Exercises: The normal distribution (idea and concepts)

A well-known sports shoe brand analyzes the lifespan of its soles before they wear out. They have determined that this time follows a normal distribution with a mean of $\mu = 800$ miles and a standard deviation of $\sigma = 120$ miles.

Explain in your own words what these two parameters ($\mu$ and $\sigma$) mean in the context of shoe wear. What would happen to the graph of the distribution if the company managed to make all its shoes last much more similarly to each other (i.e., if there were less variability)?

Observe the following two probability density functions corresponding to two different random variables, $X_1$ and $X_2$, which represent the daily usage time (in minutes) of two different social media apps:

Knowing that both curves have the same shape (same standard deviation), reason which of the two apps has a higher mean usage time. Justify your answer based on the position of $\mu$ on the horizontal axis.

The weight of students at a high school follows a normal distribution with a mean of $\mu = 65$ pounds. Without performing calculations with tables, and based solely on the symmetry property of the normal distribution, answer the following questions:

1. a) What is the probability that a randomly chosen student weighs more than $65$ pounds?
2. b) What percentage of students are expected to weigh less than $65$ pounds?
3. c) If there are $800$ students in the school, how many of them would you expect to be above the mean weight?

Imagine we compare the performance of two computer processors, $P_1$ and $P_2$. The processing speed of both follows a normal distribution with the same mean ($\mu$), but their graphs are as follows:

Indicate which of the two processors has a greater standard deviation ($\sigma$) and explain what this means regarding the stability of its speed (i.e., which of the two is more "predictable" in its performance?).

In a normal distribution, the total area enclosed under the curve and the x-axis represents the probability of the certain event.

1. a) What is the numerical value of this total area?
2. b) If we know that the probability of a variable $X$ being less than a value $k$ is $P(X < k) = 0,72$, what will be the probability of it being greater than that same value, $P(X > k)$? Justify your answer using the properties of the area under the curve.

Identify which of the following random variables is most likely to follow a normal distribution and briefly justify why (remember the idea that values tend to cluster around the center):

1. The result of rolling a six-sided die.

2. The time it takes teenagers in a city to run a 100-meter sprint.

3. The number of times "heads" appears when flipping a coin 2 times.

A normal distribution is said to be "standard" when its parameters have specific values that facilitate the use of statistical tables.

1. a) What are the values of the mean ($\mu$) and the standard deviation ($\sigma$) for a standard normal distribution $Z \sim N(\mu, \sigma)$?
2. b) In a standard normal distribution, what is the value of the y-coordinate (height of the curve) at the point where the mean is located? (You do not need the exact number, just indicate whether it is the maximum, minimum or zero point of the function).

Given a continuous random variable that follows a $X \sim N(100, 10)$ distribution, draw a freehand sketch of the Gaussian bell curve and place on the horizontal axis the values that are one, two and three standard deviations to the right of the mean.

Calculate the numerical values corresponding to those points marked with a question mark in the sketch.

Given the two probability density functions represented in the image, corresponding to two random variables with a normal distribution, identify with reasoning which of them has a greater standard deviation and which has a greater mean. Explain how each parameter influences the shape and position of the "Gaussian bell curve".

A continuous random variable follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 10$. Using the symmetry properties of the Gaussian bell curve, and knowing that the total area under the curve is $1$, determine the value of the following probabilities without needing to consult the tables:

1. a) $P(X > 50)$
2. b) $P(X = 50)$
3. c) If we know that $P(X < 40) = 0,1587$, calculate $P(X > 60)$.

The time that teenagers at a high school spend daily on a well-known social network follows a normal distribution with a mean of $145$ minutes and a standard deviation of $20$ minutes.

1. a) Standardize the variable for a student who spends exactly $175$ minutes a day.
2. b) What does the value obtained in the previous section mean, in terms of standard deviation, with respect to the group mean?

In a normal distribution $N(0, 1)$, it is known that the critical value $z_{\alpha/2}$ that leaves an area of $0,025$ to its right is $1,96$. Use this information and the symmetry of the distribution to calculate:

1. a) $P(Z < 1,96)$
2. b) $P(Z > -1,96)$
3. c) $P(-1,96 < Z < 1,96)$

The scores of a strategy video game are distributed according to a normal distribution with a mean of $\mu = 2500$ points and $\sigma = 400$ points. The developers want to award a "legendary badge" to the $5\%$ of players with the highest scores.

