The masses, in kilograms, of 'giant Botswana cabbages' have a normal distribution with mean $\mu$ and standard deviation $0.75.$ It is given that $35.2\%$ of the cabbages have a mass of less than $3 \mathrm{kg}.$ Find the value of $\mu$ and the percentage of cabbages with masses of less than $3.5 \mathrm{kg}.$ Use $\left[{\varphi }^{-1}\left(0.352\right)=-0.38. \varphi \left(0.287\right)=0.612\right]$

The weights, $X$ grams, of bars of soap are normally distributed with mean $125$ grams and standard deviation $4.2$ grams.

Find the probability that a randomly chosen bar of soap weighs more than $128$ grams.

The weights, $X$ grams, of bars of soap are normally distributed with mean $125$ grams and standard deviation $4.2$ grams.

Find the value of $k$ such that $P\left(k<X<128\right)=0.7465.$ Use $\left[\varphi \left(0.714\right)=0.7627, {\varphi }^{-1}\left(0.0162\right)=-2.14\right]$

The weights, $X$ grams, of bars of soap are normally distributed with mean $125$ grams and standard deviation $4.2$ grams.

Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than $128$ grams. Use $\left[\varphi \left(0.714\right)=0.7623\right]$

Crates of tea should contain $200 \mathrm{kg}$, but it is known that $1$ out of $45$ crates, on average, is underweight. A sample of $630$ crates is selected at random.

Find the probability that more than $12$ but fewer than $17$ crates are underweight.

(Round answer up to $3$ decimal places)

Crates of tea should contain $200 \mathrm{kg}$, but it is known that $1$ out of $45$ crates, on average, is underweight. A sample of $630$ crates is selected at random.

Given that more than $12$ but fewer than $17$ crates are underweight, find the probability that more than $14$ crates are underweight.

(Round answer up to $3$ decimal places)

Once a week, Haziq rows his boat from the island where he lives to the mainland. The journey time, $X$ minutes, is normally distributed with mean $\mu$ and variance ${\sigma }^2$.

Given that $P\left(20⩽X<30\right)=0.32$ and that $P\left(\mathrm{X}<20\right)=0.63$, find the values of $\mu$ and ${\sigma }^2$.

Once a week, Haziq rows his boat from the island where he lives to the mainland. The time taken for Haziq to row back home, $Y$ minutes, is normally distributed and $\mathrm{P}(Y<20)=0.6532$, Given that the variance of $Y$ is $58$, calculate the mean time taken by Haziq to row back home.