Exercises: Standardised variable

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

The battery life of a smartphone model under heavy use follows a normal distribution $N(8, 1.5)$, where the mean is $8$ hours. A user notices that their phone only lasts $5$ hours after the latest update. Standardize this value and state whether the result is positive or negative. What is the meaning of the sign in this context?

Two students from different high schools want to compare their scores on an Economics exam.

• Emma scored $8.2$ at her school, where the mean was $\mu = 6.5$ and the standard deviation was $\sigma = 1.1$.

• Ethan scored $8.5$ at his school, where the mean was $\mu = 7.2$ and the standard deviation was $\sigma = 1.4$.

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

Given a continuous random variable $X$ that follows a normal distribution with a mean of $\mu = 50$ and a standard deviation of $\sigma = 8$, find the value of the original variable $X$ knowing that its standardized value is $Z = 2.25$.

In an e-sports competition, the reaction time of the players follows a normal distribution with a mean of $210$ milliseconds and a standard deviation of $15$ milliseconds. Standardize 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 movie theater follows a normal distribution $N(150, 12)$ in ounces. If we select a bag at random and we know that its standardized value is $Z = -1.5$, does this bag weigh more or less than the mean? Calculate its exact weight in ounces.

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 diagram of the situation.

The time taken by young people in a city to get to high school has been measured. The data follows a normal distribution with $\mu = 25$ minutes and $\sigma = 5$ minutes. Standardize the value $X = 35$ minutes. If the standard normal table tells us that $P(Z < 2) = 0.9772$, what percentage of students takes less than $35$ minutes to arrive?

On an online gaming platform, the players' scores in 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, in the "Master" level, the scores follow a normal distribution with $\mu = 3800$ points and $\sigma = 450$ points.

A player has obtained $5200$ points in the "Legend" level and their friend has obtained $4400$ points in the "Master" level. Use the standardized variable to determine which of the two friends has obtained a better score in relation to their gaming group.

A study has been carried out on the daily time that 17-year-olds spend 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 scores on a college entrance exam in a certain state follow a normal distribution with a mean of $6,8$ and a standard deviation of $1,2$.

The university has decided to award a special excellence scholarship to the top $5\%$ of students who have obtained the best scores. Using the standard normal distribution table $N(0,1)$, calculate the minimum score required to obtain this scholarship.

The weight of the backpacks of high school seniors follows a normal distribution with a mean of $6,5$ pounds and a standard deviation of $1,4$ pounds.

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

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

In a track meet, the time in seconds taken by the participants to complete the 400 meter sprint follows a $X \sim N(52, \sigma)$ 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 analyzes the number of "likes" their posts receive. They have observed that the number of "likes" follows a normal distribution with a mean of $1200$.

1. a) If the standard deviation is $\sigma = 150$, calculate the symmetric interval around 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 variables.

We want to compare the mental agility of two groups of students using a logic test. Group A obtains results that follow a $N(75, 8)$ and Group B a $N(70, 5)$.

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

In a test preparation center for college entrance exams, it has been observed that the scores on a practice AP exam follow a normal distribution $N(\mu, \sigma)$. It is known that $15,87\%$ of the students obtained a score higher than $8$ points, while $2,28\%$ did not reach $3$ points. Calculate the value of the mean $\mu$ and the standard deviation $\sigma$ of this distribution using the standard normal table $Z \sim N(0, 1)$.

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

• Ashley obtained $72$ points in an exam with $N(60, 8)$.

• Ben obtained $80$ points in an exam with $N(70, 12)$.

• Chloe obtained $65$ points in an exam with $N(55, 5)$.

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

A company that manufactures 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 the production.
2. b) If the company decides to replace all batteries that last less 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 beforehand that they spend more than $110$ minutes. (Note: use the definition of conditional probability along with the standardization of the variable).

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

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

In a study on the monthly leisure expenditure of families in a county, it has been proven that this follows a normal distribution. It is known that the mean expenditure is $\$250$. If $20\%$ of the families spend more than $\$320$:

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

The heights of the high school seniors at a school are distributed according to a normal distribution $X \sim N(172, 9)$. The principal wishes to select the tallest $5\%$ of the students for the basketball team and the shortest $5\%$ of the students for an ergometric study.

1. a) From 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 measures exactly $172$ cm? Justify your answer based on the nature of continuous variables.

Let $X$ be a continuous random variable that follows a normal distribution $N(\mu, \sigma)$. If we define the standardized variable as $Z = \frac{X - \mu}{\sigma}$, prove analytically, using the properties of standardization 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 table for the distribution $N(0, 1)$.