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Three experiments are performed to measure the same quantity $T$. The three results are

$$ \begin{cases} T_A = 990\pm 3\\ T_B = 920\pm 30\\ T_C = 950\pm 12 \end{cases} $$

Using a $\chi^2$ test, and assuming Gaussian errors, verify the hypothesis that the three experiments are measuring the same quantity.

Can anyone explain how a $\chi^2$ test should be performed for this task? Normally I would have used a t-test. I know how the $\chi^2$ works to verify the goodness of a fit but I don't know how it could be used for this task. Thanks!

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  • $\begingroup$ Although one could apply a $\chi^2$ test in the sense of computing a suitable statistic, it almost surely would not follow a $\chi^2$ distribution, which means the computed p-value would be wrong. The reason is that the ratio $30/3$ reflecting the range of measurement errors is strong evidence that their variances differ. The best response from a statistical perspective is to refuse to conduct a $\chi^2$ test and to offer a better procedure in its stead. $\endgroup$
    – whuber
    Apr 18, 2022 at 18:48
  • $\begingroup$ What data have you summarized to get results such as $920\pm 30?$ More to the point, what does that notation mean. Is the result $951$ rare? not yet observed? unthinkable? $\endgroup$
    – BruceET
    Apr 18, 2022 at 20:33
  • $\begingroup$ This is actually an exercise so I have no background for these measurements. The error is supposed Gaussian. $\endgroup$
    – Andrea
    Apr 19, 2022 at 10:39

1 Answer 1

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I'm not entirely sure of how relevant/correct this answer is, but it might help you towards the right direction:

If all 3 random variables are normally distributed with known mean and variance, then $\tilde T_x = \frac{\bar T_x-\mu}{\sigma_x} \sim \mathcal{N}(0,1)$ for $x \in {A,B,C}$ are standard normal distributed, where $\bar T_x$ is the sample mean. Notice how the same mean $\mu$ is assumed for all values (but not $\sigma_x$), which corresponds to the 3-equation null $H_0: \mu_A = \mu; \mu_B = \mu; \mu_C = \mu;$, i.e. are all experiments measuring the same mean? We know that the sum of $n$ squared standard normal variables is chi-squared distributed with $n$ degress of freedom, i.e. $c = (\tilde T_A^2 + \tilde T_B^2 + \tilde T_C^2) \sim \chi^2_3$. Now a right-tailed test can be performed on the test statistic $c$.

As far as the actual numbers go I would assume the values to the left of $\pm$ are the sample means $\bar T_x$ and the values to the right are the standard deviations $\sigma_x$. For $\mu$ I'm not sure, maybe the mean of all 3 sample means.

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