Exercises: Probabilities of an N(0,1)

A study on the time young people spend on a well-known short video social network shows that, once the data are standardised, the variable follows a standard normal distribution $Z \sim N(0,1)$. If we select a user at random, calculate the probability that their usage time is less than or equal to $1.45$ standardised units. Use the graphical representation to identify the required area.

In an esports tournament, the reaction time of professional players has been standardised so that it follows a distribution $N(0,1)$. If any value above $0.82$ is considered a "slow reaction time", find the probability that a randomly chosen player has a slow reaction time, that is, calculate $P(Z > 0.82)$.

Scores on a musical aptitude test for teenagers follow a standard normal distribution $Z \sim N(0,1)$. The music department of a high school decides to offer extra support to those students whose standardized score is less than $-1.2$. What is the probability that a randomly chosen student needs this extra support?

A language application logs the daily progress of its users. Vocabulary improvement, after standardising the data, fits a normal distribution $N(0,1)$. Calculate the probability that a user improves their vocabulary within an interval between $0.5$ and $1.5$ standardised units.

In a trainer factory, quality control of sole durability is performed using a standardised variable $Z \sim N(0,1)$. The engineers want to identify the positive value $k$ such that the probability that a sole has a durability less than or equal to $k$ is $0.975$. Look up this value of $k$ in the standard normal table.

The queuing time at a music festival, measured in standardised form, follows a distribution $N(0,1)$. The organisers want to determine a central interval $(-k, k)$ containing $95\%$ of the attendees. Knowing that $P(-k \le Z \le k) = 0.95$, use the symmetry of the Gaussian bell curve to find the corresponding critical value $z_{\alpha/2}$.

The marks on a mock Selectividad (EBAU) exam have been standardised, yielding a standard normal distribution. If a student wants to score above the bottom $22.66\%$ of results, they must obtain a standardised score greater than $-0.75$. Calculate the probability $P(Z > -0.75)$ using the symmetry properties of the normal distribution.

Given a continuous random variable that follows a standard normal distribution $Z \sim N(0,1)$, calculate the probability that the variable takes values in the interval $[-1,1, 0.4]$. Remember that to solve this type of interval you must apply the formula:

In a study on the performance of a new artificial intelligence, response times have been standardised following a standard normal distribution $N(0,1)$. Calculate the following probabilities using the normal distribution table:

1. The probability that a standardised time is greater than $1.32$; that is, $P(Z > 1.32)$.

2. The probability that the time is below $-0.45$; that is, $P(Z < -0.45)$.

Scores on an aptitude test for 18-year-olds have been standardised so that they follow a distribution $N(0,1)$. If we select an applicant at random, determine the probability that their standardised score lies in the interval $[-1,24, 2.13]$.

Given a continuous random variable $Z$ following a standard normal distribution $N(0,1)$, find the value of the positive constant $k$ that satisfies each of the following conditions:

1. $P(Z \leq k) = 0.8708$

2. $P(Z > k) = 0.0228$

In quality control at an electronic components factory, the distribution $N(0,1)$ is used to analyse deviations from the nominal diameter. Calculate the probability that a randomly chosen part has a deviation (in absolute value) less than or equal to $1.96$ standard units. That is, find:

A social media algorithm classifies the relevance of posts using a variable $Z \sim N(0,1)$. It is known that the top $10\%$ of most relevant posts receive a special mention. Above what value of $Z$ is a post considered to be within this top $10\%$? (Find the critical value $z_{\alpha}$ such that $P(Z > z_{\alpha}) = 0.10$).

Using the symmetry properties of the Gaussian bell curve for a standard normal distribution $Z$, calculate the following probabilities without looking up negative values in the table:

1. $P(Z > -1.5)$

2. $P(-0.8 \leq Z \leq 0)$

It has been verified that the noise level in a library, once standardised, follows a distribution $N(0,1)$. If the probability that the noise exceeds a level $k$ is $0.33$, find the value of $k$.

Determine the value of $a$ so that the following equality holds in a standard normal distribution $N(0,1)$:

Explain the graphical meaning of this result in relation to the central area of the Gaussian curve.

In a study on the efficiency of a new recommendation algorithm for a social network, it was determined that the additional time spent by users follows a standard normal distribution $Z \sim N(0,1)$. If the platform wishes to identify the group of "ultra-active" users, who are those whose standardised time lies within a symmetrical interval about the mean containing exactly 88.82% of the population, calculate the critical value $k$ that defines the limits of this interval.

Calculate the value of the positive constant $a$ knowing that, in a standard normal distribution $Z \sim N(0,1)$, the probability that the variable takes values in the interval between $-a$ and $a$ is three times the probability that it takes values greater than $a$. Use the symmetry properties of the Gaussian bell curve to set up an equation before consulting the table.

A data analyst for an esports league is studying the "reflex index" of professional players, which is distributed according to an $N(0,1)$. It is known that a player is selected for the national team if their index is above a value $z_0$. If a player is chosen at random and it is known that their index is positive, the probability that they are selected for the national team is 0.3412. Determine the value of the cutoff score $z_0$ required to make the team.

Given a continuous random variable $Z$ following a normal distribution with mean $0$ and standard deviation $1$, determine the value of $k$ such that the following equality holds:

Explain the process with full justification, indicating how you use the complementary event rule and the symmetry of the distribution.

In a distribution $N(0,1)$, consider the points $z_1$ and $z_2$ such that $z_1$ is the 20th percentile and $z_2$ is the value that leaves a probability of 0.15 to its right. Calculate the probability that the variable $Z$ lies in the interval $[z_1, z_2]$. Round intermediate values from the table to four decimal places.

Let $Z$ be a random variable with a standard normal distribution. Determine the value of the constant $c > 0$ so that the following condition regarding the modulus of the variable is satisfied:

Next, using the value of $c$ found, calculate the conditional probability $P(Z > -c \text{ | } Z < c)$.

An academic aptitude test has been designed whose results follow an $N(0,1)$. The scholarship awarding institution decides that:

• The top 10% of students with the highest marks will receive the "Excellence Scholarship".

• The bottom 25% of students with the lowest marks will be excluded from any aid.

Calculate the probability that a randomly chosen student is in neither of these two extreme groups and that their standardised mark is, moreover, above the mean.

Let $Z \sim N(0,1)$. Find the value of $k$ knowing that the area of the shaded region in the figure below (which corresponds to the probability $P(k < Z < 2k)$) is equal to 0.1359.

Note: You must express the probabilities in terms of the cumulative distribution function $\phi(z)$ and look up in the table the value that yields this difference in areas.