Normal Distributions (DP IB Maths: AA SL)

A scientist is studying the movement of snails and has observed that the distribution of their speeds, S, follows a normal distribution with a mean of 48 m/h and a standard deviation of 1.5 m/h.

Sketch a diagram to represent this information.

Find the probability that a randomly selected snail has a speed of less than 46.5 m/h.

From a sample of 80 snails, calculate the expected number of snails that would have a speed of less than 46.5 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:

The heights, , of male giraffes are also normally distributed, where .

If 68% of male giraffes are shorter than the upper quartile of the female giraffe population, find the standard deviation of the male giraffes.

If 

Find  given that:

Find  and  given that  and  and  are equal distances from the mean.

The weight, W, of pumpkins purchased are normally distributed with a mean of 11.3 kg and a standard deviation of 2.1 kg.

In one year, a farmer grows 350 pumpkins on his farm.

Predict the number of pumpkins that weigh between 7.2 kg and 12.5 kg from the farm.

The heaviest 7% of pumpkins are classified as large and are sold at a higher price.

Find the range of weights of pumpkins that can be sold for a higher price.

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.

Show that the babies born at this hospital are not representative of the population.

Given that that the standard deviation is the same as the population, find the mean birth weight of a newborn at this particular hospital. Give your answer to the nearest gram.

A reaction time test has been used to collate data about how quickly a person can react to a given signal by pressing the space bar on a computer.  It is found that the results of the test, , are distributed normally with a mean of 273 milliseconds and a variance of 121 milliseconds.

Use a sketch of the distribution to show that .

A result that is lies outside of two and a half standard deviations from the mean is considered to be extreme.  A group of 145 students decide to measure their reaction times using the test.

Estimate the number of students that will receive a result that would be considered extreme. Give your answer to the nearest integer.

A software firm has recorded the distance, D, that their users have been scrolled on a computer by a mouse over the course of one year. The data is distributed normally with a mean of 18.4 miles and a standard deviation of 5.8 miles.

Calculate the interquartile range of the data.

It is known that 8% of the users scroll more than  miles but less than 22 miles.

Find the value of .

A company states that the lifespan in hours, H, of the bulbs that they manufacture follows a normal distribution with the parameters .

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

Given that , find .

From a batch of 4000 lightbulbs, calculate the expected number of lightbulbs that will have a lifespan greater than  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.

A second company also manufactures lightbulbs, whose lifespan again follows a normal distribution, .

Given that P and P , find the values of  and .

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  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 .

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 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 length of carrots in centimetres is normally distributed with mean . The following table shows the probabilities for values of .

Values of

Any carrots that have a length longer than 20.3 cm are classed as oversized carrots.

Write down the value of .

Show that the value of is 17.8 cm.

Find the probability that when picking a carrot at random, an oversized carrot is chosen.

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.

State the mean volume of ml found in a can and the standard deviation.

The probability that a can picked at random has a volume that is greater than  is 0.19. Find .

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  . The probability that a customer spends more than 25 minutes on hold is 0.0228.

Find the mean length of time that a customer spends 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 stem heights, in cm, of a particular variety of tulip follow a distribution where .

The probability that a randomly selected tulip will have a stem height that is greater than  is 0.72.

Find .

A tulip is selected at random. It is known that the height of the stem is more than 62 cm.

Find the probability that the stem height is taller 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.

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

State which vacuum cleaner brand is more reliable. Give a reason for your answer.

Using the information for the brand of vacuum cleaner than you stated in part (a), find the probability that the life expectancy of the vacuum will be between 1550 hours and 1700 hours.

When sold, each brand B vacuum cleaner comes with a 5-year warranty, where the manufacturer will replace the vacuum in the event of failure.  It is assumed that a typical customer uses the vacuum cleaner for approximately 4 hours each week.

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

The distance that a honeybee will travel from the hive to collect pollen is normally distributed with an average distance of 1.1 miles. The probability that a honeybee will travel further than 1.6 miles from the hive is 0.00621.

Find the variance of the distances that the honeybees will travel 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.

A teacher sets the same test every year for which the results follow a normal distribution with a mean score of 53 points.  The teacher decides that any student who scores lower than one standard deviation below the mean will have to re-sit the test.  The diagram below shows the distribution of test scores and the boundary for those that need to re-sit.

For a class that complete the test, four students are instructed that they will need to re-sit.

Find:

the standard deviation of the test scores.

A second class of 28 students sit the same test. If they achieve a score of greater than 65, they will achieve a commendation.

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

The masses, M, of cockroaches are normally distributed with an average mass of 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:

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

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 33 of 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.

Within a given population, the length of time that a person can hold their breath is normally distributed. 42% of the population can hold their breath for up to a maximum of 1.2 minutes but only 0.5% of the population can hold their breath for longer than 2.8 minutes.

Find

the mean,

In order to be able to access and swim through a particular underwater cave without breathing equipment a person must be able to hold their breath for longer than 3.5 minutes.

A sample of 1200 people from the population in part (a) are surveyed to determine how long they can hold their breath for.

Find the probability that 2 or more people would be able to hold their breath long enough to be able to swim through the underwater cave.

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 variance of 

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.

A teenager is selected at random from the group of 50 teenage boys in part (b) and the group of 25 teenage girls in part (c).

Find the probability that a teenager picked at random is a boy, given that they spend less than 250 minutes on a smart phone each day.