Frequency tables

Classify the following statistical variables, stating whether they are qualitative, discrete quantitative or continuous quantitative:

• Eye color of the students in a class.

• Number of goals scored by a soccer team in a season.

• Time taken by runners to complete a marathon.

• Brand of cell phone of teachers in a high school.

A municipality wants to know the opinion of its $50\,000$ inhabitants regarding the construction of a new urban park. To do this, it conducts a telephone survey of $1\,200$ randomly selected residents. Clearly identify the population, the sample, the individual and the sample size.

In a study on the type of transport used to get to work, the relative frequency of respondents using the underground is $0.35$. If the study was conducted with a total of $N = 400$ people, calculate the absolute frequency of this group.

In a florist's shop, the number of roses sold on each of the last $15$ days has been recorded, obtaining the following raw data:

$12, 15, 12, 14, 15, 12, 13, 14, 15, 12, 14, 13, 15, 12, 14$

Construct the table of absolute frequencies ($f_i$) and relative frequencies ($h_i$) corresponding to this variable.

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

A student has constructed a relative frequency table for exam marks and obtained the following values for the different grades: $0.15$; $0.25$; $0.35$; $0.30$. Without needing to know the total number of students, explain why it can be asserted with absolute certainty that the student made a mistake in their calculations.

Given the following absolute frequency table for the number of pets owned by $20$ families, calculate and add the column of cumulative absolute frequencies ($F_i$).

• $x_i = 0 \implies f_i = 5$

• $x_i = 1 \implies f_i = 8$

• $x_i = 2 \implies f_i = 4$

• $x_i = 3 \implies f_i = 3$

The following table shows the relative frequencies ($h_i$) of the number of siblings of students at a school. Calculate the cumulative relative frequencies ($H_i$) and express them also as a cumulative percentage ($P_i$).

• $x_i = 0 \implies h_i = 0.25$

• $x_i = 1 \implies h_i = 0.45$

• $x_i = 2 \implies h_i = 0.20$

• $x_i = 3 \text{ o más} \implies h_i = 0.10$

When analysing the marks on a Mathematics exam marked out of $10$, it is observed that the cumulative relative frequency up to mark $6$ (inclusive) is $H_6 = 0.65$. If there are $80$ students in total in the class, how many students scored a mark strictly greater than $6$? What percentage do they represent of the whole class?

Group the following weights (in kg) of $20$ packages from a courier company into $4$ class intervals of equal width, starting at $2.0$ kg and ending at $10.0$ kg. Construct the frequency table indicating the intervals, class midpoints ($x_i$) and absolute frequencies ($f_i$).

Data: $2.5$; $3.1$; $4.8$; $5.2$; $2.9$; $6.5$; $7.4$; $8.8$; $9.1$; $4.5$; $5.5$; $6.2$; $3.8$; $7.1$; $8.2$; $9.5$; $4.2$; $5.9$; $6.8$; $7.7$.

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

A dataset of $150$ continuous data values on the duration (in hours) of certain industrial batteries is available. Use the empirical square root rule ($\sqrt{N}$) to determine the recommended approximate number of class intervals. If the minimum recorded value is $12$ hours and the maximum is $87$ hours, calculate the width that each interval should have (rounding up to the next integer) and write down what these intervals would be, starting from the minimum value.

The salaries at a company are grouped into $3$ intervals of equal width. It is known that the class midpoint of the first interval is $24\,000$ pesos and the class midpoint of the third interval is $40\,000$ pesos. Determine the width of the intervals and write down the lower and upper limits of each of the three intervals.

The following histogram represents the ages of visitors to a museum during a morning. Note that the intervals have different widths and that frequency density is represented on the vertical axis (height $a_i = \frac{f_i}{c_i}$, where $c_i$ is the interval width).

Reconstruct the absolute frequency table knowing that the intervals represented are $[0, 10)$, $[10, 30)$, $[30, 40)$ and $[40, 60)$.

