Frequency tables

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.

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.

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 ($h_i$):

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.

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 $N$.