Exercises: Concepts

To find out the favorite video game of the students at a middle school of $850$ students, a survey was given to $40$ students chosen at random during recess. Identify the following elements of the study:

• Population

• Sample

• Individual

Classify the following statistical variables as qualitative or quantitative:

1. a) Your friends' favorite color.
2. b) The number of WhatsApp messages you receive per 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 conduct a study on the health of its dogs. To do so, the vet records the weight of each of the $35$ dogs living there.

State the statistical variable studied 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 home to school.
3. c) The temperature of a drink can when coming out of the vending machine.
4. d) The number of books 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 were obtained:

Social network	Number of students
TikTok	$12$
Instagram	$9$
Twitter	$4$

What is the name of the value that is repeated the most in this study? State what that value is in this survey.

A study is to be carried out to find out the favorite ice cream flavor of the inhabitants of a town of $50,000$ people. To do this, the first $100$ customers entering a downtown ice cream shop on a Monday morning are asked.

Do you think this sample is representative of the entire population of the town? Briefly explain your answer.

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

1. Destination country.

2. Duration in number of days.

3. Distance traveled in kilometers.

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

If in a statistical study the population consists of all the cars in a town and the variable is "car color", identify:

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

A municipality wants to know the digital leisure habits of teenagers aged between 12 and 16. To do this, it randomly selects 400 students from three different secondary schools and asks them how many hours per day on average they spend on social media (TikTok, Instagram, etc.).

Identify the following elements of the statistical study:

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

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

$$2, 1, 5, 3, 2, 4, 1, 2, 3, 2, 5, 1, 4, 2, 3$$

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 representing the number of books read by a group of students during the summer holidays:

Based on the graph above, answer:

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

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

Time (ms)	Frequency ($n_i$)
$[150, 170)$	$6$
$[170, 190)$	$12$
$[190, 210)$	$18$
$[210, 230)$	$4$

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 most players found?

You can do it on paper and send a photo with your answer.

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

$$7.5; \quad 8.2; \quad 9.0; \quad 2.1; \quad 8.7$$

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

A survey was conducted in a 9th grade class on preferred music genre by gender. The results are shown in this two-way table:

	Pop	Rock	Trap	Total
Girls	$8$	$3$	$4$	$15$
Boys	$5$	$6$		$15$
Total		$9$		30

1. a) Complete 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 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 scoring? Justify your answer by connecting the mean with the spread of the data.

In a study on the eye color of students at a secondary school, the following data was obtained:

Color	$n_i$
Brown	$120$
Blue	$30$
Green	$50$

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 circle graph, calculate the size of the central angle (in degrees) that would correspond to the "Green eyes" sector.

A municipality wants to conduct a survey on bike lane usage 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. There are $600$ students in School A, $400$ students in School B and $500$ students in School C.

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

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

1. a) Number of devices connected simultaneously in the household.
2. b) Favorite movie 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.

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

A survey was conducted on the number of hours 9th grade students sleep on a weekday. We have the following incomplete data from the frequency table, where $x_i$ is the number of hours, $n_i$ the absolute frequency, $h_i$ the relative frequency, and $H_i$ the cumulative relative frequency:

$x_i$	$n_i$	$h_i$	$H_i$
6	4		0.16
7		0.24
8	10
9
10	2		1.00

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) it takes students to get from home to 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 it takes students to get to school using the class midpoints.
3. c) In which interval is the median of the data found? Explain your reasoning.

You can do it on paper and send a photo with your answer.

The mean mark of a group of 12 students in a mathematics test was $6.5$. After reviewing the tests, the teacher realizes that he made a mistake with two students: he gave one student a $4$ when they actually had a $7$, and he gave 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 mean of the group after correcting the errors?
3. c) Can we know if the median has changed with this data? Justify your answer.

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

1. a) Explain why the graph for Store B appears to show a much steeper price rise than the graph for 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) Describe how the vertical axis of Store B should be set up so that the comparison is objective.

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

1. a) If the mean of the five scores is $8.2$, what is the score of the fifth judge?
2. b) If the competition rules state that to calculate the final score, the highest and lowest scores must be removed and the mean of the remaining three calculated, what would the contestant's final score be with the value of $x$ found previously?
3. c) How would it affect the median of the five original scores 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 spent in front of a screen:

$$3, 5, 2, 4, 3, 2, 8, 3, 1, 4, 3, 2, 3, 5, 3, 10, 2, 3, 4, 1$$

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