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

In a two-dimensional study, $10$ people were asked for their shoe size ($X$) and their number of siblings ($Y$). The pairs $(X,Y)$ obtained are the following:

$(38, 1), (40, 2), (39, 1), (40, 1), (38, 2), (41, 0), (39, 2), (40, 1), (41, 1), (39, 1)$

Construct the two-way absolute frequency table for these two variables.

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

From the following two-way table, extract the marginal distribution table for the variable $X$ (shoe size), indicating its absolute frequencies ($f_i$) and its relative frequencies ($h_i$):

Complete the following frequency table, knowing that the total number of data values is $N=50$.

• $x_i = 1 \implies f_i = 10$

• $x_i = 2 \implies f_i = x$

• $x_i = 3 \implies f_i = 15$

• $x_i = 4 \implies f_i = 5$

Calculate the value of the unknown $x$ and determine the relative frequency of the value $x_i = 2$.

Complete the following frequency table in which some data are missing. It is known that the sample size is $N = 40$.

• $x_i = 10 \implies f_i = 8$, $F_i = 8$, $h_i = 0.20$

• $x_i = 20 \implies f_i = ?$, $F_i = 20$, $h_i = ?$

• $x_i = 30 \implies f_i = 14$, $F_i = ?$, $h_i = ?$

• $x_i = 40 \implies f_i = ?$, $F_i = 40$, $h_i = 0.15$

The following bar chart shows the number of books read during the summer by a group of students. Using the visual information from the chart, construct the complete table indicating the absolute frequencies ($f_i$), relative frequencies ($h_i$) and percentages ($p_i$).

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

A company classifies the monthly salary of its employees into intervals. Given that $15\%$ earn between $1000$ and $1500$ pesos, $45\%$ earn between $1500$ and $2000$ pesos, $30\%$ earn between $2000$ and $2500$ pesos, and the remaining $12$ employees earn between $2500$ and $3000$ pesos.

Calculate the total number of employees in the company and construct the complete absolute frequency table for the salaries.

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

An absolute frequency table has $4$ distinct values: $x_1, x_2, x_3, x_4$. It is known that the absolute frequencies depend on an unknown $x$ in the following way:

$f_1 = 2x$, $f_2 = x + 5$, $f_3 = 3x - 2$, and $f_4 = 5x$.

If the cumulative relative frequency of the third value is $H_3 = 0.60$, set up an equation, find the value of $x$, and reconstruct the complete column of absolute frequencies.

A survey has been conducted on two variables, $X$ and $Y$. The two-way table is incomplete. Complete it knowing that the total number of individuals is $N=50$, the marginal distribution of $X$ has $f(X=1) = 20$ and $f(X=2) = 30$, and the marginal distribution of $Y$ has $f(Y=A) = 15$, $f(Y=B) = 25$, and $f(Y=C) = 10$.

In addition, the following partial joint frequencies are known:

• There are $5$ individuals with $X=1$ and $Y=A$.

• There are $10$ individuals with $X=1$ and $Y=B$.

• There are $5$ individuals with $X=2$ and $Y=C$.

Construct the two-way table by filling in the missing cells.

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

In a frequency distribution grouped into intervals, the following cumulative relative frequencies are known: $H_1 = 0.12$, $H_2 = 0.38$, $H_3 = 0.70$, $H_4 = 0.90$ and $H_5 = 1$.

If the absolute frequency of the third interval is $f_3 = 48$, determine the total sample size $N$ and calculate the absolute frequencies of all the intervals.

Prove mathematically, using summation notation ($\sum$), that the sum of all relative frequencies ($h_i$) of a one-dimensional distribution of $k$ distinct values is always equal to $1$, regardless of the sample size $N$.

There are two groups of students, A and B. For group A ($30$ students), it is known that the relative frequency of passes is $0.60$. For group B ($20$ students), it is known that the absolute frequency of fails is $5$.

If both groups are combined into a single overall frequency table, what will be the cumulative relative frequency of fails with respect to the total of $50$ students? (Assume that the variable takes only two ordered categorical values: Fail and Pass, in that order).