Normal Distribution (DP IB Applications & Interpretation (AI): HL): Exam Questions

The weights, $W$ g, of chocolate bars made by a manufacturer are normally distributed with mean 200 g and standard deviation 1.75 g.

Find the probability that a randomly selected bar weighs

(i) less than 195 g;

(ii) more than 203 g.

Heledd buys a pack of 12 bars. Assume that the weights of the bars in the pack are independent.

Find the probability that every bar in the pack weighs at least 195 g.

The random variable $X~\mathrm{N}(330,{10}^2)$.

Find the value of $a$, to 2 decimal places, such that:

(i) $\mathrm{P}(X<a)=0.25$

(ii) $\mathrm{P}(X>a)=0.25$

(iii) $\mathrm{P}(315\le X\le a)=0.5$

The random variable $Y~\mathrm{N}(10,10)$.

Find the value of $b$ and the value of $c$, each to 2 decimal places, such that:

(i) $\mathrm{P}(Y<b)=0.4$

(ii) $\mathrm{P}(Y>c)=0.25$

Use a sketch of the distribution of $Y$ to explain why $\mathrm{P}(b\le Y\le c)=0.35$.

The test scores, $X$, of a group of RAF recruits in an aptitude test are normally distributed with mean 210 and standard deviation 27.8.

(i) Find the lower quartile and the upper quartile of the scores.

(ii) Hence find the interquartile range of the scores.

The recruits who score in the top 30% move on to the next stage of training. Amelia scores 231.

Determine whether Amelia moves on to the next stage of training.

A machine is used to fill cans of a particular brand of soft drink.  The volume, V ml, of soft drink in the cans is normally distributed with mean 330 ml and standard deviation $\sigma$ ml. 

It is known that approximately 16% of the cans contain more than 333.28 ml of soft drink.

Using the properties of the normal distribution, explain why 3.28 ml would provide a good approximation for the value of $\sigma$.

Using $\sigma =3.28\mathrm{ml}$, find $\mathrm{P}(320\le V\le 340)$  .

Six cans of the soft drink are chosen at random.

Again using $\sigma =3.28\mathrm{ml},$, find the probability that all of the cans contain less than 329 ml of soft drink.

The running time of films is normally distributed with a mean time of 102 minutes and a standard deviation of 13 minutes.

Find the probability that, on a randomly selected day, the feature film playing at the cinema has a running time of between 97 and 108 minutes.

Jonah watches a film on 18 different occasions.

Find the expected number of occasions on which the film he watches will last less than 95 minutes.

Find the probability that on at least 6 out of the 18 occasions, the film will last for longer than 99 minutes.

The weights, $W$ kg, of pumpkins grown on a farm are normally distributed with mean 11.3 kg and standard deviation 2.1 kg.

One year, the farmer grows 350 pumpkins.

Find the expected number of these pumpkins that weigh between 7.2 kg and 12.5 kg.

The heaviest 7% of pumpkins are classed as large.

Find the range of weights of the large pumpkins.

The random variable $X~\mathrm{N}(24,16)$.

Find $x$ given that:

(i) $\mathrm{P}(X<x)=0.7$

(ii) $\mathrm{P}(X>x)=0.15$

Find $a$ and $b$ given that $\mathrm{P}(a<X<b)=0.95$ and $a$ and $b$ are equal distances from the mean.

A scientist is studying the movement of snails. The speeds, $S$ m/h, of the snails follow a normal distribution with mean 4.8 m/h and standard deviation 0.15 m/h.

Sketch a diagram to represent this information.

Find the probability that a randomly selected snail has a speed of less than 4.65 m/h.

From a sample of 80 snails, calculate the expected number of snails that would have a speed of less than 4.65 m/h. Give your answer to the nearest integer.

The height of the average female giraffe, from a population where the heights follow a normal distribution, is 4.57 m and has a standard deviation of 1.28 m.

Find:

(i) Q₁

(ii) Q₃

(iii) IQR

The heights, H, of male giraffes are also normally distributed, where $\mathrm{H}~N(4.96,1.{01}^2)$.

Find the proportion of male giraffes that fall within the IQR of the female population.

The distribution of birth weight of a newborn can be assumed to follow a normal distribution with a mean of 3369 g and a standard deviation of 567 g.

A baby is classified as being of low birth weight if its weight is less than 2500 g.

Draw a diagram to represent the situation, labelling clearly the mean and the boundary for low birth weight.

Find the expected number of babies from a sample of 1000 that are born with a low birth weight. Give your answer to the nearest integer.

14% of babies born at a particular hospital weigh more than 4 kg.

State whether this is representative of the population. Give a reason for your answer.

A reaction time test records how quickly a person reacts to a signal by pressing the space bar on a computer. The results of the test, $X$ milliseconds, are normally distributed with mean 273 milliseconds and standard deviation 11 milliseconds.

By considering the number of standard deviations that 284 and 295 are from the mean, show that $\mathrm{P}(284<X<295)$ is approximately 0.135.

A result more than 2.5 standard deviations from the mean is classed as extreme. 145 students take the test.

Find the expected number of students with an extreme result.

A software company records the total distance, $D$ miles, that each of its users scrolls with a computer mouse in one year. $D$ is normally distributed with mean 18.4 miles and standard deviation 5.8 miles.

Find the interquartile range of $D$.

8% of users scroll more than $d$ miles but less than 22 miles.

Find the value of $d$.

A company states that the lifespan in hours, H, of the bulbs that they manufacture follows a normal distribution with the parameters $\mathrm{H}~N(3000,{720}^2)$.

The company advertises that their lightbulbs exceed a lifespan of h hours.

Given that $\mathrm{P}(H<h)=0.4,$ find h.