If the value of the standardized variable $Z$ that leaves an area of $0,05$ to its right is $z = 1,645$, what is the minimum score required to obtain this badge?

Given a variable $Z$ with distribution $N(0, 1)$, use the properties of the normal distribution to express the following probabilities in terms of $P(Z < k)$, where $k$ is a positive value:

1. a) $P(Z > k)$
2. b) $P(Z < -k)$
3. c) $P(-k < Z < k)$
4. d) $P(Z > -k)$

The battery life of a smartphone model has been measured, resulting in a normal distribution with a mean of $35$ hours. We know that the characteristic interval $(31, 39)$ contains $95\%$ of the devices.

1. a) What is the maximum permissible error of this interval with respect to the mean?
2. b) Knowing that for a confidence level of $95\%$ the critical value is $z_{\alpha/2} = 1,96$, calculate the value of the standard deviation $\sigma$ of the battery life.

Conceptually compare the following three probability distributions: $N(10, 2)$, $N(10, 5)$ and $N(15, 2)$.

1. a) Which ones have the same shape (curvature) but a different position on the horizontal axis?
2. b) Which ones are centered at the same point but have different dispersion (one is more "peaked" than the other)?
3. c) If you had to choose the distribution that best represents a group of students with very similar scores close to a passing grade ($5$), what approximate values of $\mu$ and $\sigma$ would you propose?

In an e-sports academy, it has been found that the score obtained by players in a strategy simulator follows a normal distribution. It is known that $15,87 \%$ of the players obtain more than $85$ points and that $2,28 \%$ do not reach $40$ points. Calculate, with reasoning, the mean $\mu$ and the standard deviation $\sigma$ of this population of players.

A technology company manufactures processors whose lifespan follows a normal distribution with a mean of $12.000$ hours. Due to an error in the assembly line, $10 \%$ of the processors fail before $9.500$ hours.

1. a) Calculate the standard deviation $\sigma$ of the distribution.
2. b) If the company wants to offer a warranty that only covers the $2 \%$ of processors with the shortest lifespan, how many hours of lifespan should it set as the limit for this warranty?

The time that high school seniors spend daily using productivity apps follows a $N(\mu, \sigma)$ distribution. It is known that the mean time is $45$ minutes. If the probability of a student using these apps between $30$ and $60$ minutes is $0,8664$, determine the value of the standard deviation.

In a study on mobile data consumption, it is observed that the amount of GB used monthly by young people follows a normal distribution with a mean of $25$ GB. It has been found that the characteristic interval corresponding to $95 \%$ of the users is $[15,2, \,\, 34,8]$.

1. a) Determine the standard deviation of the distribution.
2. b) What would the characteristic interval be if we wanted to cover $99,7 \%$ of the population ($3\sigma$ rule)?

Two candidates, A and B, take different exams. The scores for A's exam follow a $N(6,5; \,\, 1,2)$ distribution and those for B's exam follow a $N(7; \,\, 0,8)$ distribution. Candidate A has obtained a $8,2$ and candidate B a $8,1$. Justify mathematically, by standardizing the variable, which of the two candidates has obtained a better relative result compared to their group.

An LED bulb manufacturer claims that the lifespan of its products follows a normal distribution with a standard deviation of $200$ hours. For the marketing department to be able to say that "$90 \%$ of our bulbs last more than $T$ hours", they must calculate this value $T$. If it is known that the mean is $5.000$ hours, calculate the value of $T$ and explain its meaning graphically on the Gaussian bell curve.

The amount of coffee (in fl oz) dispensed by an automatic machine follows a normal distribution $N(\mu, \sigma)$. It is known that $5 \%$ of the cups contain less than $190$ fl oz and $10 \%$ contain more than $215$ fl oz. Set up a system of equations to find $\mu$ and $\sigma$.

In a quality control of $1$ pound bags of flour, a bag is accepted if its weight is between $990$ oz and $1010$ oz. If the weights follow a normal distribution with a mean of $1000$ oz and a standard deviation of $8$ oz:

1. a) What percentage of bags are rejected?
2. b) If the manufacturer wants only $5 \%$ of the bags to be rejected for being underweight (less than $990$ oz), to what value should the mean $\mu$ of the machine be adjusted (keeping $\sigma = 8$)?