The Normal Distribution: Mathematics 9709 (Cambridge International AS & A Level)

The resting heart rate, $H$ beats per minute, of an adult patient at a clinic is modelled by $H\sim N(72, 5^2)$.

What is the probability that a randomly chosen patient has a resting heart rate of less than $78$ beats per minute?

A phone manufacturer models the battery life, $T$ hours, of a fully charged phone under normal use by $T\sim N(11.5, 1.2^2)$.

(a) Find $P(T>13)$. [3]

(b) Find $P(10<T<12.7)$. [4]

The time, $M$ minutes, that a student takes to answer a particular exam question is modelled by $M\sim N(24, 5^2)$.

(a) Find the value of $m$ such that $P(M>m)=0.10$. [3]

(b) Find the value of $x$ such that $P(M<x)=0.05$. [3]

A drinks-bottling machine fills cans with cola so that the volume, $V$ ml, in a randomly chosen can is modelled by $V\sim N(\mu,\sigma^2)$. Quality-control records show that $5\%$ of cans contain more than $350$ ml, and $2.5\%$ of cans contain less than $306$ ml.

(a) Show that $350-\mu=1.6449\sigma$, and write down a similar equation connecting $\mu$, $\sigma$ and the boundary $306$ ml. [2]

(b) Solve these two equations to find $\mu$ and $\sigma$, giving each answer to 3 significant figures. [5]

In a large regional archery competition, the probability that a competitor hits the bullseye with a randomly chosen shot is $0.4$. A random sample of $150$ shots is analysed, and $X$ is the number of these shots that hit the bullseye.

(a) State, with justification, the normal distribution that may be used to approximate $X$. [2]

(b) Using this approximation, with a continuity correction, find $P(X>70)$. [4]

A machine fills tins of paint so that the volume, $V$ litres, in a randomly chosen tin is modelled by $V\sim N(5, 0.1^2)$.

What is the probability that a randomly chosen tin contains less than $4.85$ litres of paint?

Farmed salmon reared at a fish farm have weight, $W$ grams, modelled by $W\sim N(1200, 200^2)$.

(a) Find $P(W>1000)$. [3]

(b) Find the probability that a randomly chosen salmon's weight differs from the mean by more than $300$ grams, that is, find $P(|W-1200|>300)$. [4]

In a cycling time-trial, the time taken, $R$ minutes, by a randomly chosen rider is modelled by $R\sim N(48, 4^2)$. The fastest $10\%$ of riders (those with the shortest times) qualify for the final.

What is the qualifying time, $t$ minutes, such that $P(R<t)=0.10$?

In a large batch of electronic resistors, the probability that a randomly chosen resistor is defective is $0.05$. A random sample of $200$ resistors is taken, and $X$ is the number of defective resistors in the sample.

(a) State, with justification, the normal distribution that may be used to approximate $X$. [2]

(b) Using this approximation, with a continuity correction, find $P(8\le X\le14)$. [5]

Scores on a standardised aptitude test, $S$, for applicants to a college are modelled by $S\sim N(500, \sigma^2)$, where $\sigma$ is unknown. Records show that $5\%$ of applicants score more than $650$.

(a) Find the value of $\sigma$, giving your answer to 3 significant figures. [3]

(b) Using this value of $\sigma$, find $P(400<S<600)$. [4]