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) $P(Z \le -1,96 \cup Z \ge 1,96)$

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.

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.

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) $P(Z > k) = 0,0228$

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?

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 $m$).

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.