Parameters of a Normal Distribution

A courier company has analyzed the time it takes its delivery drivers to deliver a package from the moment they leave the warehouse. They have found that this time follows a normal distribution with a mean of $25$ minutes and a standard deviation of $4$ minutes. Identify the statistical parameters of this distribution and write it using the abbreviated mathematical notation $X \sim N(\mu, \sigma)$.

A nutritionist claims that the daily sugar consumption of teenagers in a city follows a normal distribution. If we denote this variable as $X \sim N(45, 8)$, where the values are in grams:

• What is the expected value of sugar consumption for a randomly chosen teenager?

• Which parameter indicates whether the data is closely grouped around this value or widely spread out?

In the college entrance exam, the scores in Geography followed a distribution $N(6.5, 1.2)$, while in History they followed a $N(7.2, 1.2)$.

• Which parameter is identical in both subjects?

• Explain what this parameter means in the context of the "dispersion" of the students' scores.

Observe the following graph representing the distribution of the weights of newborn kittens in an animal shelter. Knowing that the distribution is symmetrical, visually indicate the value of the population mean $\mu$.

The scores obtained on a practice AP exam follow a normal distribution $X \sim N(5,8; 1,2)$. Without using the normal distribution table for this section, and based on the symmetry properties of the Gaussian bell curve, if we know that $P(X \leq k) = 0,72$, calculate with justification:

1. a) $P(X > k)$
2. b) $P(X \leq 5,8)$
3. c) $P(5,8 - (k - 5,8) \leq X \leq k)$

The time it takes a student to complete a standardized math exam follows a normal distribution. It is known that the mean time is $70$ minutes. If the probability of a student finishing the exam in less than $60$ minutes is the same as the probability of taking more than $k$ minutes, and it is known that the variance of the distribution is $25$, determine the value of $k$ using the properties of the parameters of the normal distribution.

Two brands of light bulbs, "EverLight" and "BrightMax", have a mean lifespan of $1200$ hours. However, the standard deviation of "EverLight" is $50$ hours, while that of "BrightMax" is $150$ hours.

• Which of the two brands offers a more "homogeneous" or regular product in terms of its lifespan?

• If you had to draw both Gaussian bell curves on the same axis, which of them would be taller and narrower?

Below are two normal distribution curves representing the charging time of two different smartphone models. Both have the same standard deviation. Which of the two models (A or B) has a greater population mean $\mu$? Justify your answer based on the position of the curves on the axis.

Imagine you are comparing the precision of two machines that cut wooden slats. Machine 1 cuts with a distribution $N(100, 0.1)$ and Machine 2 with a distribution $N(100, 0.5)$ (in inches).

• Which of the two machines is more precise?

• Relate the concept of "precision" to the value of the standard deviation $\sigma$ in a normal distribution.

The Standard Normal distribution is a special case widely used in statistics to be able to use area tables. State the fixed values of its mean ($\mu$) and its standard deviation ($\sigma$). Represent it symbolically.

The full charge time of a new model of electric scooter follows a normal distribution $N(240, 20)$, where time is measured in minutes.

A teenager claims that his scooter usually takes less than $215$ minutes to charge.

1. a) Calculate the probability that a randomly chosen scooter takes less than those $215$ minutes.
2. b) If the manufacturer improves the battery and the new distribution becomes $N(230, 15)$, has the probability of it taking less than $215$ minutes increased or decreased? Justify your answer by calculating the new probability.

Below are two graphs representing the distribution of the lifespan of two types of LED bulbs from different brands, A and B. Both follow a normal distribution.

1. a) Observing the shape of the curves, which of the two brands has a greater standard deviation? Justify your answer by relating the dispersion to the shape of the bell curve.
2. b) If brand A has a distribution $N(25000, 2000)$ and brand B has $N(28000, 3500)$, calculate the probability that a brand A bulb lasts longer than the mean lifespan of brand B.

The weight of the backpacks of students at a high school follows a normal distribution with a mean of $\mu = 6,2$ pounds and a standard deviation of $\sigma = 1,4$ pounds. Determine the characteristic interval centered on the mean that comprises $90\%$ of the students' backpack weights. Interpret the result obtained in the context of the problem.

