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Grade 10 Mathematics Teachers’ Book

INNODEMS

Subsection 3.1.6 Statistics in Real Life

Learner Experience 3.1.13.

Work in Groups
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What you need
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What to do
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  1. Collect data from \(30\) students in your class or from other classes asking:
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    Which of the following school lunch options do you prefer?
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    • Option A: Rice and beans
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    • Option B: Githeri
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    • Option C: Chapati and beans
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    • Option D: Ugali and vegetables
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    Record the responses as follows:
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    A, D, C, C, B, A, D, A, B, C, A, D, A,...
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  2. Prepared an ungrouped frequency table of the responses.
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    Options Tally Frequency
    A: Rice and beans
    B: Githeri
    C: Chapati and beans
    D: Ugali and vegetables
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  3. Draw an histogram and a frequency table for the lunch preferences.
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  4. Interpret the data to determine:
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    1. Which lunch option is the most preferred?
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    2. Which lunch option is the least preferred?
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    3. If the school wanted to serve the most preferred meal next week, which meal should they choose?
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  5. Why is it important to collect data from many students rather than just a few?
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  6. How did organizing and representing the data help you in interpreting the results and making a decision about which lunch option to serve?
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  7. Share your work with fellow students.
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Definition 3.1.109.

Informed decision-making is the process of using data and evidence to make choices that are based on facts and analysis rather than assumptions or guesses.
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Key Takeaway 3.1.110.

How to make informed decisions based on data:
  1. Identify the problem or question to be addressed.
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  2. Collect relevant data through surveys, experiments, or observations.
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  3. Organize the data into useful forms such as frequency tables (both grouped and ungrouped).
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  4. Represent the data visually using tables, histograms, and frequency polygons to identify patterns, trends, and relationships.
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  5. Interpret the data to draw conclusions.
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  6. Make informed decisions based on the analysis and interpretation of the data.
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Example 3.1.111.

A class of \(60\) students was asked which type of school activity they preferred and the responses were recorded in the frequency table below.
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Activity Frequency
Football 12
Debate club 10
Basketball 8
Drama 5
Music band 10
Math club 15
  1. Which could be the method used to collect the data?
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  2. Represent the organized data using a histogram.
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  3. Identify the most preferred activity.
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  4. If the school can support only one activity next term, which activity should they choose to support? Why?
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Solution.
  1. The data could have been collected using a survey where students were asked to choose their preferred activity from a list of options.
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  3. The most preferred activity is the Math club, with a frequency of 15 students.
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  4. If the school can support only one activity next term, they should choose to support the Math club because it is the most preferred activity among the students, as indicated by the highest frequency in the data.
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Example 3.1.112.

A school principal wants to introduce an extra Mathematics clinic on Saturdays. Before making a decision, she records the number of hours \(40\) students spend studying Mathematics per week.
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The data collected (in hours) is:
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\(2, 3, 4, 5, 6, 3, 4, 5, 7, 8,\)
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\(2, 3, 4, 4, 5, 6, 7, 5, 4, 3,\)
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\(6, 7, 8, 5, 4, 3, 2, 6, 7, 5,\)
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\(4, 5, 6, 3, 2, 4, 5, 6, 7, 8\)
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  1. Construct a frequency distribution table.
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  2. Represent the data using a histogram
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  3. Interpret the results.
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  4. Based on the data, advise the principal whether an extra Mathematics clinic is necessary.
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Solution.
  1. The frequency distribution table is as follows:
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    Hours Studied Tally Frequency
    2 |||| 4
    3 |||||| 6
    4 |||||||| 8
    5 |||||||| 8
    6 |||||| 6
    7 ||||| 5
    8 ||| 3
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    1. The average study time is about 5 hours.
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    2. Most students study between 3 and 6 hours.
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    3. The most common study time is 4 and 5 hours.
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    4. Very few students study less than 2 hours or more than 7 hours.
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  4. Based on the data, it appears that a majority of students are already dedicating a reasonable amount of time to studying Mathematics. However, there is a portion of students who may benefit from additional support. Therefore, the principal could consider introducing an extra Mathematics clinic, but it may be more effective to target it towards students who are struggling or spending less time on Mathematics rather than making it compulsory.
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Example 3.1.113.

