4.6 Normal Distributions

For the random variable  find the following probabilities:

(I)

(ii)

(iii)

For the random variable  find the following probabilities:

(i)

(ii)

(iii)

The weight,W  g, of a chocolate bar produced by a certain manufacturer is modelled as .

Find:

(i)

(ii)

Heledd buys a pack containing 12 of the chocolate bars.  It may be assumed that the 12 bars in the pack represent a random sample.

Find the probability that all of the bars in the pack have a weight of at least 195 g.

The random variable . 

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

(i)

(ii)

(iii)

The random variable .

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

(i)

(ii)

Use a sketch of the distribution of Y to explain why .

The test scores, X, of a group of RAF recruits in an aptitude test are modelled as a normal distribution with . 

(i)      Find the values of a and b such that and  .

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

Those who score in the top 30% on the test move on to the next stage of training.

One of the recruits, Amelia, achieves a score of 231.Determine whether Amelia will move 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  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 .

Using , find   .

Six cans of the soft drink are chosen at random.

Again using , find the probability that all of the cans contain less than 329 ml of soft drink.

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 .

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 stem heights, H 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 h is 0.72.

Find h.

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, L, 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 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.

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:

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:

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.

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 56²

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.