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 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.

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:

A student, Irene, is calculating the probability $P(Z \le -1,35)$ for a standard normal variable $Z \sim N(0,1)$. Since the standard normal table only contains positive values, she looks up the value $1,35$, sees that $P(Z \le 1,35) = 0,9115$, and concludes on her exam: "Since it is a negative value, the cumulative probability is $P(Z \le -1,35) = -0,9115$".

1. a) Explain what serious conceptual error Irene makes regarding the definition and fundamental properties of a probability.
2. b) Relying on the symmetry of the bell curve with respect to the vertical axis $Z = 0$, deduce the correct relationship between $P(Z \le -1,35)$ and the areas of the right half of the curve.
3. c) Correct Irene's procedure and calculate the exact numerical value of $P(Z \le -1,35)$.

Consider a standard normally distributed variable $Z \sim N(0,1)$.

1. a) Before carrying out any analytical calculation, make a reasoned estimate: will the probability $P(-0,50 \le Z \le 1,20)$ be strictly greater than, less than, or equal to $0,50$? Justify your answer by analysing what portion of the total area the interval encloses to the left and to the right of the mean $\mu = 0$.
2. b) Verify your estimate by calculating the exact value of $P(-0,50 \le Z \le 1,20)$. Detail each of the symmetry transformations used.

(Data: $P(Z \le 0,50) = 0,6915$; $P(Z \le 1,20) = 0,8849$).

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.

The State Meteorological Agency (AEMET) uses standardised anomalies to classify monthly temperatures with respect to historical baseline series. Suppose that the standardised thermal index of a given month follows a standard normal distribution $Z \sim N(0,1)$. The meteorological protocol classifies a month as "very warm" if its standardised index is strictly greater than $1,40$, and as "cold" if it lies below $-0,80$.

1. a) Model the situation using formal probability notation and calculate the theoretical proportion of months that will be classified as "very warm" in the long term.
2. b) Determine the probability that a randomly chosen month records an intermediate thermal behaviour, that is, that its standardised index belongs to the interval $[-0,80; 1,40]$.

(Data from the $N(0,1)$ table: $P(Z \le 0,80) = 0,7881$; $P(Z \le 1,40) = 0,9192$).

Let $Z$ be a continuous random variable with a standard normal distribution $N(0,1)$.

1. a) Find the positive value $k$ for which $P(-k \le Z \le k) = 0,8064$ holds. Explain the deductive process: state the size of each of the two outer tails and which total cumulative probability you must look up in the body of the table.
2. b) Find the value of the real constant $c$ that satisfies the condition $P(Z \ge c) = 0,7357$. Reason beforehand, without calculating, what sign $c$ must necessarily have based on the symmetry of the bell curve with respect to $0$, and then formalise the calculation of its numerical value.

(Data from the body of the table: $P(Z \le 0,63) = 0,7357$; $P(Z \le 1,30) = 0,9032$; $P(Z \le 1,67) = 0,9525$).

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.

To calculate the probability that a standard normal variable $Z \sim N(0,1)$ deviates from its mean by more than $1,50$ standardised units in absolute value, that is, $P(|Z| > 1,50)$, two classmates propose two different strategies:

• Strategy A (using the complement): Find the area of the central interval using $P(-1,50 \le Z \le 1,50) = P(Z \le 1,50) - P(Z \le -1,50)$ and then calculate $1 - P(-1,50 \le Z \le 1,50)$.

• Strategy B (using direct tail symmetry): Identify that both tails have identical area and calculate directly $2 \cdot P(Z > 1,50) = 2 \cdot [1 - P(Z \le 1,50)]$.

1. a) Formally prove that both algebraic expressions are identical using the property $P(Z \le -z) = 1 - P(Z \le z)$.
2. b) Compare both approaches and justify which one is more advantageous to minimise the risk of making sign errors and to reduce the number of table lookups.
3. c) Using the strategy you considered optimal, calculate $P(|Z| > 1,50)$ knowing that $P(Z \le 1,50) = 0,9332$.
4. d) Without performing long arithmetic operations, deduce in a reasoned manner the value of the conditional probability $P(Z > 1,50 \mid |Z| > 1,50)$ relying exclusively on the geometric symmetry of the regions involved.

In a diagnostic assessment of competencies in the educational system, the overall scores have been standardised as a standard normal variable $Z \sim N(0,1)$.

1. a) A technical committee proposes defining a "representative performance" band using an interval of fixed width equal to $1$ unit of the form $[a; a+1]$. Prove qualitatively, from the bell-shaped and unimodal geometry of the Gaussian density function, what value $a$ must take so that this interval encloses the maximum possible probability. Calculate this maximum probability knowing that $P(Z \le 0,50) = 0,6915$.
2. b) The administration wishes to divide the assessed students into three equally likely levels ("Initial", "Medium", and "Advanced"), such that each group contains exactly one third of the population (probability $1/3 \approx 0,3333$). The thresholds are set as: Initial for $Z < z_1$, Medium for $z_1 \le Z \le z_2$, and Advanced for $Z > z_2$.

1. Prove that, by symmetry with respect to the mean $0$, it must necessarily hold that $z_1 = -z_2$.

2. Set up the cumulative probability equation that allows you to determine $z_2$ and find its value to two decimal places knowing that $P(Z \le 0,43) = 0,6664$ and $P(Z \le 0,44) = 0,6700$.

1. c) If instead of three levels the distribution were divided into four equally likely quartiles using the cut-off points $Q_1$, $Q_2$, and $Q_3$, reason geometrically why $Q_2 = 0$ and justify why it is sufficient to look up a single value in the body of the table to determine all three quartiles.