Exercises: Standardised variable

In a community of content creators, the number of daily video views follows a normal distribution with a mean $\mu = 1200$ and a standard deviation $\sigma = 300$. If an influencer has obtained $1650$ views today, calculate the value of the standardised variable $Z$ corresponding to this data item.

The battery life of a smartphone model under heavy use follows an $N(8; 1,5)$ normal distribution, where the mean is $8$ hours. A user notices that their phone only lasts $5$ hours after the latest update. Standardise this value and indicate whether the result is positive or negative. What is the meaning of the sign in this context?

Two students from different secondary schools want to compare their marks in an Economics exam.

• Nerea scored $8,2$ at her school, where the mean was $\mu = 6,5$ and the standard deviation was $\sigma = 1,1$.

• Youssef scored $8,5$ at his school, where the mean was $\mu = 7,2$ and the standard deviation was $\sigma = 1,4$.

Calculate the standardised score for both and determine which of the two obtained a better mark relative to their own group.

Given a continuous random variable $X$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 8$, find the value of the original variable $X$ knowing that its standardised value is $Z = 2,25$.

In an esports competition, players' reaction time follows a normal distribution with mean $210$ milliseconds and standard deviation $15$ milliseconds. Standardise the following reaction times and interpret the results:

1. a) $195$ milliseconds.
2. b) $210$ milliseconds.
3. c) $240$ milliseconds.

The weight of popcorn bags at a cinema follows an $N(150, 12)$ normal distribution in grams. If we select a bag at random and know that its standardised value is $Z = -1,5$, does this bag weigh more or less than the mean? Calculate its exact weight in grams.

If a standardized variable $Z$ follows a standard normal distribution $N(0, 1)$, use the symmetry of the Gaussian bell curve to explain why the probability $P(Z < -1)$ is the same as $P(Z > 1)$. Draw a sketch of the situation yourself: the Gaussian bell curve, its axis of symmetry, the values $-1$, $0$ and $1$ on the horizontal axis and the two regions whose areas represent those probabilities.

You can do it on paper and send a photo with your answer.

The time taken by young people in a city to reach school has been measured. The data follow a normal distribution with $\mu = 25$ minutes and $\sigma = 5$ minutes. Standardise the value $X = 35$ minutes. If the standard normal table tells us that $P(Z < 2) = 0,9772$, what percentage of students take less than $35$ minutes to arrive?

Marcos has a random variable $X \sim N(70, 8)$ that models the time (in minutes) taken by students at a school to complete an exam. To calculate the probability that a randomly chosen student takes less than $58$ minutes, Marcos writes in his notebook:

"Standardising: $z = \dfrac{70 - 58}{8} = \dfrac{12}{8} = 1,5$. Since from the table $P(Z \le 1,5) = 0,9332$, the required probability is $93,32\,\%$."

1. a) Without needing to do calculations, reason why Marcos's final result makes no sense based on the value of the mean.
2. b) Identify the conceptual and algebraic error Marcos made when applying the standardisation formula.
3. c) Correct the procedure and find the correct probability knowing that $P(Z \le 1,5) = 0,9332$.

According to historical climate data, the accumulated precipitation during the month of November in a region follows a normal distribution $X \sim N(80, 15)$ measured in liters per square meter ($l/m^2$). In the last month of November, $115\text{ l/m}^2$ were recorded.

1. a) Reasonably estimate, before carrying out any analytical calculation, the sign of the standardized variable $Z$ corresponding to that record and between which two consecutive integers it must lie.
2. b) Calculate the value of $Z$ accurately and check whether your previous estimate was correct.
3. c) Explain the meaning of the obtained value of $Z$ in terms of standard deviations relative to the region's average rainfall.

On an online gaming platform, players' scores at the "Legend" level follow a normal distribution with a mean of $\mu = 4500$ points and a standard deviation of $\sigma = 600$ points. On the other hand, at the "Master" level, scores follow a normal distribution with $\mu = 3800$ points and $\sigma = 450$ points.

A player obtained $5200$ points at the "Legend" level and their friend obtained $4400$ points at the "Master" level. Use the standardised variable to determine which of the two friends achieved a better score relative to their gaming group.

A study was conducted on the daily time spent by 17-year-olds watching content on streaming platforms. The results show that this time follows a normal distribution $X \sim N(110, 25)$, where time is measured in minutes.

1. a) Calculate the probability that a randomly chosen young person spends more than $135$ minutes a day on these platforms.
2. b) What is the probability that the time spent is between $90$ and $120$ minutes?

The final marks in the university entrance examination (EBAU) in a certain autonomous community follow a normal distribution with mean $6,8$ and standard deviation $1,2$.

The university has decided to award a special excellence scholarship to the top $5\%$ of students with the highest marks. Using the standard normal distribution table $N(0,1)$, calculate the minimum mark required to obtain this scholarship.

The weight of the backpacks of Sixth Form students at a secondary school follows a normal distribution with mean $6,5$ kg and standard deviation $1,4$ kg.

