Exercises: Concepts

To find out what the favorite video game of the students in a middle school of $850$ students is, a survey has been given to $40$ students chosen at random during break time. Identify the following elements of the study:

• Population

• Sample

• Individual

Classify the following statistical variables as qualitative or quantitative:

1. a) The favorite color of your friends.
2. b) The number of WhatsApp messages you receive a day.
3. c) The brand of sneakers worn by the students in your class.
4. d) The amount of money you save each month.

An animal shelter wants to carry out a study on the health of its dogs. To do this, the vet notes down the weight of each of the $35$ dogs living there.

State what the studied statistical variable is and classify it (qualitative, discrete quantitative or continuous quantitative).

Classify these quantitative variables as discrete or continuous:

1. a) The number of cousins each student has.
2. b) The exact time it takes you to get from your home to school.
3. c) The temperature of a can of soda when it comes out of the machine.
4. d) The number of books there are in your classroom library.

In a survey carried out in a class of $25$ students about which social network they prefer (TikTok, Instagram or Twitter), the following answers have been obtained. What name is given to the value that is repeated the most times in this study?

A study is to be carried out to find out the favorite ice cream flavor of the inhabitants of a city of $50.000$ people. To do this, the first $100$ customers who enter an ice cream parlor in the center on a Monday morning are asked.

Do you think this sample is representative of the whole population of the city? Briefly justify your answer.

Look at the following statistical variables related to a school trip and classify them:

1. Destination country.

2. Number of days in duration.

3. Distance traveled in miles.

4. Degree of satisfaction (Very low, Low, Medium, High).

If in a statistical study the population is made up of all the cars in a city and the variable is the "color of the car", identify:

1. a) The individual.
2. b) The type of variable (Qualitative or Quantitative).
3. c) Propose three possible values that this variable can take.

A city council wants to find out about the digital leisure habits of teenagers between 12 and 16 years old. To do this, it randomly selects 400 students from three different middle schools and asks them how many hours a day on average they spend on social media (TikTok, Instagram, etc.).

State the following elements of the statistical study:

1. a) The population.
2. b) The sample.
3. c) The individual.
4. d) The statistical variable studied and its classification.

An animal shelter has recorded the age (in years) of the dogs that have been adopted during the last month. The data obtained is as follows:

Construct a frequency table that includes:

• Variable ($x_i$)

• Absolute frequency ($n_i$)

• Relative frequency ($f_i$)

• Cumulative absolute frequency ($N_i$)

• Percentage ($\%$)

Look at the following graph which represents the number of books read by a group of students during the summer vacation:

From the graph above, answer:

1. a) How many students have been asked in total?
2. b) What is the most frequent value of the variable? What is this parameter called?
3. c) What percentage of students have read exactly 2 books?

In an e-sports club, the reaction time (in milliseconds) of its players has been measured. As it is a continuous variable, the data has been grouped into the following intervals:

1. a) Calculate the class midpoints of each interval.
2. b) Represent the data using a histogram.
3. c) In which interval is the reaction time of the majority of the players located?

A student has obtained the following marks in the Mathematics tests of the second term:

1. a) Calculate the arithmetic mean ($\bar{x}$) and the median ($Me$).
2. b) If the teacher decides that the term mark should be the median instead of the mean so that a "bad day" does not penalise them so much, what would their mark be?
3. c) Explain why the median is, in this case, more representative than the mean.

A survey has been carried out in a 7th grade class on the preferred type of music according to sex. The results are shown in this two-way table:

1. a) Fill in the missing data in the table.
2. b) What percentage of the whole class prefers Rock?
3. c) If we choose a boy at random, what is the probability that his preferred music is Pop? Use Laplace's rule.

Two basketball teams have scored the following points in their last 5 matches:

• Team A: $72, 75, 74, 73, 76$

• Team B: $50, 95, 60, 110, 55$

1. a) Calculate the mean and the range of both teams.
2. b) Looking at the results, which of the two teams is more consistent in its scores? Justify your answer by relating the mean to the spread of the data.

