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 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, $P(Z < -0,45)$.

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 secondary school decides to offer extra support to those students whose standardised 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.

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:

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 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 $z_{\alpha/2}$.

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.