Normal Distribution (Cambridge International A-Level Mathematics): Exam Questions

The random variable $Z$ has the standard normal distribution, and $Φ ( z ) = \text{P} ( Z \leq z )$.

Use the table of the normal distribution function to find

(i) $Φ ( 0 . 60 )$

(ii) $Φ ( 0 . 63 )$

(iii) $Φ ( 1 . 59 )$

(iv) $Φ ( 2 . 99 )$

Use the table of the normal distribution function to find

(i) $Φ ( 0 . 601 )$

(ii) $Φ ( 0 . 635 )$

(iii) $Φ ( 1 . 004 )$

(iv) $Φ ( 1 . 598 )$

The random variable $Z$ has the standard normal distribution, and $Φ ( z ) = \text{P} ( Z \leq z )$.

Use the table of the normal distribution function to find the value of $z$ in each case.

(i) $Φ ( z ) = 0 . 5398$

(ii) $Φ ( z ) = 0 . 9147$

(iii) $Φ ( z ) = 0 . 8642$

(iv) $Φ ( z ) = 0 . 9764$

Use the table of critical values for the normal distribution to find the value of $z$ such that

(i) $\text{P} ( Z \leq z ) = 0 . 75$

(ii) $\text{P} ( Z \leq z ) = 0 . 999$

The random variable $Z$ has the standard normal distribution.

Find

(i) $\text{P} ( Z > 0 . 74 )$

(ii) $\text{P} ( Z > - 0 . 74 )$

(iii) $\text{P} ( Z > 2 . 142 )$

(iv) $\text{P} ( Z > - 1 . 516 )$

Find the value of $z$ such that

(i) $\text{P} ( Z > z ) = 0 . 9306$

(ii) $\text{P} ( Z > z ) = 0 . 0694$

(iii) $\text{P} ( Z > z ) = 0 . 1292$

(iv) $\text{P} ( Z > z ) = 0 . 5931$

The random variable $Z$ has the standard normal distribution, and $Φ ( z ) = \text{P} ( Z \leq z )$.

Find

(i) $Φ ( - 0 . 5 )$

(ii) $Φ ( - 0 . 579 )$

(iii) $Φ ( - 2 . 785 )$

Find the value of $z$ such that

(i) $Φ ( z ) = 0 . 1093$

(ii) $Φ ( z ) = 0 . 001$

(iii) $Φ ( z ) = 0 . 0549$

(iv) $Φ ( z ) = 0 . 0005$

The random variable $Z$ has the standard normal distribution.

Find

(i) $\text{P} ( 0 . 12 < Z < 1 . 34 )$

(ii) $\text{P} ( 1 . 037 < Z < 2 . 913 )$

(iii) $\text{P} ( - 2 < Z < 1 )$

(iv) $\text{P} ( - 1 . 5 < Z < - 0 . 5 )$

Find the value of $z$ such that

(i) $\text{P} ( 0 < Z < z ) = 0 . 499$

(ii) $\text{P} ( z < Z < 2 . 093 ) = 0 . 319$

(iii) $\text{P} ( z < Z < 1 . 3 ) = 0 . 9027$

(iv) $\text{P} ( - z < Z < z ) = 0 . 6826$

The times taken, in minutes, to complete a puzzle are normally distributed with mean 21 minutes and standard deviation 4 minutes.

Find the probability that a randomly chosen person takes more than 26 minutes to complete the puzzle.

The weights of female leopards in a particular region are normally distributed with mean 54 kg and standard deviation 5 kg.

Find the probability that a randomly chosen female leopard from this region weighs

(i) less than 51 kg,

(ii) at least 58 kg.

The heights of adult males in a city are normally distributed with mean 187 cm and standard deviation 10 cm.

Find the probability that a randomly chosen adult male in this city has a height between 190 cm and 203 cm.

The random variable X \sim \text{N}(32, \; 9).

Write down the mean and the standard deviation of $X$.

Find

(i) $\text{P} ( X \leq 34 )$

(ii) $\text{P} ( X \geq 39 )$

(iii) $\text{P} ( 34 \leq X \leq 39 )$

The random variable $Z$ has the standard normal distribution.

