A group of 30 learners has the following shoe sizes: 6, 4, 7, 5, 5, 6, 4, 5, 4, 6, 7, 7, 5, 6, 4, 6, 7, 5, 6, 4, 7, 5, 4, 6, 5, 5, 4, 6, 7, 5. Construct a frequency distribution table. What size shoul

First, we list all the unique shoe sizes and count how many times each size appears in the given data.

The shoe sizes are:
6, 4, 7, 5, 5, 6
4, 5, 4, 6, 7, 7
5, 6, 4, 6, 7, 5
6, 4, 7, 5, 4, 6
5, 5, 4, 6, 7, 5

Counting the occurrences for each size:

• Size 4: Appears 7 times.
• Size 5: Appears 9 times.
• Size 6: Appears 8 times.
• Size 7: Appears 6 times.

The total number of learners is $7 + 9 + 8 + 6 = 30$.

The frequency distribution table is:

$$\begin{array}{|c|c|} \hline \text{Shoe Size} & \text{Frequency} \\ \hline 4 & 7 \\ 5 & 9 \\ 6 & 8 \\ 7 & 6 \\ \hline \text{Total} & 30 \\ \hline \end{array}$$

a) The frequency distribution table is shown above.

b)

• (i) The size that should be purchased in large quantities is the one with the highest frequency. From the table, size 5 has the highest frequency (9).
  • Size 5
• (ii) The reason is that size 5 is the mode (most frequent) shoe size among the learners.
• (iii) The size that will be purchased in less quantities is the one with the lowest frequency. From the table, size 7 has the lowest frequency (6).
  • Size 7
• (iv) The reason is that size 7 is the least frequent shoe size among the learners.

The mean shoe size is calculated by summing the product of each shoe size and its frequency, then dividing by the total frequency.

$$\text{Mean} = \frac{\sum (\text{Shoe Size} \times \text{Frequency})}{\sum \text{Frequency}}$$ $$\text{Mean} = \frac{(4 \times 7) + (5 \times 9) + (6 \times 8) + (7 \times 6)}{30}$$ $$\text{Mean} = \frac{28 + 45 + 48 + 42}{30}$$ $$\text{Mean} = \frac{163}{30}$$ $$\text{Mean} \approx 5.4333...$$

Rounding to the nearest whole number:

$$\text{Mean} = \boxed{\text{5}}$$