Use F test, assuming normal distribution and independence of populations.
$n_X$ = $10$, $n_Y$ = $12$
Processs:
In the above question, the mean for original machine is given. Thus we can construct a standard normal rv followed by a chi square random variable as:
$$ \sum_1^{n_X} (\frac{X_i - \mu}{\sigma})^2 \sim \chi^2_{n_X} $$
For the second sample a chi square random variable is:
$$ (n_Y-1)\frac{s^2_Y}{\sigma^2} \sim \chi^2_{n_Y-1} $$
F distribution is given by ratio of two chi square random variables divided by their df, thus:
$$ \frac{\sum_1^{n_X}(X_i - \mu)^2}{s^2_Y}\cdot \frac{1}{n_X} \sim F_{n_X,n_Y-1} $$
Question:
F statistic is given by:
$$ \frac{s^2_X}{s^2_Y} \sim F_{n_X-1, n_Y-1} $$
Through the above logic if i use $\mu$ to calculate $s^2_X$ then does the numerator df become $n_X$