1. a) Determine the value of the standardised variable $Z$ corresponding to a backpack of $8$ kg.
2. b) The school doctor recommends that the weight of the backpack should not exceed $9$ kg. What percentage of students is failing to follow this recommendation?

The battery life of a new smartphone model under heavy use follows a normal distribution with mean $\mu$ hours and standard deviation $\sigma = 1,5$ hours.

It is known that $25,14\%$ of the phones have a battery life of less than $8$ hours. Find the value of the mean $\mu$ of this distribution using the process of de-standardisation.

In an athletics competition, the time in seconds taken by the participants to complete the 400-metre race follows an $X \sim N(52, \sigma)$ distribution.

If it is known that $84,13\%$ of the runners take less than $55$ seconds to finish the race, calculate the standard deviation $\sigma$ of the runners' times.

A content creator on a social network analyses the number of likes their posts receive. They have observed that the number of likes follows a normal distribution with mean $1200$.

1. a) If the standard deviation is $\sigma = 150$, calculate the symmetric interval about the mean that contains $90\%$ of the posts.
2. b) What is the probability that a randomly chosen post has exactly $1300$ likes? Justify your answer based on the properties of continuous random variables.

The mental agility of two groups of students is to be compared using a logic test. Group A obtains results following an $N(75, 8)$ distribution and Group B following an $N(70, 5)$ distribution.

1. a) If a student in Group A scores $85$ points and a student in Group B scores $78$ points, which of the two is in a better relative position within their own group?
2. b) In Group B, below what score are the lowest-scoring $20\%$ of students found?

Based on housing market statistics, the rental price per square meter in a large capital city ($X$) is modeled as a normal distribution $N(18, 4)$ in euros, while in an intermediate city ($Y$) it follows a normal distribution $N(9; 1,5)$ in euros. A real estate portal highlights two properties: one in the large capital city at $24\text{ euros/m}^2$ and another in the intermediate city at $12\text{ euros/m}^2$.

1. a) Standardize both prices and determine reasonably which of the two flats is relatively more expensive compared to its own local market.
2. b) If a relocation company only recommends properties located below the $75$th percentile of their respective city, calculate the maximum allowable price per square meter in each of the two cities, knowing that in the standard normal distribution $P(Z \le 0,67) \approx 0,75$.

In a selection process for a public institution, the scores follow a normal distribution $X \sim N(\mu, \sigma)$. The examination board does not directly publish the parameters $\mu$ and $\sigma$, but provides two technical pieces of information about the scores:

• The cut-off score of $65$ points is exactly $0,75$ standard deviations below the mean.

• Only $2,28\,\%$ of the candidates scored higher than $86$ points (in the standard normal distribution, $P(Z > 2) = 0,0228$).

1. a) Write the equation that directly relates $65$ to $\mu$ and $\sigma$, translating the information about its standardized relative position.
2. b) Deduce the value of the standardized variable $Z$ corresponding to the score of $86$ points and state the second equation of the system.
3. c) Solve the system of equations to calculate the mean $\mu$ and the standard deviation $\sigma$ of the exam.

At a university entrance exam preparation academy, it has been observed that the marks on a mock Mathematics Applied to the Social Sciences exam follow a normal distribution $N(\mu, \sigma)$. It is known that $15,87\%$ of the students obtained a mark higher than $8$ marks, whereas $2,28\%$ did not reach $3$ marks. Calculate the value of the mean $\mu$ and the standard deviation $\sigma$ of this distribution using the standardised normal table $Z \sim N(0, 1)$.

Three students from different schools want to compare their results on a language aptitude test. Because the exams were different, the scores follow different normal distributions:

• Ana obtained $72$ marks in an exam with $N(60, 8)$.

• Beto obtained $80$ marks in an exam with $N(70, 12)$.

• Carla obtained $65$ marks in an exam with $N(55, 5)$.

Determine, by standardising the variables, which of the three students is in a better relative position with respect to their reference group. Justify your answer by comparing the values of the standardised variable $Z$.

A company manufacturing electronic components guarantees that the lifespan of its batteries follows a normal distribution with a mean of $800$ days. After a quality study, it is determined that the probability of a battery lasting more than $1000$ days is only $0,0062$.

1. a) Find the value of the standard deviation $\sigma$ of production.
2. b) If the company decides to replace all batteries that last fewer than $k$ days, what value must $k$ have so that they only have to replace $5\%$ of the batteries sold?

The time that teenagers in a city spend daily on social media follows a normal distribution $X \sim N(120, 20)$, where time is measured in minutes.

Calculate the probability that a randomly chosen teenager spends between $100$ and $150$ minutes on social media, knowing in advance that they spend more than $110$ minutes. (Note: use the definition of conditional probability together with the standardisation of the variable).

It is known that the weight of coffee packets packaged automatically by a machine follows a normal distribution with standard deviation $\sigma = 15$ grams. The machine has been adjusted so that $97,5\%$ of the packets weigh more than $470,6$ grams.

1. a) Calculate the mean weight $\mu$ to which the machine has been programmed.
2. b) With this setting, what is the probability that a randomly chosen packet weighs more than $510$ grams?

