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SL 4.9—Normal distribution and calculations - IB Questionbank | RevisionDojo

Practice SL 4.9—Normal distribution and calculations with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

The lengths of fish caught in a lake are normally distributed with a mean of 45 cm and a standard deviation of 6 cm . A fish is considered large if its length exceeds 50 cm .

Calculate the probability that a randomly caught fish is large.

[2]

A fish is selected at random from those that are large. Find the probability that its length is more than 55 cm .

[4]

Estimate the interquartile range of the fish lengths.

[3]

Question 2

SLPaper 2

The time taken for a train to travel from station P to station Q is normally distributed with a mean of 120 minutes and a standard deviation of 10 minutes.

A train is considered delayed if it takes longer than 130 minutes.

Calculate the probability that a train is delayed.

[2]

The train is considered early if it takes less than $k$ minutes, where the probability of being early is 0.1587 . Find the value of $k$ .

[3]

During a month, there are 60 trains traveling from P to Q , and their travel times are independent. Calculate the probability that exactly 50 trains are not delayed.

[3]

Question 3

SLPaper 2

A company produces two types of light bulbs, Type A and Type B. The lifetimes of Type A bulbs are normally distributed with a mean of 1200 hours and a standard deviation of 150 hours. The lifetimes of Type B bulbs are normally distributed with a mean of 1300 hours and a standard deviation of $\sigma$ hours. The company sells a batch of 1000 bulbs, with $60 \%$ being Type A and $40 \%$ Type B. A bulb is considered premium if its lifetime exceeds 1400 hours.

Calculate the probability that a randomly selected Type A bulb is premium.

[2]

Given that $\mathrm{P}(X>1400)$ for Type B is 0.3085 , find $\sigma$ .

[3]

A bulb is selected at random from the batch and is found to be premium. Find the probability that it is a Type A bulb.

[5]

Question 4

SLPaper 2

Two machines, A and B, produce bolts. The diameters of bolts from Machine A are normally distributed with a mean of 10 mm and a standard deviation of 0.5 mm . The diameters of bolts from Machine B are normally distributed with a mean of 10.8 mm and the same standard deviation.

Find the diameter $d \mathrm{~mm}$ such that $\mathrm{P}(X>d)$ for Machine A equals $\mathrm{P}(X<d)$ for Machine B.

[3]

Given that $\mathrm{P}(X>11)$ for Machine A is 0.0228 , find $\mathrm{P}(10<X<11.6)$ for Machine B.

[4]

Question 5

SLPaper 2

A manufacturer produces packets of flour whose weights, in grams, can be modelled by a normal distribution with mean $1000$ and standard deviation $3.5$ . A packet of flour is rejected for sale if its weight is less than $995$ grams.

Find the probability that a packet selected at random is rejected.

[2]

Estimate the number of packets which will be rejected from a random sample of ${100}$ packets.

[1]

Given that a packet is not rejected, find the probability that it has a weight greater than ${1005}$ grams.

[3]

Question 6

SL & HLPaper 2

Consider a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 10$ .

Find the probability that a randomly selected value from this distribution is less than 40.

[3]

Find the probability that a randomly selected value from this distribution is between 45 and 55.

[3]

Determine the value of $x$ such that 90% of the distribution is below $x$ .

[4]

Question 7

SLPaper 2

The time it takes Suzi to drive from home to work each morning is normally distributed with a mean of 35 minutes and a standard deviation of σ minutes. On 25% of days, it takes Suzi longer than 40 minutes to drive to work.

Find the value of σ.

[4]

On a randomly selected day, find the probability that Suzi's drive to work will take longer than 45 minutes.

[2]

Suzi will work five days next week. Find the probability that she will be late to work at least one day next week.

[3]

Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.

[4]

Suzi will work 22 days this month. She will receive a bonus if she is on time at least 20 of those days. So far this month, she has worked 16 days and been on time 15 of those days. Find the probability that Suzi will receive a bonus.

[4]

Question 8

HLPaper 2

A random variable $Y$ is normally distributed with mean $\alpha$ and standard deviation $\beta$ , such that $\text{P}(Y < 30.31) = 0.1180$ and $\text{P}(Y > 42.52) = 0.3060$ .

Find $α$ and $β$ .

[6]

Find ${\text{P}}\left( {\left| {Y - \alpha } \right| < 1.2\beta } \right)$ .

[2]

Question 9

SLPaper 2

The time worked, T, in hours per week by staff members of a large corporation is normally distributed with a mean of 42 and standard deviation 10.7.

Find the probability that a staff member selected at random works more than 40 hours per week.

[2]

A group of four staff members is selected at random. Each staff member is asked in turn whether they work more than 40 hours per week. Find the probability that the fourth staff member is the only one in the group who works more than 40 hours per week.

[3]

A staff member is selected at random from this large group. Find the probability that this staff member works less than 55 hours per week.

[2]

Ten staff members are selected at random from this large group. Find the probability that exactly five of them work less than 55 hours per week.

[5]

Question 10

SL & HLPaper 2

A factory produces light bulbs with lifetimes that are normally distributed with a mean of 800 hours and a standard deviation of 100 hours.

Calculate the probability that a randomly selected light bulb lasts more than 950 hours.

[3]

Find the probability that a randomly selected light bulb lasts between 700 and 900 hours.

[3]

Determine the lifetime that only 5% of the light bulbs exceed.

[4]

SL 4.9—Normal distribution and calculations Questionbank