Find

(i) $\text{P} ( Z < 1 . 5 )$

(ii) $\text{P} ( Z > - 0 . 8 )$

(iii) $\text{P} ( - 2 . 1 < Z < - 0 . 3 )$

The random variable $X$ is normally distributed with mean 2 and standard deviation 0.1.

Find the values of $a$, $b$, $c$ and $d$ such that

(i) $\text{P} ( X < a ) = \text{P} ( Z < 1 . 5 )$

(ii) $\text{P} ( X > b ) = \text{P} ( Z > - 0 . 8 )$

(iii) $\text{P} ( c < X < d ) = \text{P} ( - 2 . 1 < Z < - 0 . 3 )$

The random variable X \sim \text{N}(23, \; 4^{2}).

Find

(i) $\text{P} ( X < 20 )$

(ii) $\text{P} ( X \geq 29 )$

(iii) $\text{P} ( 20 \leq X < 29 )$

The random variable Y \sim \text{N}(100, \; 225).

Find

(i) $\text{P} ( Y \leq 90 )$

(ii) $\text{P} ( Y > 140 )$

(iii) $\text{P} ( 85 \leq Y \leq 115 )$

The times taken, in minutes, by employees at a large company to complete a particular task are normally distributed with mean 50 and standard deviation 7.

97.5% of employees take less than $a$ minutes to complete the task. Find the value of $a$.

Only 0.1% of employees take more than $b$ minutes to complete the task. Find the value of $b$.

The lengths of the leaves of a certain type of tree are normally distributed with mean $μ$ cm and standard deviation 6 cm. The probability that a randomly chosen leaf has a length less than 51 cm is 0.8665.

Find the value of $μ$.

The masses of peaches grown in an orchard are normally distributed with mean 93 grams and standard deviation $σ$ grams. The probability that a randomly chosen peach has a mass greater than 101 grams is 0.0735.

Find the value of $σ$.

The times, in minutes, that customers spend waiting at a bank are normally distributed with mean 10 and standard deviation 3.

(i) Given that 40% of customers wait for less than $a$ minutes, find the value of $a$ correct to 2 decimal places.

(ii) Given that 25% of customers wait for more than $b$ minutes, find the value of $b$ correct to 2 decimal places.

Find the probability that a randomly chosen customer waits for between $a$ minutes and $b$ minutes.

The lengths of a particular species of fish are normally distributed. 2.5% of these fish are longer than 36.88 cm and 10% are shorter than 27.16 cm.

Find the mean and the standard deviation of the lengths of these fish.

The scores of applicants taking a company's aptitude test are normally distributed with mean 210 and standard deviation 27.8.

Find the interquartile range of the scores.

Applicants who achieve a score in the top 30% are invited to an interview. Amelia achieves a score of 231.

Determine whether Amelia is invited to an interview.

The weights of coconuts grown by a farmer are normally distributed with mean 1.25 kg and standard deviation 0.38 kg. The lightest 10% and the heaviest 10% of these coconuts cannot be exported. All the other coconuts are exported.

Find the range of weights of the coconuts that are exported, giving your answer correct to 2 decimal places.

The masses, in grams, of the tomatoes grown in a greenhouse are normally distributed. 0.1% of these tomatoes have a mass greater than 34.451 grams and 19.66% have a mass less than 14.792 grams.

Find the mean and the standard deviation of the masses of these tomatoes.

A machine is used to fill bottles with a soft drink. The volume, $V$ ml, of soft drink in the bottles is normally distributed with mean 450 ml and standard deviation $σ$ ml. 5% of the bottles contain less than 429 ml of soft drink.

Find the value of $σ$.

Find the probability that a randomly chosen bottle contains at least 485 ml of soft drink.

Find the probability that the volume of soft drink in a randomly chosen bottle is within one standard deviation of the mean.

The masses, in grams, of plums of a certain variety have the distribution $\text{N} ( 13 , 15 )$.

