Let X i </msub> be Gaussian random variables with &#x03BC;<!-- μ --> =

Consider testing $H_0:\theta \ge 3\text{vs}{H}_a:\theta <3$ using a test in which the null hypothesis is rejected when the statistic $T=min\{X_1,...,X_n\}$, for example of size $n$ has small values. Compute the $p$-value using the sample $3.11,3.3,3.46$

p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for say n = 1000. Will my p-value still be uniformly distributed or as binomial converges to normal(Poisson), p-values mostly will be in 0.8-1?

For $H_0:\pi =0.20$, for instance, the score test statistic is $Z(s)=-2.50$, which has two-sided P-value $0.012<0.05$, so $0.20$ does not fall in the interval.
It is understood that the score test statistic is $-2.5$ using $n=25$ and $\hat{\pi }=0$ (in the previous description), but the part where the P-value is $0.012$ is not understood.

We want to make an hypothesis test for the mean value $\mu$ of a normal population with known variance ${\sigma }^2=13456$, using a sample of size $n=100$ that has sample mean value equal to $562$.
Calculate the $p$-value.
Make the test with significance level $1\mathrm{\%}$ about if the population mean value from which the sample comes from is greater than $530$ using the $p$-value.
For the first one, about the $p$-value, do we have to calculate $P\left(\frac{530-562}{\frac{\sigma }{\sqrt{n}}}\right)$?
And for the second we have to check the $p$-value with the significance level, right?

Let $\mathcal{X}$ be a sample space, $T$ a test statistic and $G$ be a finite group of transformations (with M elements) from $\mathcal{X}$ onto itself. Under the null-hypothesis the distribution of the random variable $X$ is invariant under the transformations in $G$. Let
$\hat{p}=\frac{1}{M}\sum _{g\in G}{I}_{\{T(gX)\ge T(X)\}}.$
Show that $P(\hat{p}\le u)\le u$ for $0\le u\le 1$ under the null hypothesis

I'm confused about the interpretation of P value in hypothesis testing. I know that we set significance level as 0.05 which is the threshold we set for this test so that it won't suffer from Type I error by 5%.
And we are comparing P to significance level, does it mean P is the probability of making type I error based on the sample?

Are there any theoretical considerations between p-value and the Bayesian statistics? I mean say, any theorem regarding both of these two concepts at a time.