From a batch of 4000 lightbulbs, calculate the expected number of lightbulbs that will have a lifespan greater than h found in part (a) but less than 3150 hours.

The 5% of light bulbs with the shortest life span are considered to be defective.

One of the lightbulbs that is tested has a lifespan of 2213 hours. Determine whether the lightbulb is considered to be defective. Give a reason for your answer.

From a given population it is found that the average person spends on average 122 minutes exercising each week. The data follows a normal distribution and has a standard deviation of $\sigma$ minutes.

It is known that approximately 81.5% of people spend between 50 and 158 minutes exercising each week.

Using a sketch of the distribution or otherwise, explain why 36 minutes would provide a good approximation for the value of $\sigma$.

From a sample of people within the population, it is known that 15 of them spent less than 65 minutes exercising each week.

Using the value above, of 36 minutes for the standard deviation, find the total number of people within the sample.

The diagram below shows the normal distributions of the life expectancy, $L$, in hours for two different brands of vacuum cleaner.

A customer wants a vacuum cleaner whose life expectancy is as predictable as possible.

State which brand the customer should choose. Give a reason for your answer.

For the brand chosen in part (a), find the probability that the life expectancy of a vacuum cleaner is between 1550 hours and 1700 hours.

Each brand B vacuum cleaner is sold with a 5-year warranty, under which the manufacturer replaces a vacuum cleaner that fails. A typical customer uses a vacuum cleaner for approximately 4 hours each week. Assume that there are 52 weeks in a year.

Find the probability that, from a batch of 620 brand B vacuum cleaners, more than 0.5% could be returned to the manufacturer within the warranty period.

It is known that the time spent on a smart phone per day by teenage boys is normally distributed with a mean of 306 minutes and a standard deviation of 56 minutes.

Find the proportion of teenage boys that spend between 275 minutes and 375 minutes per day on a smart phone. Sketch a diagram to show this information.

A sample of 50 teenage boys are selected from this population.

Find the expected number of boys from the sample who spend more than 7 hours on their smart phone each day.

The time spent on smart phones each day by teenage girls is also considered to follow a normal distribution with a mean of 286 minutes and a standard deviation of 74 minutes.

25 teenage girls are selected at random. Find the probability that exactly 3 of them spend less than 200 minutes on a smart phone each day.

One of the 75 teenagers in parts (b) and (c) is chosen at random.

Given that this teenager spends less than 250 minutes per day on a smart phone, find the probability that the teenager is a boy.

The stem heights, $H$ cm, of a particular variety of tulip are normally distributed with mean 60.1 cm and standard deviation 7.6 cm.

The probability that a randomly selected tulip has a stem height greater than $h$ cm is 0.72.

Find $h$.

Given that a randomly chosen tulip has a stem height of more than 62 cm, find the probability that its stem height is more than 64 cm.

Leila buys a bunch of 24 tulips.

Calculate the probability that at least 10 of them have a stem height of less than 57 cm.

A teacher sets the same test every year. The scores follow a normal distribution with mean 53 points. Any student who scores lower than one standard deviation below the mean has to re-sit the test. The diagram below shows the distribution of test scores and the boundary for those who have to re-sit.

For one class, four students have to re-sit the test.

(i) Find the standard deviation of the test scores.

(ii) Estimate the number of students in the class.

A second class of 28 students sits the same test. A student who scores more than 65 receives a commendation.

Find the probability that exactly 3 students in this class receive a commendation.

The masses, $M$ g, of cockroaches are normally distributed with mean 28.8 g.

It is known that 0.15% of the population of cockroaches has a mass greater than 51.9 g.

Find an approximate value for the variance of the population of cockroaches.

38 cockroaches are caught in a trap.

Using your value for the variance in part (a), find the probability that:

(i) exactly 15 of the cockroaches have a mass that is greater than 30 g

(ii) more than 26 cockroaches have a mass that is between 26 g and 56 g.

The graph below shows the normal distribution of the volume, in ml, of drink in a can provided by a manufacturer, with the central 68% of the distribution shaded.

Write down the mean and the standard deviation of the volume of drink in a can.

A can is chosen at random. The probability that it contains more than $v$ ml is 0.19.

Find $v$.

A sample of cans of drink were analysed and 27 were found to have a volume of less than 320 ml.

Find an estimate for the number of cans of drink that were in the sample.

It is known that the time in minutes, T, that a customer spends on hold when calling a customer service line follows a normal distribution, where $T~\mathrm{N}(17,4^2)$.

Find the probability that a customer chosen at random spends more than 25 minutes on hold.

Find the interquartile range of the hold times.

Find the time spent on hold that is exceeded by 6% of the population.

From a sample of 200 customers find the probability that exactly 25% of the customers in the sample would spend less than 15 minutes on hold.

The distance that a honeybee will travel from the hive to collect pollen is normally distributed with an average distance of 1.1 miles and a variance of 0.04.

Find the probability that a honeybee will travel further than 1.6 miles from the hive.

83% of the honeybees travel a distance that is greater than h miles.

Find the value of h.

A colony consists of 60 000 honeybees.

Calculate the probability that at least 28% but no more than 31% of the honeybees in this colony will stay within 1 mile of the hive.

Alannah cycles to school each day via one of two possible routes and the time taken to complete either journey follows a normal distribution.

The journey time for route A has a mean of 28 minutes and a standard deviation of 10 minutes.

The journey time for route B has a mean of 33 minutes and a standard deviation of 4 minutes.

Identify an advantage of each route.

The school day begins at 08:30 and Alannah leaves her house at 07:55.

Determine which route is more likely to make her late.

Route A is closed off for maintenance work so Alannah travels by route B every day from Monday to Friday.

Find the probability that she is on time for school on 2 consecutive days and late for the other 3 days.