Exercises: The normal distribution

Given a random variable following a standard normal distribution $Z \sim N(0, 1)$, calculate the probability that $Z$ takes a value less than or equal to $1.32$. Graphically represent the corresponding area under the Gaussian curve.

You can do it on paper and send a photo with your answer.

An instant messaging app analyses the time it takes for servers to deliver a message. It is known that this time follows a standard normal distribution $Z \sim N(0, 1)$ after data transformation. Calculate the probability that the standardised value is less than or equal to $-0.45$ using the symmetry properties of the Gaussian curve.

The number of "likes" a post receives on a specific network, once normalised, behaves like a variable $Z \sim N(0, 1)$. Find the probability that a randomly chosen post has a value of $Z$ greater than $1.15$.

Given a standard normal distribution $Z \sim N(0, 1)$, calculate the probability that the variable lies in the interval between $0.5$ and $1.84$; that is, calculate $P(0.5 \le Z \le 1.84)$.

The score obtained by students in a popular simulation video game follows a normal distribution $X \sim N(500, 40)$, where the mean is $\mu = 500$ points and the standard deviation is $\sigma = 40$ points. Standardise the value $X = 560$ to obtain its corresponding value in the standard normal distribution $Z$.

The battery life of a smartphone model under continuous use follows a normal distribution with mean $\mu = 14$ hours and standard deviation $\sigma = 1.5$ hours. Calculate the probability that a randomly chosen phone lasts less than $15.5$ hours.

In quality control of electronic components, the output voltage is distributed according to a $N(0, 1)$. If it is desired to discard the $10\%$ of components with the highest voltages, what is the critical value $k$ such that $P(Z > k) = 0.10$? Use the normal distribution table to find the nearest value of $k$.

The heights of a group of teenagers aged $17$ follow a normal distribution $X \sim N(172, 8)$. Determine the probability that a randomly chosen teenager is between $164$ and $180$ centimeters tall using the standard deviation rule (interval $[\mu - \sigma, \mu + \sigma]$).

Given a random variable $Z$ following a standard normal distribution $N(0, 1)$, use the cumulative probability table to calculate the following probability values:

1. a) $P(Z \leq 1.32)$
2. b) $P(Z \geq 0.74)$
3. c) $P(Z \leq -1.15)$
4. d) $P(-0.45 \leq Z \leq 1.20)$

The scores obtained by students at a sixth form college in the mock university entrance exam follow a normal distribution with mean $\mu = 6.8$ and standard deviation $\sigma = 1.2$. If a student is chosen at random, calculate the probability that:

1. a) Their score is greater than $8$.
2. b) Their score is between $6$ and $7.5$.
3. c) They scored less than $5$.

A company that manufactures components for drones has found that the lifespan of its lithium batteries follows a normal distribution with a mean of $220$ minutes and a standard deviation of $15$ minutes.

1. a) What percentage of batteries last more than $240$ minutes?
2. b) If the manufacturer wishes to guarantee $90 \%$ of the batteries, what should be the minimum lifespan to meet this guarantee?

The weight of coffee bags marketed by a brand is normally distributed with unknown mean $\mu$ and standard deviation $\sigma = 10$ grams. Knowing that $15.87 \%$ of the bags weigh more than $510$ grams, calculate the value of the mean $\mu$.

On a streaming platform, the daily time teenagers spend watching series follows a normal distribution with mean $95$ minutes and standard deviation $20$ minutes.

1. a) Calculate the probability that a randomly chosen user spends more than $120$ minutes per day.
2. b) Given that a user spends more than $100$ minutes, what is the probability that they spend less than $130$ minutes?

It is known that the fasting blood glucose level of a population of healthy adults follows a normal distribution $N(85, 8)$ mg/dL.

1. a) Determine the central interval that contains $95.44\,\%$ of the population (the $\pm 2\sigma$ interval).

1. b) What is the glucose level such that only $5\,\%$ of adults have a level above it?

Two mapping apps, "RutaRápida" and "GeoGuía", estimate the arrival time at a destination. The estimation errors for "RutaRápida" follow a $N(0, 4)$ minutes, while those for "GeoGuía" follow a $N(0, 6)$ minutes (a positive error indicates that the app predicted more time than the actual time).

