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SL 4.12—Z values, inverse normal to find mean and standard deviation - IB Questionbank | RevisionDojo

Practice SL 4.12—Z values, inverse normal to find mean and standard deviation 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 time $T$ (in minutes) taken by a commuter to travel from home to work is normally distributed with mean $\mu$ and standard deviation $\sigma$ . It is known that $P(T<45)=0.3085$ and $P(T>60)=0.0668$ . The commuter must arrive by 08:00 to be on time, leaving home at 07:00. If the commute takes more than 65 minutes, the commuter takes a taxi, which costs extra.

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

[6]

Calculate the probability that the commuter needs to take a taxi.

[2]

Given that the commuter arrives on time, find the probability that the commute took less than 50 minutes.

[4]

The commuter travels to work 240 days a year. Find the expected number of days the commuter takes a taxi.

[3]

Sketch the probability density function for $T$ , shading the region for $P(T>65)$ .

[2]

Question 2

SLPaper 2

The scores $S$ of students on a test are normally distributed. It is known that $20 \%$ of students score below 60, and $75 \%$ score below 85 .

Determine the mean $\mu$ and standard deviation $\sigma$ of the scores.

[6]

Find the score $k$ such that $P(S<k)=0.95$ .

[2]

Question 3

SLPaper 2

The random variable $Y$ , representing the weight (in kg ) of a certain species of fish, follows a normal distribution with mean $\mu=4.5 \mathrm{~kg}$ . It is known that $10 \%$ of fish weigh more than 5.5 kg .

Calculate the standard deviation $\sigma$ .

[3]

Find the probability that a randomly selected fish weighs less than 4 kg .

[2]

A fisherman catches 30 fish. Find the probability that exactly 5 fish weigh less than 4 kg.

[3]

Question 4

SLPaper 2

The heights of students in a school are normally distributed with mean $\mu=165 \mathrm{~cm}$ and variance $\sigma^{2}=64 \mathrm{~cm}^{2}$ .

Calculate the probability that a randomly selected student is shorter than or equal to 157 cm .

[2]

Find the value of $k$ such that $P(165-k<H<165+k)=0.64$ .

[3]

Sketch the probability density function for the heights, shading the region for $P(H<0)$ .

[2]

Question 5

SLPaper 2

The random variable $Y$ follows a normal distribution with mean $\mu=75$ and standard deviation $\sigma=6$ .

Sketch the probability density function for $Y$ , and shade the region representing $P(69<Y<81)$ .

[2]

Calculate $P(69<Y<81)$ .

[2]

Find the value of $a$ such that $P(Y<a)=0.85$ .

[3]

Question 6

SLPaper 2

A factory produces metal rods with diameters $D$ (in mm ) normally distributed with mean $\mu=10 \mathrm{~mm}$ . The probability that a rod has a diameter less than 9.5 mm is 0.00135 . Rods with diameters between 9.8 and 10.2 mm are accepted; others are rejected.

Find the standard deviation $\sigma$ .

[3]

Calculate the probability that a rod is accepted.

[2]

Given that a rod is rejected, find the probability that its diameter is less than 9.8 mm.

[4]

Out of 1000 rods, find the probability that more than 850 are accepted.

[4]

Sketch the probability density function for $D$ , shading the region for $P(9.8<D<$ 10.2).

[2]

Question 7

SLPaper 2

The lifetime $T$ (in hours) of a type of light bulb is normally distributed with mean $\mu=1500$ and standard deviation $\sigma=200$ .

Find the probability that a randomly selected bulb lasts between 1200 and 1700 hours.

[2]

A batch of 50 bulbs is tested. Calculate the probability that at least 45 bulbs last more than 1300 hours.

[4]

Sketch the probability density function for $T$ , shading the region for $P(1200<T<$ 1700).

[2]

Question 8

SLPaper 2

A machine produces metal rods with lengths $L$ (in cm ) that are normally distributed with mean $\mu=50 \mathrm{~cm}$ . The probability that a rod is shorter than 48 cm is 0.0228 .

Find the standard deviation $\sigma$ .

[3]

Calculate $P(|L-50|<1.5 \sigma)$ .

[2]

Out of 100 rods, find the expected number of rods with lengths between 47 and 53 cm.

[3]

Question 9

SLPaper 2

The time $S$ (in seconds) for a machine to process a task is normally distributed with mean $\mu=120$ seconds. The probability that a task takes less than 100 seconds is 0.0668 . Tasks taking longer than 150 seconds are flagged for review. On a given day, 80 tasks are processed.

Find the standard deviation $\sigma$ .

[3]

Calculate the probability that a task is flagged for review given that it takes more than 110 seconds.

[4]

Find the probability that fewer than 5 tasks are flagged for review.

[3]

Sketch the probability density function for $S$ , shading the region for $P(S>150)$ .

[2]

Question 10

SLPaper 2

The lengths $L$ (in cm) of fish in a lake are normally distributed with mean $\mu$ and standard deviation $\sigma$ . It is known that $P(L<30)=0.0228$ and $P(L<50)=0.8413$ . Fish shorter than 25 cm are released, and those longer than 55 cm are considered trophy fish.

Find $\mu$ and $\sigma$ .

[6]

Find the probability that a caught fish is a trophy fish given that it is not released.

[4]

A fisherman catches 100 fish. Find the expected number of trophy fish.

[3]

SL 4.12—Z values, inverse normal to find mean and standard deviation Questionbank