[Statistics] measuring values against known value

Chickenpoxpie
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Homework Statement


You use a standard gram weight of known weight (x) to check the accuracy of a balance. For one balance, this standard is weighed 3 times yielding the following values; a, b, and c. Calculate the 95% confidence limit for the mean


Homework Equations


general statistics and mean equations
ts/n^1/2

The Attempt at a Solution


I am confused how to do this problem because of the known value. At first I did the problem ignoring the known value and going through all the motions (figuring the mean, standard dev, and then plugging into ts/n^1/2 for 2 degrees of freedom) but how do things change because of the known weight and "testing for the accuracy of the balance"?
 
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Nevermind y'all, was a silly mistake. I can easily do a one sample t test
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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