Exercises: Mean and standard deviation of the BIN

A recent study on digital habits indicates that $85\%$ of Spanish teenagers aged 15 to 16 use two-factor authentication on their video game accounts. If we select a random sample of $20$ teenagers:

1. a) Identify the parameters $n$ and $p$ of the binomial distribution that models this situation.
2. b) Calculate the expected number (mean) of teenagers who use this security system in the sample.

A music streaming platform has verified that the probability of a user "skipping" an algorithm-recommended song before it ends is $0,3$. If one afternoon you listen to a playlist of $12$ recommended songs:

1. a) Determine the mean and variance of the number of songs you are expected to skip.
2. b) Calculate the standard deviation and interpret what information it provides regarding data dispersion in this context.

In a vocational training college, $40\%$ of the students use public transport to get to class. A random sample of $50$ students from the college is taken. Calculate the mean and standard deviation of the number of students who use public transport in this sample.

A professional basketball player has a free-throw success rate of $80\%$. If she takes a series of $15$ shots in a training session:

1. a) What is the expected mean number of baskets she will score?
2. b) Calculate the standard deviation. What would happen to the standard deviation if her probability of scoring were $50\%$? Verify this by calculation.

Consider a discrete random variable $X$ that follows a binomial distribution $B(n, p)$. If we know that the number of trials is $n = 100$ and that the mean of the distribution is $\mu = 60$:

1. a) Find the value of the probability of success $p$.
2. b) Calculate the variance of this distribution.

In a factory manufacturing smartphone components, it is known that $2\%$ of the touchscreens have a small cosmetic defect. The screens are sent to shops in boxes of $250$ units.

1. a) Calculate the mean number of defective screens found in a box.
2. b) Calculate the standard deviation of the variable "number of defective screens per box".

A fair six-sided die is rolled a total of $30$ times, and the number of times the number "6" comes up is recorded.

1. a) Define the random variable and its distribution.
2. b) Calculate the expected value of the variable and its standard deviation. Round the results to two decimal places.

A multiple-choice exam consists of $10$ questions, each with three possible answers, of which only one is correct. If a student answers all questions at random:

1. a) What is the mean number of correct answers?
2. b) If the student needs at least $5$ correct answers to pass, is the mean value above or below the pass mark? Justify your answer.

Sofía is analysing a batch of $n = 100$ tax returns selected at random from a tax office branch, where the probability that a return contains incorrect deductions is $p = 0{,}2$. To summarise the behaviour of the sample, she performs the following calculation in her notebook:

«Mean number of tax returns with errors: $\mu = 100 \cdot 0{,}2 = 20$.

Standard deviation: $\sigma = \sqrt{100 \cdot 0{,}2} = \sqrt{20} \approx 4{,}47$ returns».

1. a) Identify the conceptual error that Sofía made and explain why her reasoning is incorrect.
2. b) Correct the calculation and find the true standard deviation $\sigma$ of the distribution.
3. c) Determine the interval of typical values $(\mu - \sigma,\ \mu + \sigma)$ using the correct value and interpret its meaning for the branch inspector.

According to data from the National Statistics Institute (INE), $30\,\%$ of Spanish households live in rented accommodation. For a local sociological study, a random sample of $n = 80$ households from a municipality is selected.

1. a) Define the random variable $X$ that counts the number of rented households in the sample and state the parameters of the binomial distribution that models it.
2. b) Calculate the expected mean number of rented households and the standard deviation of the distribution.
3. c) Determine the interval $(\mu - \sigma,\ \mu + \sigma)$ of most likely outcomes. If $35$ rented households are finally found in the chosen sample, can it be considered an atypical result according to this interval? Justify your answer.

On a popular social network, it has been estimated that the probability of a user interacting with a post containing cultural content is $p = 0,18$. If an influencer launches a campaign that reaches a group of $n = 150$ followers:

1. a) Calculate the expected number of followers who will interact with the post and its standard deviation.

