Topic 4.12 · SL and HL

Standardisation of normal variables: notes and practice questions

Q: What does Standardisation of normal variables cover in IB Maths AA?

Standardising converts a normal variable X with mean μ and standard deviation σ to a standard normal variable Z:. Z = (X - μ)/(σ). The standard normal distribution has mean 0 and standard deviation 1.

Q: Is Standardisation of normal variables SL or HL?

Both. SL and HL students study Standardisation of normal variables to the same depth.

Q: How do I revise Standardisation of normal variables for IB Maths AA?

Start from the core idea: standardising converts a normal variable X with mean μ and standard deviation σ to a standard normal variable Z:. In the exam: this is the subtopic where a probability is given and μ or σ is wanted: inverse normal for z, then solve z = (x-μ)/(σ). Two unknowns needs two given probabilities and simultaneous equations, which is the harder version. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Standardisation of normal variables?

FourtyFive has 6 Standardisation of normal variables questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Standardisation of normal variables practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Standardisation of normal variables answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

4 min read Jump to 6 practice questions

Summary

Standardising converts a normal variable $X$ with mean $\mu$ and standard deviation $\sigma$ to a standard normal variable $Z$ :

$Z = \frac{X - \mu}{\sigma}$

The standard normal distribution has mean 0 and standard deviation 1.

Use standardisation to calculate probabilities using the standard normal table.

How it is examined

This is the subtopic where a probability is given and $\mu$ or $\sigma$ is wanted: inverse normal for $z$ , then solve $z = \frac{x-\mu}{\sigma}$ . Two unknowns needs two given probabilities and simultaneous equations, which is the harder version. Paper 2, 5 to 7 marks.

Given in the booklet

$z = \dfrac{x - \mu}{\sigma}$ is given.

Key ideas

Standardization of normal variables ( $z$ -values).
Inverse normal calculations where mean or standard deviation are unknown.

Linking questions

Links to other subjects: the normal distribution (biology); descriptive statistics (psychology).

Practice questions

6 questions · 4 medium · 2 hard

Showing 6 of 6

Question 1

MediumPaper 2 · calculator15 marks

(a)

The delivery times, $T$ minutes, for packages from a logistics hub to a regional distribution center can be modelled by a normal distribution with a mean of 120 minutes and a standard deviation of $\sigma$ minutes.

Given that 3% of the delivery times are longer than 135 minutes, find the value of $\sigma$ .

[3]

(b)

Find the probability that a randomly selected package will have a delivery time of more than 130 minutes.

[2]

(c)

Given that a package delivery takes longer than 130 minutes, find the probability that it takes less than 135 minutes.

[4]

(d)

On a particular day, there are 80 packages scheduled for delivery from the hub.

Find the expected number of packages that will have a delivery time of more than 130 minutes.

[3]

(e)

Find the probability that more than 8 of the packages on this particular day will have a delivery time of more than 130 minutes.

[3]

Question 2

HardPaper 2 · calculator16 marks

(a)

(a) The random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$ .

Find $P(\mu - 1.2\sigma < X < \mu + 1.2\sigma)$ .

[3]

(b)

(b) The diameters of ball bearings produced by a factory, in mm, are normally distributed with mean $\mu$ and standard deviation $\sigma$ . The ball bearings are categorized as defective, standard, large, or premium, according to their diameter. The following table shows the probability a ball bearing is classified into each category.

Category	Probability
Defective	0.03
Standard	0.65
Large	0.25
Premium	0.07

The maximum diameter of a defective ball bearing is 14.8 mm.

The minimum diameter of a premium ball bearing is 16.5 mm.

Find the value of $\mu$ and of $\sigma$ .

[6]

(c)(i)

Find the probability that a ball bearing chosen at random from those sold is categorized as

(i) standard;

[1]

(c)(ii)

(ii) large;

[1]

(c)(iii)

(iii) premium.

[1]

(d)

(d) The selling prices of the different categories of ball bearings at this factory are shown in the following table:

Category	Selling Price ($)
Standard	1.50
Large	1.80
Premium	2.50

The factory incurs a fixed cost of $300 for the production run and assumes it will sell the accepted ball bearings in exactly the same proportion as calculated in part (c).

According to this model, find the minimum number of accepted ball bearings that must be sold so that the net profit for the factory is at least $550.

[4]

Question 3

MediumPaper 2 · calculator15 marks

(a)

(a) The lifespan of a new type of battery, $L$ hours, can be modelled by a normal distribution with a mean of 1200 hours and a standard deviation of $\sigma$ hours.

Given that 5% of the batteries last longer than 1280 hours, find the value of $\sigma$ .

[3]

(b)

(b) Find the probability that a randomly selected battery will have a lifespan of less than 1150 hours.

[2]

(c)

[4]

(d)

(d) A batch of 500 batteries is produced. Find the expected number of batteries that will have a lifespan of less than 1150 hours.

[3]

(e)

(e) Find the probability that more than 85 of the batteries in this batch will have a lifespan of less than 1150 hours.

[3]

Question 4

HardPaper 2 · calculator7 marks

The masses of a certain type of melon are normally distributed with mean $\mu\text{ g}$ and standard deviation $\sigma\text{ g}$ .

The probability that a randomly chosen melon has a mass greater than $850\text{ g}$ is $0.3$ .

If a melon with a mass greater than $850\text{ g}$ is chosen at random, the probability that its mass is greater than $900\text{ g}$ is $0.4$ .

Find the value of $\mu$ and the value of $\sigma$ .

Question 5

MediumPaper 2 · calculator5 marks

(a)

(a) The lifespan of a certain brand of LED bulbs, $L$ , is normally distributed with a mean of $50000$ hours and a standard deviation of $4000$ hours.

Find the probability that a randomly selected LED bulb lasts less than $1.2$ standard deviations below the mean lifespan.

[2]

(b)

(b) The manufacturer wants to determine a minimum lifespan threshold such that only $5\%$ of bulbs last longer than this threshold. Find the number of standard deviations, $k$ , above the mean that corresponds to this threshold.

[3]

Question 6

MediumPaper 2 · calculator6 marks

The lifespan of a new model of LED light bulb, $X$ , is modelled by a normal distribution with mean $\mu$ hours and standard deviation $\sigma$ hours.

It is observed that 15% of these bulbs fail before 8000 hours, and only 5% of the bulbs last longer than 12000 hours.

Find the value of $\mu$ and $\sigma$ .

Every Standardisation of normal variables question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Rounding an intermediate value and then using it.
Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three.
Writing the answer and nothing else.

FAQ

Questions,
answered.

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What does Standardisation of normal variables cover in IB Maths AA?

Standardising converts a normal variable X with mean μ and standard deviation σ to a standard normal variable Z:. Z = (X - μ)/(σ). The standard normal distribution has mean 0 and standard deviation 1.

Is Standardisation of normal variables SL or HL?

Both. SL and HL students study Standardisation of normal variables to the same depth.

How do I revise Standardisation of normal variables for IB Maths AA?

Start from the core idea: standardising converts a normal variable X with mean μ and standard deviation σ to a standard normal variable Z:. In the exam: this is the subtopic where a probability is given and μ or σ is wanted: inverse normal for z, then solve z = (x-μ)/(σ). Two unknowns needs two given probabilities and simultaneous equations, which is the harder version. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Standardisation of normal variables?

FourtyFive has 6 Standardisation of normal variables questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Standardisation of normal variables practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Standardisation of normal variables answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Standardisation of normal variables: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Every Standardisation of normal variables question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Every Standardisation of normal variables question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.