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 .
1.[2]
Calculate the probability that a randomly caught fish is large.
2.[4]
A fish is selected at random from those that are large. Find the probability that its length is more than 55 cm .
3.[3]
Estimate the interquartile range of the fish lengths.
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.
1.[2]
Calculate the probability that a train is delayed.
2.[3]
The train is considered early if it takes less than minutes, where the probability of being early is 0.1587 . Find the value of .
3.[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.
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 hours. The company sells a batch of 1000 bulbs, with being Type A and Type B. A bulb is considered premium if its lifetime exceeds 1400 hours.
1.[2]
Calculate the probability that a randomly selected Type A bulb is premium.
2.[3]
Given that for Type B is 0.3085 , find .
3.[5]
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.
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.
1.[3]
Find the diameter such that for Machine A equals for Machine B.
2.[4]
Given that for Machine A is 0.0228 , find for Machine B.
Question 5
SLPaper 2
The time customers spend in a coffee shop is normally distributed with a mean of 45 minutes and a standard deviation of 8 minutes.
A customer is considered to stay long if they spend more than 50 minutes.
1.[2]
Calculate the probability that a customer stays long.
2.[3]
Find the time such that of customers spend less than minutes.
3.[3]
On a given day, 40 customers visit the shop, and their times are independent. Calculate the probability that fewer than 10 customers stay long.
Question 6
SLPaper 2
The diameters of oranges from a farm are normally distributed with a mean of 8 cm and a standard deviation of 1.2 cm . An orange is considered premium if its diameter is greater than 9 cm .
1.[2]
Calculate the probability that a randomly selected orange is premium.
2.[3]
A premium orange is selected at random. Find the probability that its diameter is greater than 10 cm .
3.[2]
The farm sells 100 oranges, and their diameters are independent. Estimate the number of premium oranges.
Question 7
SLPaper 2
The daily rainfall in a town during the rainy season is normally distributed with a mean of 12 mm and a standard deviation of 3 mm . A day is classified as dry if the rainfall is less than 8 mm .
1.[2]
Find the probability that a randomly selected day is dry.
2.[3]
Find the rainfall amount such that the probability of rainfall being between 12 mm and is 0.35 .
Question 8
SLPaper 2
The time taken to complete a puzzle is normally distributed with a mean of 30 minutes and a standard deviation of 5 minutes. A participant is considered quick if they complete the puzzle in less than 25 minutes.
1.[2]
Find the probability that a randomly selected participant is quick.
2.[3]
Find the time minutes such that of participants take longer than minutes.
3.[3]
A group of 20 participants is selected, and their completion times are independent. Estimate the expected number of quick participants.
Question 9
SLPaper 2
The weights of apples produced by a farm are normally distributed with a mean of 150 grams and a standard deviation of 20 grams.
An apple is considered large if its weight exceeds 170 grams.
1.[2]
Find the probability that a randomly chosen apple is large.
2.[3]
The farm packs apples in boxes of 25, and the weights are independent. Calculate the probability that at least 5 apples in a box are large.
3.[4]
The farm wants to classify apples as medium if their weights are between grams and 170 grams, such that of apples are medium. Find .
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Question 10
SLPaper 2
The battery life of a certain brand of smartphone is normally distributed with a mean of 18 hours and a standard deviation of 2.5 hours. A battery is considered reliable if it lasts more than 20 hours.
1.[2]
Calculate the probability that a randomly selected smartphone has a reliable battery.
2.[3]
Find the battery life hours such that of smartphones have a battery life less than hours.
3.[3]
A company tests 50 smartphones, and their battery lives are independent. Calculate the probability that at least 10 of them have reliable batteries.