$X$ has a normal distribution in which $\sigma =\mu -30$ and $P(X⩾12)=0.9$. Find the value of $\mu$ and of $\sigma$.

(Write final answer up to $1$ decimal place)

For the variable $V~\mathrm{N}\left(\mu ,{\sigma }^2\right)$, it is given that $P(V<8.4)=0.7509$ and $P(V>9.2)=0.1385$. Find the value of $\mu$ and of $\sigma$, and calculate $P(V⩽10)$.

(Round answer up to $2$ decimal places)

(Round answer up to $2$ decimal places)

(Round answer up to $2$ decimal places)

The waiting times, in minutes, for patients at a clinic are normally distributed with mean $13$ and variance $16.$

Calculate the probability that a randomly selected patient has to wait for more than $16.5$ minutes.

The waiting times, in minutes, for patients at a clinic are normally distributed with mean $13$ and variance $16.$

Last month $468$ patients attended the clinic. Calculate an estimate of the number who waited for less than $9$ minutes. Use $\varphi \left(1\right)=0.84134$

Tomatoes from a certain producer have masses which are normally distributed with mean $90$ grams and standard deviation $17.7$ grams. The tomatoes are sorted into three categories by mass, as follows:
Small: under $80 \mathrm{g};$ Medium: $80 \mathrm{g}$ to $104 \mathrm{g};$ Large: over $104 \mathrm{g}.$

Find, correct to $2$ decimal places, the percentage of tomatoes in each of the three categories. Use $\left[\varphi \left(0.565\right)=0.71226, \varphi \left(0.79\right)=0.78524\right]$