5. The caretaker of a mountain hut has been observing for several years how long it takes hikers to travel from the base in the valley to the mountain hut just below the peak of the mountain. The hikers recorded the time spent themselves, and the caretaker collected and organized the data into a frequency table, where he recorded the relative frequencies.

| time spent in minutes | relative frequency (in \%) |
| :--- | :---: |
| $90-105$ | 7.9 |
| $105-120$ | 19.4 |
| $120-135$ | 37.2 |
| $135-150$ | 16.3 |
| $150-165$ | 12.7 |
| $165-180$ | $?$ |

Determine the unknown relative frequency and calculate the average time.

## You have 120 minutes to solve the problems.

Solve the problems independently on the provided paper, with each problem on a separate page. Do not sign the sheets; write only your code. Write your answers in ink, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

Each problem will be graded on a scale of 0 to 6 points.

The national competition committee wishes you great success.

## 10th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 17, 2010

## Solutions to the problems and scoring guide

## A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the problem statement,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points anticipated. However, if the solution is guessed (and cannot be arrived at without calculation), such a solution is scored with 0 points.

A contestant who partially solves a problem, with correct steps but no clear path to the final solution, cannot receive more than half of the possible points.

## First Year
5. The sum of relative frequencies must be $100 \%$. Thus, we calculate $f_{6}=6.5$. For grouped data, the arithmetic mean is calculated using the formula $\bar{x}=\frac{f_{1} \cdot x_{1}+f_{2} \cdot x_{2}+f_{3} \cdot x_{3} \ldots f_{f} \cdot x_{k}}{N}$. The values $x$ are the midpoints of the classes and are obtained as the average of the upper and lower class limits $x_{i}=\frac{s_{i}+z_{i}}{2}$. Since for relative frequencies it holds that $f_{i}^{0}=\frac{f_{i}}{N} \cdot 100$, the arithmetic mean can be calculated as $\bar{x}=\frac{f_{1} \cdot x_{1}}{N}+\frac{f_{2} \cdot x_{2}}{N}+\ldots=\frac{f_{1} \cdot x_{1}}{100}+\frac{f_{2} \cdot x_{2}}{100}+\ldots$ Substituting our data, we get $\bar{x}=131.4$ minutes.

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-14.jpg?height=60&width=1642&top_left_y=461&top_left_x=207)

Calculation of class midpoints | 97.5 | 112.5 | 127.5 | 142.5 | 157.5 | 172.5 |
| :--- | :--- | :--- | :--- | :--- | :--- |$\ldots \ldots \ldots .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-14.jpg?height=71&width=1636&top_left_y=567&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-14.jpg?height=65&width=1642&top_left_y=630&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-14.jpg?height=52&width=1642&top_left_y=682&top_left_x=207)

Calculation of $\bar{x}=131.4$ minutes.................................................................................