In a study on the eye color of the students in a middle school, the following data has been obtained:

1. a) What type of variable is "Eye color"?
2. b) Can the mean of this data be calculated? Why?
3. c) If we wanted to represent this data in a pie chart, calculate the size of the central angle (in degrees) that would correspond to the "Green eyes" sector.

A city council wants to carry out a survey on the use of the bike lane among young people. To do this, it selects $450$ students from three different middle schools proportionally to the total number of students in each school. In School A there are $600$ students, in School B there are $400$ students and in School C there are $500$ students.

1. a) State what the population, the sample and the individual are in this statistical study.
2. b) How many students must be selected from each school so that the sample is representative and proportional?
3. c) Classify the variable "daily time (in minutes) that each young person uses the bicycle" according to its nature.

A streaming platform analyzes the data of its teenage users to improve its algorithm. Classify the following variables collected in the study:

1. a) Number of devices connected simultaneously in the home.
2. b) Favorite film genre (horror, comedy, action, etc.).
3. c) Total weekly viewing time (in hours and minutes).
4. d) Rating given to a series (1 star, 2 stars, 3 stars, 4 stars or 5 stars).
5. e) Brand of the main device used to watch content.

Justify whether any of these quantitative variables could be treated as qualitative in a given context.

A survey has been carried out on the number of hours that 7th grade students sleep on a weekday. We have the following incomplete data from the frequency table, where $x_i$ is the number of hours, $n_i$ is the absolute frequency, $h_i$ is the relative frequency and $H_i$ is the cumulative relative frequency:

1. a) Calculate the total sample size ($N$).
2. b) Fill in all the gaps in the frequency table.
3. c) What percentage of students sleep 8 hours or more?

The following histogram represents the time (in minutes) that students take to get from their home to the middle school. The data has been grouped into intervals of the form $[a, b)$.

1. a) Construct the absolute frequency table from the graph, identifying the class midpoints of each interval.
2. b) Calculate the mean time students take to get to school using the class midpoints.
3. c) In which interval is the median of the data located? Explain your reasoning.

The arithmetic mean of the marks of a group of 12 students in a mathematics exam was $6,5$. After reviewing the exams, the teacher realizes that he has made a mistake with two students: he had given one a $4$ when they actually had a $7$, and he had given another a $9$ when they actually had a $6$.

1. a) What is the total sum of the marks of the 12 students before the correction?
2. b) What is the new arithmetic mean of the group after correcting the mistakes?
3. c) Can we know if the median has changed with this data? Justify your answer.

Look at the following two graphs which represent the evolution of the price of a model of sneakers in two different stores during the same 4-month period.

1. a) Explain why the graph for Store B seems to show a much sharper price rise than that of Store A, even though the actual price data is identical in both.
2. b) What is this visual effect called in statistics and why is it considered a misleading practice?
3. c) Mentally redraw (or describe) what the vertical axis of Store B should look like for the comparison to be objective.

In a talent show, a participant receives the following scores from 5 judges: $8, 7, 9, 8$ and a fifth unknown mark ($x$).

1. a) If the mean of the five marks is $8,2$, what is the score of the fifth judge?
2. b) If the rules of the show state that to calculate the final mark the highest and lowest scores must be removed and the mean of the remaining three calculated, what would be the participant's final mark with the value of $x$ found previously?
3. c) How would it affect the median of the five original marks if the fifth judge had given them a $10$ instead of the value $x$?

A study on digital leisure in 13-year-old teenagers yields the following data on the number of daily hours in front of a screen:

1. a) Calculate the three measures of central tendency: mean, median and mode.
2. b) Looking at the values of the mean and the median, would you say that the distribution is symmetrical or that there are extreme values that shift the mean? Justify your answer by naming these values.
3. c) If we remove the two users with the highest consumption of hours, what would the new median be? Explain whether this parameter is sensitive to extreme values in the same way as the mean.