Hypothesis Testing model My teacher posted the following in his notes, and they're not sitting right with me. Here is the question and the answer he gave:
$\text{Question Statement:}$ An old model battery has a lifetime having a Gaussian Distribution with mean 150 hr and variance 16 hr. Now, the company claims that that the new batteries have improved lifetime (assume these batteries possess the same variance). We measure the lifetime of 9 batteries with sample mean of 155 hrs. Find a test at $\alpha = 1\%$.
$\text{His Solution:}$ Let $H_0$ be the battery life unchanged. Then
$$\alpha = 0.01 = P(\hat{X}_9>150+c|H_0)$$
$$=P\Bigg[\frac{\hat{X}_9-150}{\sqrt{\frac{16}{9}}}>\frac{c}{\sqrt{\frac{16}{9}}}\Bigg]=Q\Bigg[\frac{c}{\sqrt{\frac{16}{9}}}\Bigg]$$
$$\therefore \ \ \  c=Q^{-1}(0.01)\cdot\sqrt{\frac{16}{9}}$$
$$=3.10$$
Since $155>153.10$, we want to reject $H_0$; i.e., the battery life is changed (or in this case, improved).
$\text{My thoughts:}$ I don't agree with his conclusion.  I feel like since we get that $153.10<155$ we should accept the null hypothesis. Either way, could someone smarter than myself please explain whether or not he's right?
Thank you!
 A: Recall that $ \alpha $ refers to the probability of making a type-1 error, which means that the null hypothesis ($ H_0 $) is true, but it is rejected anyways.
In this example, the company claims that the new batteries have a longer lifetime than the old batteries. As such, the null hypothesis ($ H_0 $) can be formulated as follows: "The new batteries do NOT have a longer lifetime than the old batteries." 
For this problem, you want to find a test to determine whether or not the new batteries have a longer lifetime than the old batteries. If the new batteries do not have a longer lifetime than the old batteries, then you want this test to fail only 1 % of the time. In other words, you want to find c such that:
$$ P(\hat{X}_9>150+c | H_0) = 0.01 $$
You found that $ c = 3.10 $. This means that when $ \hat{X}_9 > 150 + 3.10 = 153.1 $, you can reject the null hypothesis $ (H_0) $ and be confident that there is only a 1 % chance that you have incorrectly rejected the null hypothesis $ (H_0) $.
Since $ \hat{X}_9 = 155 > 153.10 $ (given in the problem), then you should reject the null hypothesis $ (H_0) $.