(i) 20% of these plums have a mass greater than $a$ grams. Find the value of $a$.

(ii) 50% of these plums have a mass between $b$ grams and 14 grams, where $b$ is less than 14. Find the value of $b$.

(iii) The probability that the mass of a randomly chosen plum differs from the mean mass by less than $c$ grams is 0.9372. Find the value of $c$.

Explain why there is no value of $d$ such that 50% of these plums have a mass between 14 grams and $d$ grams, where $d$ is greater than 14.

The time, in minutes, that Paul spends solving sudoku puzzles in a month is normally distributed with mean 2048 and standard deviation 64.

Find the probability that, in a randomly chosen month, Paul spends less than 1945 minutes solving sudoku puzzles.

On how many months of the year (12 months) would you expect Paul to spend between 2019 and 2091 minutes solving sudoku puzzles?

It takes Paul exactly 10 minutes to solve each sudoku puzzle.

Find the greatest integer $n$ for which the probability that Paul solves fewer than $n$ puzzles in a month is less than 0.01.

A machine fills bags with potatoes. The weights of potatoes in the bags are normally distributed with mean 3 kg and standard deviation $σ$ kg. 7% of the bags contain more than 3.05 kg of potatoes.

Find the value of $σ$.

Find the probability that a randomly chosen bag contains between 2.9 kg and 3.1 kg of potatoes.

Twelve of the bags are chosen at random.

Find the probability that not more than one of the twelve bags contains less than 2.96 kg of potatoes.

The scores of candidates in a university entrance test are normally distributed with mean 520 and standard deviation 89.9.

Find the interquartile range of the scores.

Candidates who score more than 670 are invited to an interview. Only candidates who score more than 750 are offered a scholarship.

Mervyn is one of the candidates. Given that Mervyn is invited to an interview, find the probability that he is offered a scholarship.

The scores in an aptitude test taken by applicants to a training scheme are normally distributed with mean 120 and standard deviation 26.5. Scores in the test are whole numbers.

Applicants with scores in the highest 10% go straight through to the training scheme. Applicants whose scores are not among the lowest 40% or the highest 10% of scores may take the test again.

Find the range of scores of the applicants who may take the test again.

(i) Find the probability that a randomly chosen applicant scores more than 180.

(ii) The maximum possible score in the test is 180. Give a reason why the normal distribution may not be a suitable model for the scores.

The masses, in kg, of the feed in sacks produced by a manufacturer have the distribution \text{N}\left(20, \; \left(\dfrac{1}{60}\right)^{2}\right).

Find the probability that 12 randomly chosen sacks all contain feed with a mass within 35 grams of the mean.

The masses of the vases made by an ancient Greek pottery workshop are normally distributed. 5% of the vases have a mass greater than 2.237 kg. 1% of the vases have a mass less than 1.906 kg.

Find the mean and the standard deviation of the masses of the vases.

The vases with a mass less than $m$ kg are fragile. A random sample of 10 of these vases is chosen. The probability that all 10 vases in the sample are fragile is $9 . 766 \times 10^{-24}$.

Find the value of $m$.

The masses, in kg, of the turkeys on Nicholas's farm have the distribution $\text{N} ( 4 . 7 , 1 . 9 )$. Nicholas classes a turkey as 'holiday ready' if its mass is greater than 5.5 kg.

A turkey is chosen at random. Find the probability that its mass is less than 6 kg, given that it is 'holiday ready'.

Nicholas takes a large random sample of turkeys that are 'holiday ready'.

Find an estimate of the median mass of the turkeys in his sample.

The heights, in metres, of the trees on a plantation are normally distributed with mean $μ$ and standard deviation $σ$. The shortest 10% of the trees and the tallest 2.5% of the trees are not sold. All the other trees are sold.

Find, in terms of $μ$ and $σ$, the least height and the greatest height of the trees that are sold.

It is given that 20% of the trees are shorter than $( μ - 8 )$ m.

Find the probability that a randomly chosen tree is taller than $( μ + 8 )$ m, given that it is taller than $( μ - 8 )$ m.