If X and Y are random variables and c is any constant, show that E(X+Y)=E(X)+E(Y).

A population of values has a normal distribution with ${\displaystyle \mu =133.5}$ and ${\displaystyle \sigma =5.2}$. You intend to draw a random sample of size ${\displaystyle n=230}$.
Find the probability that a single randomly selected value is between 133.6 and 134.1.
${\displaystyle P\left(133.6<X<134.1\right)=}$?
Write your answers as numbers accurate to 4 decimal places.

We wish to estimate what percent of adult residents in a certain county are parents. Out of 600 adult residents sampled, 192 had kids. Based on this, plot a $99\mathrm{\%}$ confidence interval for the proportion of adult residents who are parents in a given county.
Express your answer in the form of three inequalities. Give your answers in decimal fractions up to three places ${\displaystyle <p<}$ Express the same answer using a point estimate and a margin of error. Give your answers as decimals, to three places.
${\displaystyle p=\pm }$

You are conducting a Goodness of Fit hypothesis test for the claim that all 5 categories are equally likely to be selected. Complete the table. Report all answers correct to three decimal places.
$\begin{array}{|cccc|}\hline \text{ Category }& \text{ Observed Frequency }& \text{ Expected Frequency }& \text{ (obs-exp)^2/exp }\\ \text{ A }& 15\\ \text{ B }& 19\\ \text{ C }& 23\\ \text{ D }& 18\\ \text{ E }& 5\\ \hline\end{array}$
What is the chi-square test-statistic for this data?${\displaystyle {\chi }^2=}$
At the alpha = 0.05 level, what is the conclusion for this test?
Fail to reject the null hypothesis
Reject the null hypothesis
Report all answers accurate to three decimal places.

A population of values has a normal distribution with ${\displaystyle \mu =129.7}$ and ${\displaystyle \sigma =7.7}$. You intend to draw a random sample of size ${\displaystyle n=10}$.
Find the probability that a single randomly selected value is less than 130.9.
${\displaystyle P\left(X<130.9\right)=}$?
Write your answers as numbers accurate to 4 decimal places.