IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Statistics & Probability-Probability Calculations
Topic :Statistics & Probability- Weightage : 21 %
All Questions for Topic : Measure of dispersion: standard deviation,Correlation, quantitative handling, using technology,Histograms for continuous fixed interval groups,Addition and multiplication rule-conditional probability,Probability calculations,Dependent and independent events
Question
In this question you will predict the reaction times of sprinters based on previously acquired data and their sleep pattern.
Sprinters competing in a 100 metre race should ensure they are well rested before competitions.
Studies have shown that a good sleeping habit can improve reaction time.
The start of the race is one of the most important factors for an overall fast time. Sprinters need to react as quickly as possible to the start signal.
In this question you will explore the reaction times of sprinters with different sleeping habits and how this affects their probability of winning a race.
These sprinters take a test that records their reaction time. The table below shows the results. 8 hours sleeping habit
Question (a)
Write down the mode and median reaction times.
▶️Answer/Explanation
Ans:
To find the mode and median reaction times, let’s first sort the reaction times in ascending order:
0.75, 0.76, 0.77, 0.78, 0.79, 0.80
Mode:
The mode is the value that appears most frequently in a set of data. In this case, the reaction time 0.78 appears six times, which is the highest frequency among all the reaction times. Therefore, the mode for the given data is 0.78 seconds.
Median:
The median is the middle value in a set of data when it is arranged in ascending or descending order. Since we have an even number of data points (6 in this case), the median is calculated by taking the average of the two middle values. In this case, the middle values are 0.77 and 0.78. Therefore, the median reaction time is (0.77 + 0.78) / 2 = 0.775 seconds.
So, the mode reaction time is 0.78 seconds, and the median reaction time is 0.775 seconds.
Question (b)
Show that the mean reaction time is $0.77 \mathrm{~s}$, for this group of sprinters.
▶️Answer/Explanation
Ans:
To find the mean reaction time, we calculate the average of all the reaction times in the data set.
Given the reaction times and their corresponding frequencies:
\begin{align*}
\text{Reaction times:} &\ 0.75, 0.76, 0.77, 0.78, 0.79, 0.80 \\
\text{Frequencies:} &\ 4, 3, 5, 6, 1, 1
\end{align*}
We can calculate the sum of the products of each reaction time and its frequency, and then divide it by the total number of sprinters:
\begin{align*}
\text{Mean} &= \frac{0.75 \cdot 4 + 0.76 \cdot 3 + 0.77 \cdot 5 + 0.78 \cdot 6 + 0.79 \cdot 1 + 0.80 \cdot 1}{4 + 3 + 5 + 6 + 1 + 1} \\
&= \frac{3.00 + 2.28 + 3.85 + 4.68 + 0.79 + 0.80}{20} \\
&= \frac{15.40}{20} \\
&= 0.77 \, \text{s}
\end{align*}
Therefore, the mean reaction time for this group of sprinters is 0.77 seconds.