# Hypothesis testing, is sample solution wrong and question strange or am I wrong

Question: Conder the following hypothesis test: $H_0: μ <= 45$ $H_1: μ > 45$ For a sample size of $36$ and a significance level $α = 0.01$, what is your conclusion for the following sample result: a) sample mean = $43$ and $s = 4.6$ Sample solution says: Reject $H_0$, but that can't be right, because

a)$t = ($sample mean $- μ_0) / (s/6) = (43 - 45) / (4.6/6) -2.61$ $t_0.01 = 2.4377$ Because $-2.61 < 2.4377$, do not reject $H_0$, or am I wrong?

Also the second question: b) Find the highest possible sample mean for which the null hypothesis can be rejected at a $5%$ significance level, assuming the sample standard deviation is $5.0$

So in this case the biggest sample mean should be infinite.

If your sample mean $\overline{x} = 43$ is less than $\mu_0 = 45$, clearly you cannot reject the null hypothesis $\mu \leq \mu_0$. You must have $\overline{x} > \mu_0$ to have a chance to reject $H_0$.
• If $\overline{x} < \mu_0$, then you cannot reject the null hypothesis $\mu\leq \mu_0$ whatever the significance level is. Commented Jan 3, 2018 at 9:46
• "whatever the significance level is" I dont think so. Taking any kind of action has a cost and that will give a certain $\alpha$. Commented Jan 3, 2018 at 9:51
• I'm not an expert, so I'll be grateful if you can show me an explicit example of t-test (just like the one proposed by the OP), where $\overline{x} < \mu_0$, and you can reject, at some significance level, the null hypothesis $\mu\leq \mu_0$. Commented Jan 3, 2018 at 9:58