Binomial Distribution

The survey is conducted in a large factory. If $27\%$ of the factory workers weigh less than $65$ kg and that $25\%$ of the factory workers weigh more than $96$ kg.  [ use the standard values $P(Z<-0.6128\dots )=0.27\text{ and }P(Z<0.6744\dots )=0.75$]

Then the assumed weights of the factory workers is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Determine two simultaneous linear equations satisfied by $\mu$ and $\sigma$.

Find the values of $\mu$ and $\sigma$.

The national park houses of a large number of tall trees. The heights of the given trees are normally distributed with mean of $45$ metres and a the standard deviation of $9$ metres. A selection of $50$ trees are chosen randomly then find the probability that at least $5$ of these trees are taller than $55$ meters.

Max tosses four coins a total of $80$ times. The following table shows the distribution of the number of heads obtained in each toss of four coins.

$\begin{array}{|lccccc|}\hline \begin{array}{l}\text{ Number of }\\ \text{ heads }\end{array}& 0& 1& 2& 3& 4\\ \text{ Frequency }& 10& 24& 26& 17& 3\\ \hline\end{array}$

Using the binomial distribution $B\left(4,0.5\right)$, find the expected probabilities of obtaining $0,1,2,3$ or $4$ heads respectively

The number of boys in $90$ families with three children is shown in the table.

If the probability of having a boy is $0.5$, use the binomial expansion $B\left(3,0.5\right)$ to find the expected probabilities.

Percy sews three seeds in each of $50$ different pots. The probability that a seed will germinate is $0.75$. The number of seeds that germinated in each pot is shown in the table.

Using the binomial expansion $\mathrm{B}(3,0.75)$, find the expected probabilities of $0,1,2 \text{or} 3$ seeds germinating.

If $E\left(X\right)=5$ and  $Var\left(X\right)=2.5$, find $n$ and $p$.