Exercises: Statistical parameters of a normal distribution

A local council has analysed 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 $\mu$.

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?

In the Selectividad exam for Geography, the marks followed a distribution $N(6,5; 1,2)$, whereas 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' marks.

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.

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.

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?

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 centimetres).

• 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.

Based on climate series from AEMET, the daily maximum temperature during the month of May in two peninsular cities is modeled using normal distributions: in coastal city A it follows $X_A \sim N(21, 2)$ in °C, while in inland city B it follows $X_B \sim N(21, 5)$ in °C.\n\na) Interpret the meaning of the parameters $\mu$ and $\sigma$ in both cities: in which of them is the climate more regular or predictable in May? Relate your answer to the width and height of their respective bell curves.\nb) If an association wants to organize a fun run and needs the temperature not to exceed $25$ °C to prevent heat strokes, in which of the two cities is there a higher probability of exceeding that limit? Reason your answer by comparing how many standard deviations the value $25$ °C lies from the mean in each city, without needing to consult tables.

Lucía analyzes the monthly electricity expenditure of households in her municipality, which follows a normal distribution $X \sim N(60, 12)$ in euros. For next year, the power company announces a tariff update that linearly transforms the monthly cost according to the formula $Y = 1,05 \cdot X + 8$ (a $5\%$ increase in consumption plus a fixed maintenance fee of $8$ €).\n\nLucía states in her report: "Since each bill is multiplied by $1,05$ and $8$ € are added to it, both parameters change in the same way, so the new distribution will be $Y \sim N(1,05 \cdot 60 + 8,\ 1,05 \cdot 12 + 8) = N(71,\ 20,6)$".\n\na) Identify the conceptual error made by Lucía and explain, from the perspective of data dispersion, why adding a fixed amount does not alter the standard deviation.\nb) Correct the calculation and write down the correct normal distribution followed by variable $Y$.

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 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.

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.

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

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) $P(5,8 - (k - 5,8) \leq X \leq k)$

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.

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.

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 $x$?

An olive oil cooperative bottles jugs whose volume in milliliters follows a normal distribution $X \sim N(\mu, 15)$. Labeling regulations require guaranteeing that at least $95\%$ of the jugs contain $1000$ ml or more (that is, $P(X \ge 1000) = 0,95$, or equivalently, $P(X < 1000) = 0,05$).\n\na) Estimate: Before calculating, reason whether the mean $\mu$ to which the machine should be calibrated must be greater than, equal to, or less than $1000$ ml. Knowing that in any normal distribution the interval $(\mu - \sigma,\ \mu + \sigma)$ concentrates $68,26\%$ of the production and $(\mu - 2\sigma,\ \mu + 2\sigma)$ concentrates $95,44\%$, estimate between what approximate values $\mu$ should lie.\nb) Check: Calculate precisely the value that $\mu$ must have knowing that in a standard normal distribution $Z \sim N(0,1)$ it holds that $P(Z \le 1,645) = 0,95$ (and by symmetry, $P(Z \le -1,645) = 0,05$). Compare the result with your previous estimate and assess whether it is consistent with the cooperative's safety margin.

The delivery time (in minutes) for rush-hour deliveries of a logistics company follows a distribution $X \sim N(40, 6)$. Management establishes an operating cost index due to delay defined by the linear transformation $Y = 1,5 \cdot X - 30$ (in euros). We wish to calculate the probability that the cost of a delivery exceeds $45$ €.\n\nTwo team members propose different strategies:\n- Method 1: First determine the parameters of the new variable $Y$ by applying the linear transformation $Y \sim N(a\mu+b,\ a\sigma)$ and then standardize the value $45$ directly on the distribution of $Y$.\n- Method 2: Solve from the inequality $1,5 \cdot X - 30 > 45$ which delivery time in the original variable $X$ is equivalent to that cost, and standardize that value on $X \sim N(40, 6)$.\n\na) Apply both methods detailing the intermediate steps and verify that both lead to exactly the same standardized $Z$ value. Calculate this probability knowing that $P(Z \le 1,67) = 0,9525$.\nb) Discuss with reasoning which of the two methods would be more advantageous and efficient if the company needed to calculate not only that probability, but also the $10$th, $50$th, and $90$th percentiles of the cost $Y$.

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 behaviour on this social network.

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.

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?

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

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 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.

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 $\sqrt{\mu}$).

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"?

In a study on the monthly salary of recent graduates in Social Sciences in Spain, it is assumed that remuneration follows a normal distribution $X \sim N(\mu, \sigma)$. After processing information from two job placement surveys via standardization, the research team sets up the following system of linear equations to calculate the parameters of the distribution:\n$$\begin{cases} \mu - 1,28 \cdot \sigma = 1100 \\ \mu + 0,84 \cdot \sigma = 1600 \end{cases}$$\n\na) Inverse problem (reconstruction): Interpret and deduce which two probabilistic or percentage data about the salaries of $1100$ € and $1600$ € gave rise to this system. Rely on the fact that in a standard normal distribution $Z \sim N(0,1)$, $P(Z \le 1,28) = 0,90$ and $P(Z \le 0,84) = 0,80$ hold.\nb) Solve the system to find the mean salary $\mu$ and standard deviation $\sigma$ of the group.\nc) If the Ministry of Labor were to consider a salary threshold of $1300$ € and it was verified that $25\%$ of graduates receive a salary below that figure, write the linear equation in terms of $\mu$ and $\sigma$ that would model this new data point. (Standard normal datum: $P(Z \le 0,67) = 0,7486 \approx 0,75$).

The waiting time (in days) for scheduled surgery in a regional health service follows a normal distribution $X \sim N(90, 20)$. The guarantee law establishes a maximum period of $120$ days; if the patient waits longer than that time, the administration must refer them to a partnered private center with the resulting public extra cost.\n\na) Model the current situation by calculating the percentage of patients who must be referred for exceeding $120$ days. (Datum: $P(Z \le 1,5) = 0,9332$).\nb) Hospital management has the budget to implement only one of the following two measures:\n- Plan A: Hire operating room reinforcements to reduce the mean by $10$ days, maintaining the same organizational dispersion ($X_A \sim N(80, 20)$).\n- Plan B: Standardize scheduling and preoperative preparation protocols to reduce the standard deviation by half, maintaining the mean ($X_B \sim N(90, 10)$).\nCalculate the new referral percentage that would be obtained with each of the plans. (Data: $P(Z \le 2) = 0,9772$; $P(Z \le 3) = 0,9987$).\nc) Justify mathematically why reducing the standard deviation (Plan B) is much more effective for containing the referral percentage than reducing the mean (Plan A), analyzing how quickly probabilities decay in the tail of a bell curve as you move away from the mean in units of $\sigma$.