1. b) If the cost of the campaign is a fixed $50$ € plus $2$ € for each expected interaction (based on the mean), what is the expected budget for this advertising action?

A discrete random variable follows a binomial distribution $X \sim B(n, p)$. It is known that its mean is $\mu = 12$ and its standard deviation is $\sigma = 3$.

1. a) Determine the values of the parameters $n$ (number of trials) and $p$ (probability of success).

1. b) Justify whether this distribution could be approximated by a normal distribution, based on the values obtained.

Two basketball players, Alba and Bruno, practice free throws. Alba has a scoring probability of $0,8$ and shoots $50$ throws. Bruno has a probability of $0,6$ but shoots $80$ throws.

1. a) Which of the two players has a higher expected number of baskets?

1. b) Which of the two players shows greater dispersion (standard deviation) in their results? Interpret what this means in terms of consistency in the game.

A logistics company uses drones to deliver packages in urban areas. The probability that a drone suffers a delay due to weather conditions is $5\%$. If $200$ independent flights are conducted on a given day:

1. a) Calculate the value of the mean and variance of the number of flights that arrive late.

1. b) The maintenance protocol is activated if on a given day the number of delays exceeds the value $\mu + 2\sigma$. Calculate this threshold value.

In a survey on reading habits conducted with $60$ teenagers, it is known that the expected number of regular digital press readers is $21$.

1. a) Find the probability $p$ that a randomly chosen teenager is a regular digital press reader.

1. b) Calculate the standard deviation of this distribution.

1. c) If we select another sample where the number of teenagers is doubled ($n=120$), how does the standard deviation change relative to the original sample? Justify your answer with calculations.

It has been verified that $40\%$ of students at a language centre sit the official B2 level exam. If there are $100$ students enrolled this year:

1. a) Determine the parameters of the variable $X$ defining the number of students sitting the exam.

1. b) In binomial distributions, the most "typical" or likely values are considered to lie within the interval $(\mu - \sigma, \mu + \sigma)$. Calculate this interval for this specific case.

The quality control department of an LED light bulb factory knows that the probability of a unit being defective is $p = 0,02$. The light bulbs are packaged in boxes of $n$ units.

1. a) If it is desired that the mean number of defective light bulbs per box is exactly $1,6$, how many light bulbs should each box contain?

1. b) With that value of $n$, calculate the probability that a box contains no defective light bulbs.

Let $X$ be a random variable following a binomial distribution. If the number of trials is $n = 25$ and the variance is $Var(X) = 4$:

1. a) Calculate the two possible values of the probability of success $p$.

1. b) For each value of $p$ found, calculate its corresponding mean $\mu$.

1. c) If we know that the distribution is positively skewed (that is, success is less likely than failure), which value of $p$ would you choose? Justify your answer.

In an electronic components factory, the probability of a microchip being defective on the assembly line is $p = 0{,}4$. A quality control sample of $n = 5$ microchips is taken.

To calculate the mean and standard deviation of the number of defective parts in the sample, two analysts propose different methods:

• Method A: Develop the probability distribution table by calculating $P(X = k)$ for each $k \in \{0, 1, 2, 3, 4, 5\}$ using the binomial formula, and then apply the general formulas for discrete variables: $\mu = \sum x_i \cdot P(X = x_i)$ and $\sigma = \sqrt{\sum x_i^2 \cdot P(X = x_i) - \mu^2}$.

• Method B: Directly apply the expressions $\mu = n \cdot p$ and $\sigma = \sqrt{n \cdot p \cdot q}$.

1. a) Apply Method B to calculate the mean and standard deviation of this batch.
2. b) Compare the efficiency of both approaches: do they lead to exactly the same numerical value? Justify why Method B is preferable and explain what operational difficulties would arise with Method A if the sample size were $n = 150$.
3. c) If the machinery is adjusted and the probability of defect drops to $p = 0{,}1$ keeping $n = 5$, reason (without calculating the numerical square root) whether the dispersion $\sigma$ will increase or decrease compared to the previous case by analyzing the product $p \cdot q$.