A company conducted a study on the daily number of customers visiting two supermarkets (A and B) over \(30\) days.
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The grouped data is shown below:
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Table 3.1.114. Supermarket A
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Number of customers Frequency
0 - 50 2
50 - 100 5
100 - 150 8
150 - 200 10
200 - 250 5
Table 3.1.115. Supermarket B
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Number of customers Frequency
0 - 50 6
50 - 100 8
100 - 150 10
150 - 200 4
200 - 250 2
  1. Calculate the estimated mean number of customers for each supermarket
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  2. State which supermarket is more consistent and justify your answer using the distribution.
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  3. Draw comparative histograms for both supermarkets
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  4. The company wants to expand one supermarket. Using statistical evidence, advise which one should be expanded.
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Solution.
  1. The estimated mean number of customers for Supermarket A is calculated as follows:
    \begin{align*} \text{Mean}_A =\amp \frac{(25 \times 2) + (75 \times 5) + (125 \times 8) + (175 \times 10) + (225 \times 5)}{30} \\ =\amp \frac{50 + 375 + 1000 + 1750 + 1125}{30} \\ =\amp \frac{4500}{30} \\ =\amp 150 \end{align*}
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    The estimated mean number of customers for Supermarket B is calculated as follows:
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    \begin{align*} \text{Mean}_B =\amp \frac{(25 \times 6) + (75 \times 8) + (125 \times 10) + (175 \times 4) + (225 \times 2)}{30}\\ =\amp \frac{150 + 600 + 1250 + 700 + 450}{30}\\ =\amp \frac{3150}{30} \\ =\amp 105 \end{align*}
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  2. Supermarket A is more consistent because it has a higher frequency of customers in the middle range, while Supermarket B has a more spread out distribution with higher frequencies in the lower range.
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  4. Based on the estimated mean number of customers, Supermarket A has a higher average (150) compared to Supermarket B (105). Additionally, Supermarket A is more consistent with a higher frequency in the middle ranges. Therefore, it would be advisable for the company to expand Supermarket A as it has a larger customer base and more consistent traffic.
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Exercises Exercise

1.

The number of students who visited the school library over \(10\) days is shown below:
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\(20, 25, 30, 35, 25, 40, 30, 20, 25, 35\)
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  1. Construct a frequency table.
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  2. Calculate the mean number of students.
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  3. State the mode.
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  4. Should the school increase the number of library seats from 25 to 40? Give a reason.
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Answer.
  1. Frequency Table:
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    Number of Students Frequency
    20 2
    25 3
    30 2
    35 2
    40 1
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  2. Mean \(= 28.5\)
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  3. Mode = \(25\)
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  4. Since the most common number is \(25\) students and the average is about \(28\text{,}\) increasing seats to \(40\) is not necessary at the moment. Increasing to around \(30\) seats would be reasonable.
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2.

The marks obtained by \(30\) students in a Mathematics test are grouped below:
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Marks Frequency
0-20 3
20-40 5
40-60 10
60-80 8
80-100 4
  1. Estimate the mean mark.
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  2. Draw a histogram.
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  3. Comment on the general performance.
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Answer.
  1. Mean \(= 53.3\)
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  3. The histogram shows a relatively uniform distribution of marks, with the highest frequency in the \(40\) to \(60\) range and the average mark is about \(53\text{.}\)
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    This suggests that most students performed moderately well.
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3.

A dairy shop recorded daily milk sales (litres) below for \(7\) days:
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\(80, 90, 100, 120, 110, 95, 105\)
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  1. Find the mean.
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  2. Find the median.
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  3. Should the shop stock \(85\) litres daily? Give a reason.
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Answer.
  1. Mean = \(\frac{700}{7} = 100\)
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  2. Median = \(100\)
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  3. The shop should stock about \(100\) to \(110\) litres to avoid shortage, since average sales are \(100\) litres and stocking only \(85\) litres may lead to running out on busy days.
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4.

The number of passengers using a bus route over 8 days is shown below:
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\(45, 50, 55, 60, 50, 45, 65, 70\)
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  1. Construct a frequency table.
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  2. Calculate the mean.
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  3. Represent the data using a histogram.
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  4. Should the company use a 50-seater bus? Give a reason.
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Answer.
  1. Frequency Table:
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    Passengers Frequency
    45 2
    50 2
    55 1
    60 1
    65 1
    70 1
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  2. Mean \(= 55\)
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  3. The histogram shows that passenger numbers are distributed between 45 and 70, with the highest frequency in the 45 and 50 passenger values.
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  4. Since the average number of passengers is 55, a 50-seater bus is not sufficient. A larger bus is recommended.
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