The 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.

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 $Z = 2,25$.

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?

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 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 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.

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