1. a) What is the probability that "RutaRápida" makes an overestimation error of more than $3$ minutes?
2. b) Which app is more likely to make an error, whether overestimating or underestimating, of less than $1$ minute?

A tyre factory produces a model whose mileage lifespan follows a normal distribution. It is known that $10 \%$ of the tyres last less than $35,000$ km and $5 \%$ last more than $55,000$ km. Calculate the mean and standard deviation of the lifespan of these tyres.

The reaction time of pilots in a state-of-the-art flight simulator follows a normal distribution $X \equiv N(\mu, \sigma)$. The developer company has determined that $15.87 \%$ of pilots take more than $0.8$ seconds to react, whereas only $2.28 \%$ of pilots manage to react in less than $0.4$ seconds.

1. a) Calculate the parameters $\mu$ and $\sigma$ of this distribution.

1. b) If a pilot is selected at random, what is the probability that their reaction time is between the mean time and $0.75$ seconds?

The battery life of a high-end smartphone model follows a normal distribution with a mean of $3500$ charge cycles. It is known that the probability of a battery lasting more than $3800$ cycles is $0.1587$.

1. a) Determine the standard deviation of the distribution.

1. b) If the manufacturer offers a replacement guarantee for the $5 \%$ of batteries that last the shortest time, how many charge cycles should this guarantee cover?

1. c) If a battery has already exceeded $3400$ charge cycles, what is the probability that it functions correctly up to at least $3600$ cycles?

In a manufacturing process for electronic components, the diameter of a circular part follows a normal distribution $X \equiv N(20, \sigma)$. A part is considered "optimal" if its diameter is between $19.7$ mm and $20.3$ mm. It is known that $25.1 \%$ of the parts produced are discarded for being too large (diameter greater than $20.3$ mm).

1. a) Calculate the value of the standard deviation $\sigma$.

1. b) If the parts are packed in batches of $10$ units, what is the probability that a randomly chosen batch contains exactly $6$ optimal parts?

The scores on a university entrance exam follow a normal distribution. The mean of the scores is $6.2$ points. It is known that the probability of obtaining a score greater than $8$ is $0.0668$, which is the same as the probability of obtaining a score lower than $x$:

1. a) Calculate the standard deviation of the scores.

1. b) Find the value of $x$.

1. c) What minimum score would a student need to achieve to be among the top $10 %$ of academic records?

An artificial intelligence algorithm classifies satellite images. The time it takes to process each image (in milliseconds) is normally distributed $N(\mu, \sigma)$. It has been observed that the mean processing time is $120$ ms and that $75 \%$ of the images are processed in less than $135$ ms.

1. a) Calculate the value of the standard deviation.

1. b) The system enters "alert" mode if the processing time of an image exceeds $150$ ms. If the algorithm independently processes a batch of $5$ images, what is the probability that the system enters "alert" mode at least once?

In an archery competition, the signed deviation of the impact from the center of the target (in cm) follows a normal distribution $X \equiv N(0, \sigma)$. Since the distance cannot be negative, the model takes it as the absolute value of this deviation: $d = |X|$.

1. a) If it is known that $50 \%$ of the impacts are at a distance of less than $4$ cm from the center, calculate the value of $\sigma$.

1. b) Elite archers are those who manage to have $95 \%$ of their shots land less than $2$ cm from the center. What should be the maximum standard deviation of an archer to be considered elite?

The weight of apples from a harvest is normally distributed. It has been verified that $20 \%$ of the apples weigh less than $150$ g and $20 \%$ weigh more than $230$ g.

1. a) Calculate the mean weight and standard deviation of the harvest, reasoning about the symmetry of the curve.

1. b) The apples are classified into three categories: "Small" (less than $130$ g), "Standard" (between $130$ and $250$ g) and "Extra" (more than $250$ g). If a farmer picks $2000$ apples, how many are expected to belong to the "Standard" category?

A study is to be carried out on the height of students on a university campus. It is assumed that height follows a normal distribution $N(\mu, \sigma)$. After preliminary sampling, the following data are obtained:

• $10.56 \%$ of students are less than $165$ cm tall.

• The probability that a student is taller than $180$ cm is $0.3085$.

1. a) Set up a system of equations and calculate the mean height and standard deviation of the students.

1. b) If $3$ students are chosen at random, what is the probability that at least one of them is taller than $190$ cm?