Use a two-way table to explore the probabilities related to each group or category in the table.

The following definition is discussed in advanced mathematics courses.

$f(x)=\{\begin{array}{ll}0& ifxisarationalnumber\\ 1& ifxisanirrationalnumber\end{array}$
Evaluate  $f(-\frac{3}{4}),f(-\sqrt{2}),andf(\pi )$.

Complete the given table below by writing directly the needed data. Show complete solution in solving the Mean, Variance and Standard Deviation.
$$\begin{array}{|cccccc|}\hline x& P(x)& x\cdot P(x)& x-\mu & (x-\mu {)}^2& (x-\mu {)}^{2}P(x)\\ 0& 0.11\\ 1& 0.22\\ 2& 0.33\\ 3& 0.25\\ 4& 0.09\\ \hline\end{array}$$

Let T be the time needed to complete a job at a certain factory. By using the historical data, we know that
$$P(T\le t)=\{\begin{array}{ll}\frac{1}{16}{t}^2& for0\le t\le 4\\ 1& fort\ge 4\end{array}$$
a. Find the probability that the job is completed in less than one hour, i.e., find ${\displaystyle P(T\le 1)}$.
b. Find the probability that the job needs more than 2 hours.
c. Find the probability that ${\displaystyle 1\le T\le 3}$.

With ${\displaystyle \alpha =.05}$ and df = 8, the critical values for a two-tailed t test are ${\displaystyle t=\pm 2.306}$. Assuming all other factors are held constant, if the df value were increased to df = 20, what would happen to the critical values for t?
a. They would increase (move farther from zero).
b. They would decrease (move closer to zero).
c. They would stay the same.
d. Not enough information to answer

Calculating the position of the median
I know this is a simple question, but I cannot find a straight answer anywhere.
When calculating the medium of listed data, the formula is (n+1)/2. My Statistics teacher said for grouped data the position of the medium is n/2. However, this seems contradictory as for discrete grouped data, the data could be written in a list if the original values were known. Therefore two different values for the median are found.
I know this question is probably going to get flagged as it has already been asked, however, it has never been answered. I find it frustrating that such a fundamental concept in Statistics, what is supposed to be precise and never subjective has a wishy-washy answer.
Rant over, I think the position should be (n+1)/2 for discrete data and n/2 for continuous data. Moreover, this then leads into the obvious question of how we should calculate quartiles as well.
It would be nice if everyone could agree on a certain method for calculating the medium as all my textbooks are saying different things.
Someone please dispell the confusion and this statistical mess.

How to prove that a normally distributed assumption was fulfilled for the Two-Proportional z-test?
Select all relevant assumptions.
Residual graphs do not have a pattern
Scattered plots are linear
Scattered plots do not have a pattern
Normal sample distribution: npge5&n(1-p)ge5
Bell-shaped histograms
Histograms are uniform
Residual graphs are linear
Residual graphs are linear box plots have less than two outliers
Normal quantile plots do not have a pattern
Normal sample distribution: n,p,ge5,n,(1-p)ge5,n_2p_2ge5&91-p_2)ge5
Normal sample distribution: n'sge30 or populations usually are distributed