A transport consultancy firm audits the punctuality of urban bus lines in a Spanish provincial capital. In the final executive report, due to a layout error, the original parameters $n$ and $p$ of the binomial distribution $B(n, p)$ for the number of buses arriving late per shift have been erased. However, two statistical figures can still be read in the summary of conclusions: the mean number of delays is $\mu = 20$ buses and the standard deviation is $\sigma = 4$ buses.

1. a) Prove algebraically that, from the ratio $\frac{\sigma^2}{\mu}$, it is possible to directly deduce the value of $q = 1 - p$ in any binomial distribution.
2. b) Use this relationship to calculate the probability $p$ that a bus is delayed and the total number $n$ of buses inspected per shift in the audit.
3. c) Suppose that the report had only indicated that $\sigma = 4$ and that the sample size was $n = 100$. Write the equation that allows solving for $p$, find its two possible solutions, and explain why there are two different values of $p$ that produce exactly the same standard deviation.

A social media content creator knows that the probability that a new user subscribes to their channel after watching a "short" is $p = 0,12$. The algorithm informs them that, in their latest campaign, the mean number of new subscribers was $18$ people.

1. a) Determine the sample size $n$ (number of views) analysed by the algorithm.
2. b) Calculate the standard deviation of this distribution.
3. c) What is the probability that the number of subscribers is exactly equal to the mean?

A logistics company uses drones for package delivery in smart cities. It is known that the variable $X$ defining the number of failed deliveries in a batch of $n$ shipments follows a binomial distribution. After an internal study, it has been verified that the mean number of failures is $4$ and the variance is $3,2$.

1. a) Calculate the parameters $n$ and $p$ of the distribution.
2. b) If an individual shipment is made, what is the probability that the delivery is successful?
3. c) What is the probability that in a batch of $n$ shipments, none of them fail?

A cybersecurity analyst is studying phishing attacks. The probability that an employee clicks on a malicious link is $0,005$. The analyst wants to send a test campaign to a group of employees of size $n$.

Calculate the minimum number of employees to whom the email must be sent so that the probability of at least one of them clicking on the link is greater than $90\%$.

In a study on the use of sustainable transport, it has been observed that the number of students who come to school by electric scooter, in groups of $50$ people, follows a binomial distribution $B(50,\,p)$, where $p$ is the probability that a student uses an electric scooter. The standard deviation of that number of users is $\sigma = 3$.

The figure shows how the standard deviation $\sigma=\sqrt{50\,p\,(1-p)}$ varies as a function of $p$, along with the line $\sigma = 3$.

1. a) Find the two possible values of the probability $p$ that a student uses an electric scooter.

1. b) If it is known that the scooter is a minority means of transport ($p < 0,5$), calculate the mean number of users in a sample of $100$ students.

A strategy video game has a probability $p$ of obtaining a "legendary item" when opening a chest. A player opens $200$ chests and observes that the standard deviation of their reward is double what it would be if the probability of success were $0,01$.

1. a) Set up an equation to find the value of $p$.
2. b) Determine the expected mean number of legendary items in the $200$ chests with that $p$.

A biased coin is tossed $n$ times. It is known that the probability of getting heads is $p$. If the mean number of heads obtained is $15$ and the standard deviation is $\sqrt{10}$, calculate:

1. a) The probability $p$ of getting heads.

1. b) The number of tosses $n$ performed.

1. c) The probability of getting exactly $2$ heads in only $5$ tosses.

In a quality control check at an LED light bulb factory, the number of defective units in boxes of a given size is analysed. The variance of defective light bulbs per box is $3,75$ and it is known that the probability of a light bulb being defective is $5\%$.

1. a) How many light bulbs does each box contain?
2. b) If the manufacturer wants to halve the standard deviation, how should the number of light bulbs per box change while maintaining the same manufacturing quality ($p$)?

