I am given the following information:
$$\alpha = 0.025$$ $$n = 15$$ $$\sigma^2 = 100$$
I need to find the critical values.
My attempt: I chose the t-test. $$+/-t_{(n-1,\frac{\alpha}{2})}$$ $$+/-t_{(14,0.0125)}$$ $$+/-2.864$$
My question: Is this the correct way to do it? Being given the variance is making me question myself and I am left wondering if I am misunderstanding the question.
Without knowing the hypothesis being tested or the distribution of the test statistic under the null hypothesis, it's unreasonable to state the critical value of the test, so I consider this question ill-posed. For example, without knowing whether the test is one-tailed or two-tailed, the critical value is impossible to state. Even the way the sample was collected will affect the choice of test.
That said, if we assume that the test is for a location parameter and is two-tailed; e.g., $$H_0 : \mu = \mu_0 \quad \text{vs.} \quad H_a : \mu \ne \mu_0,$$ and a simple random sample of size $n = 15$ was taken, and that the observations are assumed to have been drawn from a distribution with known variance $\sigma^2 = 15$, then the test statistic $$Z \mid H_0 = \frac{\bar x - \mu_0}{\sigma/\sqrt{n}}$$ is approximately standard normal (unless the observations were drawn from a normal distribution, in which case $Z \mid H_0$ is exactly standard normal).
This test statistic will reject $H_0$ at $\alpha = 0.025$ if $|Z| > z^*_{\alpha/2}$, where $\Phi(z^*_{\alpha/2}) = 1 - \alpha/2 = 0.9875$; that is to say, $z^*_q$ is the upper $q^{\rm th}$ quantile of the standard normal distribution. Using a computer or statistical table, $$z^*_{0.0125} \approx 2.2414.$$ This is the critical value of the test.
Note that this is where the two-tailed nature of the hypothesis test comes into play. Had the test been one-tailed, e.g. $$H_a : \mu > \mu_0,$$ then you would reject $H_0$ in favor of $H_a$ if $Z > z^*_{\alpha}$, with no absolute values, and the critical value is now $$z^*_{0.025} \approx 1.95996.$$
And if the test were one-tailed in the other direction, $$H_a : \mu < \mu_0,$$ then your critical value would be $z_{0.025} = -1.95996$, and you would reject $H_0$ if $Z < z_{0.025}$.
So as you can see, I have insufficient information to uniquely identify the appropriate critical value for your test. One thing I can say is that a $t$-test is not appropriate if the variance is assumed to be known.