In a study on the monthly leisure expenditure of families in a province, it was found that it follows a normal distribution. The mean expenditure is known to be $250$ euros. If $20\%$ of families spend more than $320$ euros:

1. a) Calculate the standard deviation of the distribution.
2. b) Determine the expenditure interval, symmetrical about the mean, in which the central $90\%$ of families in the province lies.

The heights of Year 13 students (2nd Baccalaureate) at a school are normally distributed as $X \sim N(172, 9)$. The headteacher wishes to select the tallest $5\%$ of students for the basketball team and the shortest $5\%$ of students for an ergometric study.

1. a) Above what height are students selected for the basketball team?
2. b) What is the maximum height to take part in the ergometric study?
3. c) What is the probability that a student is exactly $172$ cm tall? Justify your answer based on the nature of continuous variables.

Let $X$ be a continuous random variable following a normal distribution $N(\mu, \sigma)$. If we define the standardised variable as $Z = \frac{X - \mu}{\sigma}$, prove analytically, using the properties of standardisation and the symmetry of the Gaussian bell curve, that the following probabilities are equivalent:

1. $P(\mu - \sigma < X < \mu + 2\sigma)$

2. $P(-1 < Z < 2)$

Calculate this value using the distribution table for $N(0, 1)$.

To analyze energy consumption in an autonomous community during the winter season, the daily electricity demand is modeled using a variable $X \sim N(3200, 400)$ in megawatts (MW). The goal is to calculate the percentage of days on which demand falls within the moderate consumption interval, between $2600\text{ MW}$ and $3800\text{ MW}$. Two analysts propose different procedures:

• Method 1 (Central symmetry): Observe that the endpoints are equidistant from the mean $\mu = 3200$, standardize the upper endpoint $z_2$, and calculate the probability as $2 \cdot [P(Z \le z_2) - 0,5]$.

• Method 2 (Standard subtraction of cumulative distribution functions): Standardize both endpoints $z_1$ and $z_2$ and directly apply $P(z_1 \le Z \le z_2) = P(Z \le z_2) - P(Z \le z_1)$, then transform the left tail using $P(Z \le z_1) = 1 - P(Z \le -z_1)$.

1. a) Fully develop both methods numerically, knowing that $P(Z \le 1,5) = 0,9332$, and verify that both lead to the same probability.
2. b) Justify which of the two methods minimizes the risk of making errors with negative signs when working with tables that only include positive values of $Z$.
3. c) If the interval of study changed to $[2800, 4000]\text{ MW}$, explain whether Method 1 would still be valid or if it would be mandatory to resort to Method 2.

The travel time in minutes of a high-speed train line in Spain follows a normal distribution $X \sim N(\mu, 12)$. The operating company maintains a punctuality commitment: if the trip exceeds $165$ minutes, it compensates passengers by refunding their ticket price. Management sets a quality target that at most $1\,\%$ of trains require compensation, for which the critical value $P(Z \le 2,33) \approx 0,99$ is used.

1. a) Model the situation using the standardized variable and calculate what the scheduled mean travel time $\mu$ must be to fulfill this commitment with the current dispersion ($\sigma = 12$ minutes).
2. b) The engineering team proposes two alternative modernization projects:

• Project A: Reduce the mean travel time to $\mu_A = 140$ minutes, although the dispersion would increase to $\sigma_A = 11$ minutes due to new stops.

• Project B: Keep the mean $\mu$ obtained in part a), but invest in track technology to reduce the standard deviation to $\sigma_B = 9$ minutes.

Compare the standardized variables associated with the compensation threshold ($165$ minutes) in both proposals and explain which of the two projects achieves a greater reduction in the percentage of compensated trains.

1. c) Explain what practical implication it has, from the passenger's point of view, that two railway lines with different means and standard deviations have exactly the same standardized value $Z$ with respect to their delay threshold.

Unit Challenge. Does your city's climate fit a Gaussian bell curve? You are going to check it by analysing real temperatures.

1. Data collection and parameters: Go to AEMET (or your region's meteorological portal) and download the daily maximum temperatures for the last $30$ days. Enter the data into a spreadsheet and find the mean $\mu$ and standard deviation $\sigma$ to define your model $X \sim N(\mu, \sigma)$.

2. Standardisation and probabilities: Set a meaningful heat threshold (for example, $x = 28,5\,^\circ\text{C}$). Standardise the variable using $Z = \frac{X-\mu}{\sigma}$ and use the $N(0,1)$ table to calculate $P(X \ge x)$. Compare this theoretical probability with the actual relative frequency obtained in your sample.

3. Visualisation and conclusions: In GeoGebra, plot the probability density function of your $N(\mu, \sigma)$, shading the calculated area. Write a short report (one page) analysing whether the normal distribution adequately models the temperature behaviour in your area.

Evaluation criteria:

• Accuracy in estimating the parameters $\mu$ and $\sigma$ using the spreadsheet.

• Analytical correctness when standardising and calculating probabilities with the $N(0,1)$ table.

• Quality of the graphical plot in GeoGebra and critical comparison between the theoretical model and the empirical data.