square**of this factor.

3) The standard deviation is the (non-negative) square root of the variance, and so the variance is the square of the standard deviation.

Let the random variables X_1, X_2, X_3, ... , X_n represent a random sample of n data values, each of which has standard deviation sigma >= 0 (and therefore variance sigma^2). If we assume the population is very large, then it's reasonable to call these n random variables independent.

The sample mean is (X_1 + X_2 + X_3 + ... + X_n)/n, and the standard error of the mean is the standard deviation of the sample mean. Therefore, the standard error of the mean is

standard deviation[(X_1 + X_2 + X_3 + ... + X_n)/n]

= sqrt{variance[(X_1 + X_2 + X_3 + ... + X_n)/n]}

= sqrt[variance(X_1 + X_2 + X_3 + ... + X_n)/(n^2)]

= sqrt{[variance(X_1) + variance(X_2) + variance(X_3) + ... + variance(X_n)]/(n^2)}

= sqrt[n sigma^2 / (n^2)]

= sqrt(sigma^2 / n)

= sigma/sqrt(n).

Have a blessed, wonderful day!(3 votes)

- Hey

can anyone explain what is the difference between True population Proportion and True Population mean.

...

I am bit confused(3 votes)- If its a Yes/No Question that we are answering, it's a comment on the population behaviour and hence a True Population Proportion.

If its numeric value(measuring something), it's a comment on the measure of subjects in the population, irs a True Population Mean

Reference :

https://www.youtube.com/watch?v=DSYPMs1jSN0(1 vote)

## 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.