Let $X$ be a random variable following a binomial distribution $B(n, p)$. It is known that its mean is $\mu = 30$.

1. a) What is the minimum value its standard deviation can take? Justify your answer by reasoning about the possible values of $p$.
2. b) If the standard deviation is $\sigma = 5$, calculate the probability of obtaining at least one success in a sample of only $3$ individuals from that same population.

Three demographic research institutes analyze the acceptance of three social measures in Spain using independent samples of the same size, $n = 400$ people each:

• Measure 1: has an estimated proportion of supporters of $p_1 = 0{,}10$ (very small minority).

• Measure 2: has an estimated proportion of supporters of $p_2 = 0{,}48$ (very closely divided).

• Measure 3: has an estimated proportion of supporters of $p_3 = 0{,}85$ (broadly consensual).

1. a) Estimate before calculating: Without performing numerical operations, predict and order with reasoning the three measures from lowest to highest dispersion (standard deviation $\sigma$) in the count of people in favor. Justify your prediction by analyzing the properties of the product $p \cdot (1 - p)$ and its symmetry.
2. b) Calculate: Obtain the exact value of the standard deviation $\sigma$ for each of the three distributions.
3. c) Check and assess: Contrast the results with your initial prediction. Next, calculate the theoretical maximum value that $\sigma$ could reach for any sample of $n = 400$ and deduce for which value of $p$ it would be reached. Why does a very close vote ($p \approx 0{,}5$) show greater variability from one sample to another than a nearly unanimous vote?

An e-commerce platform analyzes complaints about damaged parcels during shipments in mainland Spain. The daily number of damaged parcels follows a binomial model $B(n, p)$. The logistics management requires that the coefficient of variation of the distribution, defined as the ratio between dispersion and average ($CV = \frac{\sigma}{\mu}$), does not exceed $5\,\%$ ($0{,}05$) to consider that the daily process is statistically regular and predictable. From historical records, it is known that the probability of a parcel being damaged is $p = 0{,}02$.

1. a) Obtain a simplified expression for the coefficient of variation $CV$ solely in terms of $n$ and $p$.
2. b) Determine the minimum number of daily shipments $n$ that the platform must manage to meet the regularity criterion ($CV \le 0{,}05$). With that minimum value of $n$, calculate the mean and standard deviation of damaged parcels per day.
3. c) Suppose that during a sales period a fixed volume of $n = 3600$ parcels per day is managed, but transport conditions worsen and the rate $p$ can vary between $0{,}02$ and $0{,}50$. For what value of $p$ would the number of incidents reach its highest standard deviation? What would that value of $\sigma_{\max}$ be? Explain with reasoning what operational impact a large $\sigma$ has on customer service staff planning.

Unit Challenge. How does the labour market impact young people in your area? You are going to prepare a quantitative report for a municipal employment proposal using official data from the INE.

1. Data and model: Go to the INE website (EPA) and check the youth unemployment rate in your autonomous community to set the probability $p$ as a decimal (for example, $p = 0,28$). Define the variable $X$: "number of unemployed young people in a random sample of $n = 8$ young people", modelling it as $X \sim B(8, p)$.

2. Calculation and parameters: Justify by hand the calculation of $P(X = 2)$ using combinations: $\binom{8}{2} \cdot p^2 \cdot (1-p)^6$. Then, calculate the statistical parameters of the distribution: the mean $\mu = n \cdot p$ and the standard deviation $\sigma = \sqrt{n \cdot p \cdot (1-p)}$.

3. Final product: Use GeoGebra or a spreadsheet to generate the bar chart of the distribution. Write a short report (one page) interpreting $\mu$ and $\sigma$ in the real context and proposing two career guidance measures based on the results.

Evaluation criteria:

• Rigour in obtaining the real data from the INE and in defining the binomial model.

• Correctness in the manual calculation using combinations and in finding $\mu$ and $\sigma$.

• Quality of the digital chart and coherence in the socioeconomic interpretation of the report.