Probabilities with the Standard Normal Distribution

A study on the time young people spend on a well-known short-video social network determines that, once the data is standardized, 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 $1,45$ standardized units. Use the graphical representation to identify the requested area.

In an esports tournament, the reaction time of professional players has been standardized 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 $P(Z > 0,82)$.

The score in a musical aptitude test for teenagers follows a standard normal distribution $Z \sim N(0,1)$. The music department of a high school decides to provide support to those students whose standardized score is below $-1,2$. What is the probability that a randomly selected student needs this support?

The scores of a practice college entrance exam (such as the SAT) have been standardized, obtaining a standard normal distribution. If a student wants to be above the bottom $22,66\%$ of the results, they must obtain a standardized score higher than $-0,75$. Calculate the probability $P(Z > -0,75)$ using the symmetry properties of the normal distribution.

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

1. The probability that a standardized time is greater than $1,32$; that is, $P(Z > 1,32)$.

2. The probability that the time is below $-0,45$; that is, $P(Z < -0,45)$.

A language app records the daily progress of its users. The improvement in vocabulary, after standardizing the data, fits a normal distribution $N(0,1)$. Calculate the probability that a user improves their vocabulary in an interval between $0,5$ and $1,5$ standardized 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 these types of intervals you must apply the formula:

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. $P(-0,8 \leq Z \leq 0)$

In a sports shoe factory, the quality control of the sole's resistance is carried out using a standardized variable $Z \sim N(0,1)$. The engineers want to identify the positive value $k$ such that the probability of a sole having a resistance less than or equal to $k$ is $0,975$. Look up this value of $k$ in the standard normal table.

Given a continuous random variable $Z$ that follows 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. $P(Z > k) = 0,0228$

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 the most relevant posts receive a special mention. From what value of $Z$ is a post considered to be within that top $10\%$? (Find the critical value $z_{\alpha}$ such that $P(Z > z_{\alpha}) = 0,10$).

It has been proven that the noise level in a library, once standardized, follows a distribution $N(0,1)$. If the probability that the noise exceeds a level $k$ is $0,33$, find the value of $k$.

The waiting time in line for a music festival, measured in a standardized way, follows a distribution $N(0,1)$. The organizers want to determine a central interval $(-k, k)$ that contains $95\%$ 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 the quality control of an electronic components factory, the distribution $N(0,1)$ is used to analyze the 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:

Determine the value of $a$ so that the following equality is satisfied 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 has been determined that the additional time users spend follows a standard normal distribution $Z \sim N(0,1)$. If the platform wishes to identify the group of "ultra-active" users, which are those whose standardized time lies in a symmetrical interval around the mean that encloses 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.

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

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

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 of them being selected for the national team is 0.3412. Determine the cutoff score $z_0$ required to join the team.

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

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

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 the intermediate values from the table to four decimal places.

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

• The top 10% of students with the best scores will receive the "Excellence Scholarship".

• The bottom 25% of students with the lowest scores will be excluded from any financial support.

Calculate the probability that a randomly chosen student is not in either of these two extreme groups and their standardized score is also 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 distribution function $\phi(z)$ and look up the value in the table that makes the difference in areas possible.