Standardized Variable (Standardization)

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?

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.

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.

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.

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?

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 $Z$.

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?

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 weight of the backpacks of high school students follows a normal distribution with mean $6.5$ kg and standard deviation $1.4$ kg.

1. a) Determine the value of the standardized 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 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).

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 $N(0, 1)$.

The final scores on the university admission exam in a certain state 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 scores. Using the standard normal distribution table $N(0,1)$, calculate the minimum score required to obtain this scholarship.

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 a track and field competition, the time in seconds taken by the participants to complete the 400-meter 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.

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 $Z \sim N(0, 1)$.

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?

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 state, it was found that it follows a normal distribution. The mean expenditure is known to be $5\,000$ pesos. If $20\%$ of families spend more than $6\,400$ pesos:

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