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I am studying now t-scores. As far as I understand, t-scores are used when we don't know true population parameters (such as: standard deviation and population mean) and cant use z-scores. Here is formula that is in books and in Internet for calculating t-score: $$t= \frac{\bar{X}-\mu }{\frac{S}{\sqrt{n}}}$$

As far as I know μ is used to define true population mean. So in formula above I need true population mean μ to calculate t-score. But as I said before when calculating t-score we don't know true population parameters, in this case true population mean μ. So what number I should use in μ and how to calculate it?

Also to make it clear, it will be very helpful if you provide example of actual t-score calculation.

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  • $\begingroup$ In a hypothesis testing, the u is the hypothesis value. $\endgroup$ – SmallChess Sep 4 '14 at 8:24
  • $\begingroup$ Basically, when doing a t-test, you assume something for u. Calculate the sample's average. Use those values to do the test. You simply don't need the true population mean. $\endgroup$ – SmallChess Sep 4 '14 at 8:25
  • $\begingroup$ @Student T You mean I should use for μ the mean of many other samples? But if I have only one sample (consisting of 30 elements)? $\endgroup$ – vasili111 Sep 4 '14 at 8:34
  • $\begingroup$ Are you talking about a t-test? In a t-test, the u is the value specified in the null hypothesis. This statistic is really trying to transform your sample average into the standard normal to significance testing. I don't think it has anything to do many other samples. $\endgroup$ – SmallChess Sep 4 '14 at 8:44
  • $\begingroup$ @Student T I am talking about t-score to find estimated standard error when we don't have true population parameters (standard deviation and population mean). $\endgroup$ – vasili111 Sep 4 '14 at 8:49
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As far as I know μ is used to define true population mean.

Not quite, and here's the rub. μ represents whatever the true mean is. It's defined by the problem for which this little bit of statistical inference is the analysis, not by the data itself (that would make it an estimate, not a hypothesis)

So in formula above I need true population mean μ to calculate t-score.

You need a hypothesis about what it is, that is: a possible value for it. You dont need to know what that value really is.

But as I said before when calculating t-score we don't know true population parameters, in this case true population mean μ. So what number I should use in μ and how to calculate it?

An example, done a few ways

Assume for a moment that you ask that a pool of subjects estimate the price of something - say a new college textbook, for concreteness - and you're interested whether they over- or underestimate the true price.

Here you can look up the true price, so if it's 45 dollars and the price guesses are in dollars too, then the μ=45. If the subjects average guess is 60 then your t-test is testing whether there is enough evidence that they are systematically overestimating the price or whether their guesses could have have come from a population of subjects that neither under nor overestimated the textbook price.

Looking at this another completely equivalent way, you might subtract the true price from each subject's guess. Then you are looking at deviations from the correct price and the test would set μ=0 (unbiased price guessing)

Looked at a third way, you might think about running this test for all values of μ (you wouldn't really do this, but bear with me). For μs near the subjects' average, the test will 'not reject', but for μs quite far away from the subjects' average, the test will reject that the data came from a distribution with that value of μ. The region of μ values for which the test does not reject is, in a sense, the region of μ values that are 'reasonable' in the light of the data. This is one way to motivate the idea of (and sometimes actually construct) a confidence interval. When the confidence interval (the region of non-rejected μs) does not overlap 45 (or zero in the second formulation), then we reject the hypothesis that this population is unbiased in its textbook price guessing.

Each of these approaches get you to the same place in a different way. None of them require knowing the true value of μ. The first two are the ones to consider in your case.

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  • $\begingroup$ Thank you for detailed explanation. One more clarification, the t-test and finding value of t for the our sample is different right? For t-test we use formula that is at my question and for finding value of t for our sample we are using abbreviated t score table that shows the values of t corresponding to different areas under the normal distribution for various sample sizes (degrees of freadom), am I right? So for finding value of t for our sample we need only sample size n, the percentage of area in tail (or tails) and abbreviated t score table, am I right? $\endgroup$ – vasili111 Sep 4 '14 at 15:22
  • $\begingroup$ Here is screenshot of abbreviated t score table from my textbook: i.imgur.com/Odbm0Qc.png $\endgroup$ – vasili111 Sep 4 '14 at 15:23
  • $\begingroup$ From the sample you compute a) the degrees of freedom, which here is one less than the number of observations (n), b) the average vaue of the sample (X-bar), the sample standard deviation (s). When you make a hypothesis about the population mean (μ) then you have everything ready to compute the statistic (t). The 't-score table' allows you to choose from some different 'levels of significance' for your test. $\endgroup$ – conjugateprior Sep 4 '14 at 22:23
  • $\begingroup$ Following my example, hypothesise that the population mean was 45 (μ=45). You get prices from ten people (n=10) and these guesses average fifty (X-bar=50) with standard deviation five (s=5). So the statistic t is 3.16. The middle column gives numbers that t should be larger in absolute value than to reject (that μ=45) in a two-tailed test at 'level' 0.05 for various degrees of freedom. Here you have n-1=9 so the number to be larger than is 2.262. 3.16 is larger than this, so you can reject p<.05 that μ=45 in the population from which this is a sample. $\endgroup$ – conjugateprior Sep 4 '14 at 22:39
  • $\begingroup$ I can also calculate t score for individual element of my sample, right? Which formula to use for it t=(X-μ)/S or t=(X-μ)/estimated standard error ? I think I need to use first one, am I right? In that formulas μ is sample size, X is element value, S sample standard deviation. $\endgroup$ – vasili111 Sep 5 '14 at 11:03
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There are two different $\mu$'s involved here:

  1. the hypothesized mean that you use in the numerator of your t-statistic for a t-test (sometimes denoted as $\mu_0$), and
  2. the true population mean, $\mu$.

The t-test is actually to see whether the true population mean differs from the hypothesized mean -- that is, it's a test for a null hypothesis $H_0\!:\, \mu=\mu_0$.

Don't confuse $\mu$ with $\mu_0$. Only one of the two is known.

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