How many strings are there of lowercase letters

Determine whether the given set S is a subspace of the Vector space V.

(there are multiple)

Find k$k$ such that the following matrix M$M$ is singular.

M=⎡⎣⎢−225+k−5933−4114⎤⎦⎥

Let a and b be coprime integers, and let m be an integer such that a | m and b | m. Prove that ab | m

|-10+e^x|<9

Assume that when adults with smartphones are randomly selected, 57​% use them in meetings or classes. If 5 adult smartphone users are randomly selected, find the probability that at least 3 of them use their smartphones in meetings or classes.
 

Find a 95 confidence interval for ${\displaystyle \theta }$ based on inverting the test statistic statistic ${\displaystyle \hat{\theta }}$.
For our data we have ${\displaystyle Y_i\sim N(\theta x_i,1)\text{for}i=1,\dots ,n.}$
Therefore it can be proven that the MLE for ${\displaystyle \theta }$ is given by
${\displaystyle \hat{\theta }=\frac{\sum x_i{Y}_i}{\sum x_i^2}}$
To find the confidence interval I should invert the test statistic ${\displaystyle \hat{\theta }}$.
The most powerful unbiased size ${\displaystyle \alpha =0,05}$ test for testing
${\displaystyle H_0:\mu ={\mu }_0\text{vs.}{H}_1:\mu \ne {\mu }_0}$
where ${\displaystyle X_1,\dots ,X_n\sim \text{ iid }n(\mu ,{\sigma }^2)}$ has acceptance region
$A\left({\mu }_0\right)=\left\{\mathbf{x}:|\overline{x}-{\mu }_0|\le 1,96\sigma /\sqrt{n}\right\}.$
Substituting my problem (I think) we get that the most powerful unbiased size ${\displaystyle \alpha =0,05}$ test for testing
${\displaystyle H_0:\theta =\hat{\theta }\text{vs.}{H}_1:\theta \ne \hat{\theta }}$
has acceptance region $\{A\left(\hat{\theta }\right)=\{\mathbf{y}:|\bar{y}-\hat{\theta }|\le 1,\frac{96}{\sqrt{n}}\}$
or equivalently, $A\left(\hat{\theta }\right)=\left\{\mathbf{y}:\frac{\sqrt{n}\overline{y}-1,96}{\sqrt{n}{x}_i}\le \hat{\theta }\le \frac{\sqrt{n}\overline{y}+1,96}{\sqrt{n}{x}_i}\right\}$
Substituting ${\displaystyle \hat{\theta }=\sum x_{i}Y\frac{i}{\sum }{x}_i^2}$ we obtain
$A\left(\hat{\theta }\right)=\left\{\mathbf{y}:\frac{\sqrt{n}\overline{y}-1,96}{\sqrt{n}{x}_i}\le \frac{\Sigma x_{i}Yi}{\Sigma x_i^2}\le \frac{\sqrt{n}\overline{y}+1,96}{\sqrt{n}{x}_i}\right\}$
This means that my ${\displaystyle 1-0,05=0,95\left(95\mathrm{\%}\right)}$ confindence interval is defined to be
${\displaystyle C\left(y\right)=\{\hat{\theta }:y\in A\left(\hat{\theta }\right)\}}$
But I can't seem to find anything concrete and I feel that I've made mistakes somewhere. What to do?