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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.

## 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.
• 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'.
• 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.
• why when calculating p hat (sample proportion), we dont use t score?
• When calculating phat, we know sigma. However, now we don't, as mentioned in , 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.
• Why is the expression at 'not so good'? Where can I get to read the math behind calculating z and t?
• 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)... ?
• 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)
• At , 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?
• 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".
(1 vote)
• Hey
can anyone explain what is the difference between True population Proportion and True Population mean.
...
I am bit confused
• 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 :
(1 vote)
• Could someone help reason through this test question? The instructions say to use the Z table. My confusion: We know the sample size, so why do we use the Z table instead of the T table? How do we know when to use T table?

Elena wants to estimate what proportion of computers produced at a factory have a certain defect. A random sample of 200 computers shows that 12 computers have the defect. She is willing to assume independence between computers in the sample.
Based on this sample, which of the following is a 95%, percent confidence interval for the proportion of computers that have the defect?
(1 vote)
• We use a Z table because we are interested in a proportion.