Continuous Probability Distributions and the Normal Curve

It is known that the time it takes for a smartphone model to charge its battery from 0% to 100% follows a continuous probability distribution whose probability density function is constant in the interval $[80, 120]$ minutes, and is $0$ at any other point.

1. a) Determine the value of the probability density function in this interval for it to be valid.
2. b) Calculate the probability that the phone takes less than 90 minutes to charge.
3. c) Represent the probability density function graphically.

The time it takes a group of young people to completely disconnect after turning off their cell phones before sleeping is being studied. The probability density function of this continuous random variable, $x$, expressed in minutes, is given by the following graph:

Calculate, justifying your answer:

1. a) The value of the maximum ordinate $k$ for it to be a valid probability density function.
2. b) The probability that a young person takes between 15 and 20 minutes to disconnect.
3. c) What is the probability that they take exactly 22 minutes?

Consider the piecewise continuous probability function of a random variable $X$, which represents the time (in hours) that a group of teenagers spends studying math over a weekend:

$f(x) = \begin{cases} kx & \text{if } 0 \le x < 2 \\ k(4 - x) & \text{if } 2 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}$

1. a) Determine logically the value of the constant $k$ so that $f(x)$ is a probability density function.
2. b) Calculate the probability that a student chosen at random studies between 1.5 and 3 hours.
3. c) What is the probability that they study exactly 2 hours? Justify your answer based on the nature of the variable.

Observe the following two curves corresponding to normal distributions. Both represent the weight of two different breeds of domestic cats.

1. a) Do both distributions have the same mean $\mu$? Justify your answer by observing the position of the maximum.
2. b) Which of the two distributions has a larger standard deviation $\sigma$? Explain how this parameter influences the shape of the "bell".

Using the standard normal distribution table $Z \sim N(0, 1)$, obtain the following direct probabilities:

1. a) $P(Z < 1,25)$
2. b) $P(Z \leq 0,47)$
3. c) $P(Z < 2,03)$

Briefly explain what the value obtained in part (a) represents with respect to the total area under the Gaussian curve.

In a standard normal distribution $Z \sim N(0, 1)$, calculate the following probabilities using the properties of symmetry and the complementary event:

1. a) $P(Z > 1,12)$
2. b) $P(Z < -0,54)$
3. c) $P(Z \geq -1,96)$

Draw a Gaussian bell curve for part (b), shading the area you are calculating.

Calculate the area enclosed in the following intervals for a standard normal distribution $Z \sim N(0, 1)$:

1. a) $P(0,5 < Z < 1,5)$
2. b) $P(-1,2 < Z < 1,2)$

Represent the interval from part (b) graphically, indicating why the area is symmetrical with respect to the vertical axis.

The score in a popular mental agility test for teenagers follows a normal distribution $X \sim N(100, 15)$.

1. a) Standardize the value $X = 115$ and explain what the obtained result means (how many standard deviations from the mean it is).
2. b) Standardize the value $X = 80$. What does the negative sign indicate in this case?

Two friends, Lucy and Mark, play on different esports teams. Lucy's score in the game "Galaxy Quest" follows a distribution $N(1200, 15)$, while Mark's in "Pixel Arena" follows a distribution $N(850, 10)$.

In the last match, Lucy scored 1,235 points and Mark scored 875 points. Use standardization of variables to determine which of the two obtained a more meritorious result compared to their group.

The battery life of mid-range wireless headphones follows a normal distribution with a mean of 6 hours and a standard deviation of 0.5 hours ($X \sim N(6; 0,5)$).

Calculate the probability that, when choosing headphones at random, the battery lasts less than 7 hours. (Remember to standardize the variable first to be able to use the table).

The height of 11th-grade students at a high school follows a normal distribution with a mean of $172$ inches and a standard deviation of $8$ inches. If we choose a student at random, find the probability that:

1. a) They are taller than $185$ inches.
2. b) They are shorter than $160$ inches.
3. c) Their height is between $165$ and $180$ inches.

A company that manufactures wireless headphones has verified that the battery life follows a normal distribution with a mean of $24$ hours and a standard deviation of $3,5$ hours.

1. a) What percentage of headphones have a battery that lasts between $20$ and $28$ hours?
2. b) If the company manufactures a batch of $5000$ headphones, how many are expected to last less than $18$ hours?

Look up the value of the x-coordinate $k$ in the standard normal distribution table $N(0, 1)$ in the following cases (inverse problem):

1. a) $P(Z < k) = 0,8413$
2. b) $P(Z < k) = 0,9750$
3. c) $P(Z < k) = 0,5000$

What happens to the value of $k$ as the probability approaches 1? Justify your answer by observing the shape of the bell curve at its tails.

