Statistical parameters of a normal distribution

A local council has analysed the time taken by buses on the circular route to complete their journey. They have confirmed that this time follows a normal distribution with a mean of $25$ minutes and a standard deviation of $4$ minutes. Identify the statistical parameters of this distribution and write it using the abbreviated mathematical notation $X \sim N(\mu, \sigma)$.

Look at the following graph, which represents the weight distribution of newborn kittens in an animal shelter. Knowing that the distribution is symmetrical, visually state the value of the population mean $\mu$.

Two brands of lightbulbs, "LuzEterna" and "BrilloMax", have a mean lifespan of $1200$ hours. However, the standard deviation of "LuzEterna" is $50$ hours, whereas that of "BrilloMax" is $150$ hours.

• Which of the two brands offers a more "homogeneous" or consistent product in terms of lifespan?

• If you had to draw both Gaussian bell curves on the same axis, which one would be taller and narrower?

Imagine you are comparing the precision of two machines that cut wooden slats. Machine 1 cuts with a distribution $N(100; 0,1)$ and Machine 2 with a distribution $N(100; 0,5)$ (in centimetres).

• Which of the two machines is more precise?

• Relate the concept of "precision" to the value of the standard deviation $\sigma$ in a normal distribution.

Two graphs are presented below showing the distribution of the lifespan of two types of LED lightbulbs from different brands, A and B. Both follow a normal distribution.

1. a) Looking at the shape of the curves, which of the two brands has a larger standard deviation? Justify your answer by relating the dispersion to the shape of the bell curve.
2. b) If brand A has a distribution $N(25000, 2000)$ and brand B a $N(28000, 3500)$, calculate the probability that a lightbulb from brand A lasts longer than the mean lifespan of brand B.

The Standard Normal distribution is a special case widely used in statistics to enable the use of area tables. State the fixed values of its mean ($\mu$) and its standard deviation ($\sigma$). Represent it symbolically.

In a popular strategy video game, the experience points obtained by players on a specific mission follow a normal distribution with a mean of $\mu = 4500$ points and a standard deviation of $\sigma = 800$ points. If the developers want to award a special medal only to the $15\%$ of players who achieve the highest scores, from how many points upwards will this medal be awarded?

A study has been carried out on the time that young people aged between $16$ and $18$ spend daily on a well-known short-video social network. It is known that this time follows a normal distribution with a mean of $\mu = 110$ minutes. Given that $33\%$ of users spend more than $125$ minutes a day on the platform, calculate the value of the standard deviation $\sigma$ of this distribution.

A sustainable courier company uses electric scooters for its deliveries. The battery range of these scooters follows a normal distribution with a mean of $60$ km. It is known that the probability of a scooter having a range greater than $75$ km is $0,0668$. Due to a software update, the range of all scooters increases by $10\%$ plus a fixed extra of $2$ km. Determine the original standard deviation of the batteries and the new parameters (mean and standard deviation) after the software update.

A manufacturer of components for space telescopes produces lenses whose diameter follows a normal distribution. Through quality control, it is known that the mean is $120$ mm. For a lens to be accepted, its diameter must be in the range $[118, 122]$ mm. If the manufacturer wants $99,73\%$ of its production to be acceptable in order to avoid financial losses, what must be the maximum value of the standard deviation of the manufacturing process?