Statistical Parameters of a Normal Distribution

A local municipality has analyzed the time taken by buses on the circular route to complete their journey. They have confirmed 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 $X \sim N(\mu, \sigma)$.

Look at the following graph, which represents the weight distribution of newborn kittens in an animal shelter. Knowing that the distribution is symmetrical, visually state the value of the population mean $\mu$.

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 are closely clustered around that value or widely spread out?

Two normal distribution curves are shown below, 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 larger population mean $\mu$? Justify your answer based on the position of the curves on the axis.

Two brands of lightbulbs, "LuzEterna" and "BrilloMax", have a mean lifespan of $1200$ hours. However, the standard deviation of "LuzEterna" is $50$ hours, whereas that of "BrilloMax" is $150$ hours.

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

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

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 centimeters).

• 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.

Two graphs are presented below showing the distribution of the lifespan of two types of LED lightbulbs from different brands, A and B. Both follow a normal distribution.

1. a) Looking at the shape of the curves, which of the two brands has a larger 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 a $N(28000, 3500)$, calculate the probability that a lightbulb from brand A lasts longer than the mean lifespan of brand B.

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

The marks obtained in a mock exam of Mathematics Applied to the Social Sciences follow a normal distribution $X \sim N(5.8; 1.2)$. Without using the normal table for this section, and based on the symmetry properties of the Gaussian bell curve, if we know that $P(X \leq k) = 0.72$, work out with full 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 taken by a student to complete a University Entrance Exam in Mathematics follows a normal distribution. It is known that the mean time is $70$ minutes. If the probability that a student finishes the exam in less than $60$ minutes is the same as the probability that they take 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.

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

A teenager claims that their 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 taking less than $215$ minutes increased or decreased? Justify your answer by calculating the new probability.

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

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

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 exceeding $65$ daily minutes of listening:

1. On which platform is a "highly active" user more likely to be found?

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

A study has been carried out on the time that young people aged between $16$ and $18$ 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. Given 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.

A courier company guarantees that the delivery time of its parcels in a city follows a normal distribution with a standard deviation of $12$ minutes. After analysing thousands of shipments, it is observed that only $2.5\%$ of the parcels take more than $75$ minutes to be delivered. From this information, calculate the mean delivery time ($\mu$) of the company.

In a popular strategy video game, the experience points obtained by players on 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 upwards will this medal be awarded?

The algorithm of a well-known social network analyses the time users spend watching "reels". It has been shown 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, giving clear reasoning, the mean watch time ($\mu$) and standard deviation ($\sigma$) of user behavior on this social network.

The annual carbon footprint (in metric tons of $CO_2$) of households in a European city follows a normal distribution. It has been observed that $33\%$ of households emit less than $4.5$ metric tons and $33\%$ of households emit more than $6.2$ metric 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.

A study on adolescent sleep indicates that the daily number of 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 $\sqrt{\mu}$).

The weight of the backpacks of students at a high school follows a normal distribution with mean $\mu = 6.2$ kg and standard deviation $\sigma = 1.4$ kg. Determine the characteristic interval centered on the mean that contains $90\%$ of the student backpack weights. Interpret the result obtained in the context of the problem.

In an e-sports championship, the score obtained by players in a qualifying phase follows a normal distribution $N(\mu, \sigma)$. The organising committee has established that the characteristic interval centred 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 reaching the final requires being in the top $5\%$ of players, what is the minimum required score?

A manufacturer of components for space telescopes produces lenses whose diameter follows a normal distribution. Through quality control, it is known that the mean is $120$ mm. For a lens to be accepted, its diameter must be in the range $[118, 122]$ mm. If the manufacturer wants $99.73\%$ of its production to be acceptable in order to avoid financial losses, what must be the maximum value of the standard deviation of the manufacturing process?

A sustainable courier company uses electric scooters for its deliveries. The battery range of these scooters follows a normal distribution with a mean of $60$ km. It is known that the probability of a scooter having a range greater than $75$ km is $0.0668$. Due to a software update, the range of all scooters increases by $10\%$ plus a fixed extra of $2$ km. Determine the original standard deviation of the batteries and the new parameters (mean and standard deviation) after the software update.