The Standard Normal Distribution N(0, 1)

Observe the following representation of the standard normal distribution. If we know that the area of the shaded region to the right of $z = 1,96$ is $0,025$, reason what the cumulative probability up to that point will be, $P(Z \leq 1,96)$.

In a reflex training app, users' results have been standardized following a standard normal distribution $Z \sim N(0, 1)$. If a player obtains a standardized score of $z = 1,45$, calculate the probability that a randomly chosen user has a lower score than theirs; that is, calculate $P(Z < 1,45)$.

The time it takes students to complete a logic challenge on TikTok follows a standard normal distribution once the data has been standardized. Calculate the probability that a student takes more time than the mean corresponding to a value of $Z = 0,82$. That is, find $P(Z > 0,82)$.

Given a random variable $Z$ that follows a standard normal distribution $N(0,1)$, use the probability table to calculate the following areas under the curve:

1. a) $P(Z \leq 1,48)$
2. b) $P(Z < 0,35)$
3. c) $P(Z \geq 2,12)$
4. d) $P(Z > 1,96)$

Graphically represent the area corresponding to part c) on a bell curve.

A study has been carried out on the monthly savings of young people in a city. After standardizing the variable, it is observed that it follows a $N(0, 1)$ distribution. Using the symmetry properties of the normal distribution, calculate the probability that a young person has a savings level lower than a standardized value of $z = -1,26$.

In an entrance exam for a vocational training program, the scores fit a standard normal distribution. Knowing that the curve is symmetrical with respect to the y-axis, determine the probability that a candidate obtains a standardized score greater than $-0,54$.

Making use of the symmetry properties of the standard normal distribution $N(0,1)$, determine the following probabilities for negative values of the variable:

1. a) $P(Z \leq -0,74)$
2. b) $P(Z < -1,25)$
3. c) $P(Z \geq -0,58)$
4. d) $P(Z > -2,03)$

Calculate the probability that a random variable $Z \sim N(0, 1)$ takes values within the interval $[0,45, \, 1,88]$. Remember that to calculate the area between two positive values $a$ and $b$, you must subtract their cumulative probabilities: $P(a < Z < b) = P(Z < b) - P(Z < a)$.

In the quality control of some electronic components, it is accepted that the measurement error follows a $N(0, 1)$ distribution. Find the probability that the standardized error lies in the symmetrical interval $[-1,5, \, 1,5]$.

Calculate the probability that the random variable $Z \sim N(0,1)$ takes values within the following intervals:

1. a) $P(0,45 \leq Z \leq 1,82)$
2. b) $P(-1,05 < Z < 2,15)$
3. c) $P(-2,30 \leq Z \leq -0,90)$

Explain by means of a drawing why in case c) the result must be equal to the probability of the corresponding positive interval.

Calculate in a reasoned way, using the properties of the standard normal distribution $Z \sim N(0, 1)$, the value of the following probabilities without directly using the table for negative values (only use the table for positive values):

1. a) $P(|Z| \ge 1,25)$
2. b) $P(-0,5 \le Z \le 0,5)$
3. c) $P(Z \le -1,96 \cup Z \ge 1,96)$

A study has been carried out on the reaction time of teenagers to a visual stimulus. After standardizing the data to a $N(0, 1)$, it is observed that $15,87\%$ of the subjects have a very slow reaction time (high values of $Z$).

If we choose two teenagers at random, what is the probability that both have a value of $Z$ between $-1$ and $1$? Assume that the reactions of both are independent.

Prove algebraically that for any variable $Z \sim N(0, 1)$ and any real value $k > 0$, it holds that $P(-k \le Z \le k) = 2 \cdot \Phi(k) - 1$.

Subsequently, use this formula to find the probability that $Z$ is at a distance from the mean greater than $1,645$ standard deviations.

Given a standard normal distribution $Z \sim N(0, 1)$, use the table in reverse to find the value of the constant $k$ such that the probability of obtaining a value less than it is exactly $0,9750$. That is, look for $k$ in: $P(Z < k) = 0,9750$.

In the standard normal distribution $N(0,1)$, it is common to have to take the "reverse path": knowing the probability and looking for the value of the abscissa. Find the value of the constant $k$ in each of the following cases:

1. a) $P(Z \leq k) = 0,8749$
2. b) $P(Z < k) = 0,9913$
3. c) $P(Z \leq k) = 0,0516$
4. d) $P(Z < k) = 0,3336$

Find the value of the abscissa $k$ in a distribution $N(0,1)$ knowing that:

1. a) The area to the right of $k$ is $0,1515$.
2. b) The area to the right of $k$ is $0,9418$.
3. c) $P(Z > k) = 0,0228$

In advanced statistical studies, symmetrical intervals with respect to the origin that enclose a given probability are often sought. Find the positive value of $k$ such that:

1. a) $P(-k \leq Z \leq k) = 0,90$
2. b) $P(-k < Z < k) = 0,95$
3. c) $P(-k \leq Z \leq k) = 0,99$

Note: These values of $k$ are known as critical values and are fundamental for calculating confidence intervals.

A streaming platform classifies its users according to their "loyalty index", which follows a distribution $N(0, 1)$. The system considers "VIP Users" to be those who are in the central interval that concentrates $68,26\%$ of the population. However, for an aggressive marketing campaign, they want to redefine this group.

Find the symmetrical values $-z_{\alpha/2}$ and $z_{\alpha/2}$ that enclose a probability of $0,98$. Explain what percentage of users would be left out of this new group at each end.

Find a positive value $m$ such that the area enclosed under the standard normal curve $N(0, 1)$ between $-m$ and $m$ is three times the total area of the two outer tails (the sum of the areas to the left of $-m$ and to the right of $m$).

Two students from different high schools want to compare their results on an English exam. As it is not the same exam, they use the standardized score $Z$ (which follows a $N(0,1)$) to know who is in a better position with respect to their group.

• Student A has a standardized score of $z_A = 1,2$.

• Student B has a standardized score of $z_B = 0,8$.

Calculate the percentage of students below each of them in their respective school and interpret who has achieved a more outstanding result.

In the world of esports, the skill of players of a famous video game is measured with a standardized score that follows a $N(0, 1)$. The developer company has decided to award an exclusive skin only to players who exceed the 99th percentile. On the other hand, players who are below the value $-2,33$ enter a "basic training" program.

What is the probability that a randomly chosen player does not receive the skin but also does not have to enter the training program?

Given a variable $Z \sim N(0,1)$, calculate the value of $a$ knowing that the probability of the variable being greater than $a$ is seven times greater than the probability of it being less than $a$, where $a$ is a negative value.

Use the relationship $P(Z < a) + P(Z > a) = 1$ and the standard normal tables to help you.

In a standard normal distribution $N(0, 1)$, it is known that there is a positive value $k$ such that the probability that the variable $Z$ is greater than $k$ is exactly a quarter of the probability that $Z$ is less than $k$. Using the symmetry properties of the bell curve and the distribution tables, calculate the value of this parameter $k$ to two decimal places.

Given a random variable $Z \sim N(0, 1)$, calculate the value of the abscissa $a$ knowing that the following probability relationship holds:

$P(Z < a) = 3 \cdot P(a < Z < 1,5)$

Note: You will need to set up an equation using the distribution function $\Phi(z)$ and solve by looking it up in the table.