Normal distribution

2 14 It is given that X ~ N(9, 1.5) and P(a ≤ X ≤ b) ≈ 95%

Identify the correct pair of values of a and b.

Tick () one box.

a = 4.5 and b = 13.5

a = 6 and b = 12

a = 7 and b = 11

a = 7.5 and b = 10.5

The cholesterol level, X, of an adult can be modelled by a normal distribution with mean 5.7 mmol/l and standard deviation 1.2 mmol/l.

The lower quartile of X is a and the upper quartile is b

It is given that the P(X ≤ a) = 0.25 and P(X ≥ b) = 0.25

Find the interquartile range of X

A group of scientists believe that a new dietary supplement can reduce the mean cholesterol level.

A random sample of 160 adults were given the new dietary supplement for a trial period of one year.

After the trial, their cholesterol level was found to have a mean of 5.6 mmol/l.

Carry out a hypothesis test at the 5% significance level to investigate whether the mean cholesterol level has reduced after taking the dietary supplement.

It can be assumed that the cholesterol level after taking the dietary supplement follows a normal distribution with an unchanged variance.

The oxides of nitrogen emissions, F g/km, of a car registered in 2000 can be modelled by a normal distribution with mean 0.41 and standard deviation 0.07

Find P(F < 0.39)

Find P(0.3 < F < 0.5)

Find P(F > 0.6)

Explain why P(F ≥ 0.6) = P(F > 0.6) in this model.

The oxides of nitrogen emissions, F g/km, of a car in 2010 can be modelled by a normal distribution with mean 0.36 and standard deviation 0.09

Compare the oxides of nitrogen emissions in 2000 with those in 2010.

A researcher collected data on the oxides of nitrogen emissions from the Large Data Set for cars registered in 2002 and 2016.

The researcher cleaned the data by removing the details for some cars.

Using your knowledge of the Large Data Set, give one reason why the researcher cleaned the data.

The researcher wished to make a comparison between the oxides of nitrogen emissions for a sample of Nissan cars in their local town from 2016 with the data from all cars registered in the same year in the Large Data Set.

Using your knowledge of the Large Data Set, give one reason why a meaningful comparison could not be made.

A farm supplies apples to a supermarket.

The diameters of the apples, D centimetres, are normally distributed with mean 6.5 and standard deviation 0.73

Find P(D 52)

Find P(D 7)

The supermarket only accepts apples with diameters between 5 cm and 8 cm.

Find the proportion of apples that the supermarket accepts.

The farm also supplies plums to the supermarket.

These plums have diameters that are normally distributed.

It is found that 60% of these plums have a diameter less than 5.9 cm.

It is found that 20% of these plums have a diameter greater than 6.1 cm.

Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm.

X N(14, 0.35)

Find the standard deviation of X, correct to two decimal places.

Circle your answer.

0.12 0.35 0.59 1.78

In a particular year, the height of a male athlete at the Summer Olympics has a mean 1.78 metres and standard deviation 0.23 metres.

The heights of 95% of male athletes are between 1.33 metres and 2.22 metres.

Comment on whether a normal distribution may be suitable to model the height of a male athlete at the Summer Olympics in this particular year.

You may assume that the height of a male athlete at the Summer Olympics may be modelled by a normal distribution with mean 1.78 metres and standard deviation 0.23 metres.

Find the probability that the height of a randomly selected male athlete is 1.82 metres.

Find the probability that the height of a randomly selected male athlete is between 1.70 metres and 1.90 metres.

Two male athletes are chosen at random.

Calculate the probability that both of their heights are between 1.70 metres and 1.90 metres.

The summarised data for the heights, h metres, of a random sample of 40 male athletes at the Winter Olympics is given below.

Use this data to calculate estimates of the mean and standard deviation of the heights of male athletes at the Winter Olympics.

Using your answers from part (c), compare the heights of male athletes at the Summer Olympics and male athletes at the Winter Olympics.

A factory produces jars of jam and jars of marmalade.

The weight, X grams, of jam in a jar can be modelled as a normal variable with mean 372 and a standard deviation of 3.5

Find the probability that the weight of jam in a jar is equal to 372 grams.

Find the probability that the weight of jam in a jar is greater than 368 grams.

The weight, Y grams, of marmalade in a jar can be modelled as a normal variable with mean and standard deviation

Given that P(Y 346) 0975, show that

Fully justify your answer.

Given further that

The lifetime of Zaple smartphone batteries, X hours, is normally distributed with mean 8 hours and standard deviation 1.5 hours.

Find P(X 8)

Find P(6 X 10)

Determine the lifetime exceeded by 90% of Zaple smartphone batteries.

A different smartphone, Kaphone, has its battery’s lifetime, Y hours, modelled by a normal distribution with mean 7 hours and standard deviation .

25% of randomly selected Kaphone batteries last less than 5 hours.

Find the value of , correct to three significant figures.

Elizabeth’s Bakery makes brownies.

It is known that the mass, X grams, of a brownie may be modelled by a normal distribution.

10% of the brownies have a mass less than 30 grams.

80% of the brownies have a mass greater than 32.5 grams.

Find the mean and standard deviation of X.

Find P(X 35)

Find P(X 35)

Brownies are baked in batches of 13.

Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams.

You may assume that the masses of brownies are independent of each other.

A survey of 120 adults found that the volume, X litres per person, of carbonated drinks they consumed in a week had the following results:

Calculate the mean of X.

Calculate the standard deviation of X.

Assuming that X can be modelled by a normal distribution find

P(0.5 X 1.5)

P(X 1)

Determine with a reason, whether a normal distribution is suitable to model this data.

It is known that the volume, Y litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21

Given that P(Y 075) 010, find the value of , correct to three significant figures.