Using the standard normal distribution table $N(0, 1)$, calculate the following probabilities:

1. a) $P(Z < 1,48)$
2. b) $P(Z > 0,72)$
3. c) $P(Z < -1,15)$
4. d) $P(-0,5 < Z < 1,2)$

Furthermore, find the value of the positive constant $a$ such that $P(-a < Z < a) = 0,90$.

On a music streaming platform, the time users spend listening to a "Top 50" playlist follows a distribution $N(45, 12)$ minutes. The platform decides to send a satisfaction survey to the $15\%$ of users who spend the most time listening to this playlist. After how many minutes of listening will a user receive the survey?

The weight of the chocolate bars produced by a factory follows a normal distribution with a mean of $200$ ounces. Quality control rejects those bars that weigh less than $195$ ounces. Knowing that $3\%$ of the bars are rejected, calculate the value of the standard deviation of this production.

In a prestigious design school, the final grade of the applicants follows a normal distribution. It is known that the mean grade is $6,8$ and that $33\%$ of the students obtain a grade higher than $7,5$.

1. a) Find the standard deviation of the grades.
2. b) If earning a merit scholarship requires being in the top $5\%$ of grades, what is the minimum grade needed to get this scholarship?

A logistics company analyzes the delivery time of its local packages. The delivery time follows a normal distribution with a standard deviation of $12$ minutes. It is known that $75\%$ of the packages are delivered in less than $50$ minutes.

1. a) Calculate the mean delivery time.
2. b) The company guarantees a shipping refund if the package takes more than an hour ($60$ minutes) to arrive. If $5000$ packages are sent in a month, how many refunds is the company estimated to have to make?
3. c) If the company wants to halve the number of refunds by improving logistics but keeping the same mean, what should the new standard deviation be?

The latency (ping) in a popular online video game follows a normal distribution $N(\mu, \sigma)$. After taking thousands of measurements, the developers have observed that $15,87\%$ of the games have a latency greater than $110$ ms, while $6,68\%$ of the games have a latency of less than $75$ ms.

1. a) Calculate the parameters $\mu$ (mean) and $\sigma$ (standard deviation) of this distribution.
2. b) If a player considers a game to be "unplayable" when the latency exceeds $125$ ms, what percentage of games will fall into this category?

A technology company manufactures precision sensors for drones. The margin of error in the height measurement of these sensors follows a normal distribution with a mean of $0$ inches and a standard deviation of $1,5$ inches. For safety reasons, the company discards all sensors whose absolute error is greater than a critical value $k$.

1. a) If it is known that $10\%$ of the total production is discarded, calculate the value of this margin of error $k$.
2. b) What is the probability that, when choosing an accepted (not discarded) sensor, its error is positive?

Two models of batteries for electric scooters are being compared. Model A has a mean range of $40$ miles with a standard deviation of $4$ miles. Model B has a mean range of $42$ miles with a standard deviation of $8$ miles. Both durations follow normal distributions.

1. a) If a user needs to make a trip of $48$ miles without recharging, which battery model offers them greater reliability (higher probability of success)? Justify the calculations.
2. b) For what distance is the probability of both models failing the same?

A survey on social media use has been carried out and it is known that $30\%$ of American teenagers prefer to use TikTok over other platforms. If we select a random sample of $200$ teenagers:

1. a) Justify whether it is possible to approximate the distribution of the variable "number of teenagers who prefer TikTok" using a normal distribution.
2. b) Using this approximation (with Yates's continuity correction), calculate the probability that more than $70$ teenagers prefer TikTok.

A music streaming app has launched a new recommendation algorithm. It is estimated that the probability of a user liking a recommended song is $65\%$. If in one day the algorithm recommends songs to a group of $400$ users:

1. a) Justify whether it is possible to approximate the distribution of the number of users who like the recommendation using a normal distribution.
2. b) Using the appropriate approximation (with continuity correction), calculate the probability that the recommendation is successful for a number of users between $250$ and $275$ (inclusive).

The weight of the recycled plastic bottles collected at a sustainability center follows a distribution $N(120, 15)$ ounces. The center classifies the bottles into three categories: "Light" (the lightest $20\%$), "Standard" (the middle $60\%$), and "Heavy" (the heaviest $20\%$).

1. a) Determine the weights that delimit each of the three categories.
2. b) If we choose $5$ bottles at random, what is the probability that exactly $3$ of them are classified as "Standard"?