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 $\left[\varphi \left(0.565\right)=0.71226, \varphi \left(0.79\right)=0.78524\right]$

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 the value of $k$ such that $\mathrm{P}(k⩽X<104)=0.75,$ where $X$ is the mass of a tomato in grams. Use $\left[\varphi \left(0.79\right)=0.78524, {\varphi }^{-1}\left(0.0352\right)=-1.805\right]$

The mass of a certain species of fish caught at sea is normally distributed with mean $5.73 \mathrm{kg}$ and variance $2.56{ \mathrm{kg}}^2.$ Find the probability that a randomly selected fish caught at sea has a mass that is: less than $6.0 \mathrm{kg}$. Use $\left[\varphi \left(0.169\right)=0.567\right]$

The mass of a certain species of fish caught at sea is normally distributed with mean $5.73 \mathrm{kg}$ and variance $2.56{ \mathrm{kg}}^2.$ Find the probability that a randomly selected fish caught at sea has a mass that is: more than $3.9 \mathrm{kg}$. Use $\left[\varphi \left(1.144\right)=0.874\right]$

The mass of a certain species of fish caught at sea is normally distributed with mean $5.73 \mathrm{kg}$ and variance $2.56{ \mathrm{kg}}^2.$ Find the probability that a randomly selected fish caught at sea has a mass that is: between $7.0\text{kg and} 8.0\text{kg}$. Use $\left[\varphi \left(1.149\right)=0.92220, \varphi \left(0.794\right)=0.788\right]$

The distance that children at a large school can hop in $15$ minutes is normally distributed with mean $199 \mathrm{m}$ and variance $3700{ \mathrm{m}}^2.$

Calculate an estimate of $b,$ given that only $25\%$ of the children hopped further than $b$ metres. Use $\left[{\varphi }^{-1}\left(0.75\right)=0.674\right]$

The distance that children at a large school can hop in $15$ minutes is normally distributed with mean $199 \mathrm{m}$ and variance $3700{ \mathrm{m}}^2.$

Calculate an estimate of $b,$ given that only $25\%$ of the children hopped further than $b$ metres. Find an estimate of the interquartile range of the distances hopped. Use $\left[{\varphi }^{-1}\left(0.75\right)=0.674\right]$