A manufacturer of components for space telescopes produces lenses whose diameter follows a normal distribution. From quality control, it is known that the mean is $120$ inches. For a lens to be accepted, its diameter must be in the range $[118, 122]$ inches. If the manufacturer wants $99,73\%$ of its production to be acceptable to avoid economic losses, what must be the maximum value of the standard deviation of the manufacturing process?

In an e-sports championship, the score obtained by players in a qualifying phase follows a normal distribution $N(\mu, \sigma)$. The organizing committee has established that the characteristic interval centered on the mean for $95,44\%$ of the players is $[1250, 1750]$ points.

1. Calculate the mean and variance of the players' scores.

2. If qualifying for the final requires being in the "top" $5\%$ of the best players, what is the minimum score required?

In a popular strategy video game, the experience points obtained by players in a specific mission follow a normal distribution with a mean of $\mu = 4500$ points and a standard deviation of $\sigma = 800$ points. If the developers want to award a special medal only to the $15\%$ of players who achieve the highest scores, from how many points will this medal start to be awarded?

At a ski resort, the snow depth during the month of January follows a normal distribution with a mean of $85$ inches and a standard deviation of $18$ inches.

1. a) What is the probability that on a day in January the depth is greater than $100$ inches?
2. b) If a minimum depth of $x$ inches is required to open a black diamond trail, and it is known that this trail is open on $70\%$ of the days in January, what is this minimum depth $x$?

A study has been carried out on the time that young people between $16$ and $18$ years old spend daily on a well-known short-video social network. It is known that this time follows a normal distribution with a mean of $\mu = 110$ minutes. If it is known that $33\%$ of users spend more than $125$ minutes a day on the platform, calculate the value of the standard deviation $\sigma$ of this distribution.

An express courier company guarantees that the delivery time of its packages in a city follows a normal distribution with a standard deviation of $12$ minutes. After analyzing thousands of shipments, it is observed that only $2,5\%$ of the packages take more than $75$ minutes to be delivered. From this information, calculate the mean delivery time ($\mu$) of the company.

The annual carbon footprint (in tons of $CO_2$) of households in an American city follows a normal distribution. It has been observed that $33\%$ of households emit less than $4,5$ tons and $33\%$ of households emit more than $6,2$ tons.

1. Justify whether the distribution is symmetrical and calculate the mean carbon footprint of the city.

2. Determine the standard deviation of the carbon footprint of these households.

The algorithm of a well-known social network analyzes the time users spend watching "reels". It has been found that this time follows a normal distribution $X \sim N(\mu, \sigma)$. Data analysts have determined that only $15,87\%$ of users spend more than $45$ minutes a day, while $2,28\%$ spend less than $10$ minutes. Calculate, with reasoning, the mean time spent ($\mu$) and the standard deviation ($\sigma$) of user behavior on this social network.

A study on sleep in teenagers indicates that the number of daily hours of rest follows a normal distribution. If the value of the variance is numerically equal to the mean, and it is known that the probability of sleeping more than $10,36$ hours is only $0,0228$, calculate the value of the mean ($\mu$) and the standard deviation ($\sigma$). (Note: You will need to solve a quadratic equation for $\sqrt{\mu}$).

Two music streaming platforms, "SoundCloud" and "Melody", compete for the daily usage time of their young subscribers.

• On "SoundCloud", the time follows a $N(45, 10)$.

• On "Melody", the time follows a $N(40, 15)$.

If a user is considered "highly active" when they exceed $65$ minutes of daily listening:

1. On which platform is it more likely to find a "highly active" user?

2. What value should the standard deviation of "Melody" have so that, keeping its mean at $40$, the percentage of "highly active" users is exactly the same as on "SoundCloud"?

A sustainable courier company uses electric scooters for its deliveries. The battery range of these scooters follows a normal distribution with a mean of $60$ miles. It is known that the probability of a scooter having a range greater than $75$ miles is $0,0668$. Due to a software update, the range of all scooters increases by $10\%$ plus a fixed extra of $2$ miles. Determine the original standard deviation of the batteries and the new parameters (mean and standard deviation) after the software update.