Dean Chalmers and Julian Gilbey Solutions for Chapter: The Normal Distribution, Exercise 12: END-OF-CHAPTER REVIEW EXERCISE 8

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.

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 expected number of days over a period of five years (each of $52$ weeks) on which Haziq takes more than $25$ minutes to row back home.

The time taken,$T$ seconds, to open a graphics programme on a computer is normally distributed with mean $20$ and standard deviation $\sigma$.
Given that $P\left(T>13\mid T⩽27\right)=0.8$, find the value of $\sigma$.

The time taken, $T$ seconds, to open a graphics programme on a computer is normally distributed with mean $20$ and standard deviation $7.24$. Find the value of $k$ for which $P\left(T>k\right)=0.75$.

(Round answer up to $2$ decimal spaces)

A law firm has found that their assistants make, on average, one error on every $36$ pages that they type. A random sample of $90$ typed documents, with a mean of $62$ pages per document, is selected. Given that there are more than $140$ typing errors in these documents, find an estimate of the probability that there are fewer than $175$ typing errors.