Descriptive Statistics in Action: A Case Study of Bongi’s Burger Scenario

Bongi, the owner of Bongi’s Burgers, is having trouble deciding how to purchase the tomato sauce and mustard for the restaurant for each month during 2024. He decides to look at his records to see how many of each item were used each month during 2023. He organises the data in the table below.

Month	Units
Month	Tomato Sauce (bottles)	Mustard Sauce (bottles)
January	12	8
February	11	6
March	6	4
April	7	5
May	5	4
June	6	4
July	7	21
August	8	7
September	5	8
October	9	5
November	9	6
December	15	9

Question 1: Descriptive Statistics: Calculate the total number of units used for the entire year for each item.

$$\sum_{i=1}^{n} x_i$$

$$\text{Tomato sauce} = 12 + 11 + 6 + 7 + 5 + 6 + 7 + 8 + 5 + 9 + 9 + 15 = 100 \; \text{bottles}$$

$$\text{Mustard} = 8 + 6 + 4 + 5 + 4 + 4 + 21 + 7 + 8 + 5 + 6 + 9 = 83 \; \text{bottles}$$

Question 2: Descriptive Statistics: Calculate the mean number of tomato sauce bottles used per month during 2023. Type your answer in the textbox provided. Show all working. Round final answers off to 2 decimal places.

$$\mu = \frac{\sum_{i=1}^{n} x_i}{n}$$

$$\mu = \frac{12 + 11 + 6 + 7 + 5 + 6 + 7 + 8 + 5 + 9 + 9 + 15}{n} = 8.33$$

Question 3: Descriptive Statistics: Bongi is offered a special deal by his tomato sauce supplier. If Bongi commits to ordering a single constant number of bottles every month of the year he will receive a 5% discount. Do you believe he should take the offer? Justify your answer. (3) Type your answer in the textbox provided

Yes. The supplier offers a 5% discount if Bongi orders a constant number of bottles each month. Based on the mean calculation, Bongi could order 8 bottles per month. Total for 12 months at 8 bottles/month = 8 × 12 = 96 bottles. Since this is slightly below the total usage (100), there might be a small shortfall, but with the discount, it could be worth it if the shortfall can be managed.

Question 4: Descriptive Statistics: Construct a histogram with appropriate headings for the number of tomato sauce bottles used per month by Bongi using the absolute frequency. Use a bin size of 2 units and a starting bin of (5, 7) moving onto (7, 9) and so on.

Question 5: Descriptive Statistics: Classify the distribution of tomato source bottles used per month as either normally or exponentially distributed. Select the correct answer.

Based on the histogram, the data does not follow a normal distribution. The higher concentration of values in the lower ranges (with one outlier month at 15) suggests that the distribution is skewed, so it is more indicative of an exponential distribution. The data appears to be right-skewed, which is a characteristic of an exponentially distributed set of values.

Question 6: Descriptive Statistics: Calculate the interquartile range for the number of tomato source bottles used each month. Show all working including the calculations for (Q_1) and (Q_3) (both the position as well as values). Round final answers off to 2 decimal places.

Arrange the data set in ascending order:

Position	1	2	3	4	5	6	7	8	9	10	11	12
Value	5	5	6	6	7	7	8	9	9	11	12	15

Step-by-step calculation for the first quartile (Q₁):

Step 1: Determine the position of (Q_1)

$$
Q_1\ \text{Position} = \frac{n + 1}{4} = \frac{12 + 1}{4} = 3.25
$$

Step 2: Find the lower and upper values surrounding the position

Lower value at position 3: (6)
Upper value at position 4: (6)

Step 3: Calculate the fractional leftover

$$
\text{Fractional Leftover} = 6 \; – \; 6 = 0
$$

Step 4: Find the fractional value

$$
\text{Fractional Value} = 0 \cdot (6 \; – \; 6) = 0 \cdot 0 = 0
$$

Step 5: Add the fractional value to the lower value

$$
Q_1 = 6 + 0
$$

Therefore, the first quartile (Q₁) value is:

$$
Q_1 = 6.00
$$

Step-by-step calculation for the third quartile (Q₃):

Step 1: Determine the position of (Q_3)

$$
Q_3\ \text{Position} = \frac {3 \cdot(n + 1)}{4} = \frac {3 \cdot(10 + 1)}{4} = 8.25
$$

Step 2: Find the lower and upper values surrounding the position

Lower value at position 9: (9)
Upper value at position 10: (11)

Step 3: Calculate the fractional leftover

$$
\text{Fractional Leftover} = 9.75 \; – \; 9 = 0.75
$$

Step 4: Find the fractional value

$$
\text{Fractional Value} = 0.75 \cdot (11 \; – \; 9) = 0.75 \cdot 2 = 1.5
$$

Step 5: Add the fractional value to the lower value

$$
Q_3 = 9 + 1.5 = 10.5
$$

Therefore, the third quartile (Q₃) value is:

$$
Q_3 = 10.50
$$

Interquartile Range:

$$
IQR = Q_3 \; – \; Q_1 = 10.50 \; – \; 6.00 = 4.50
$$

Question 7: Bongi ran a promotion during 2023. For one month, the “Must-Have Must-Ard Burger” was selling for only R12 and was made using an extra dollop of mustard. During which month do you think this promotion was run?

December

Question 8: Descriptive Statistics: The number of mustard bottles used during the month of the “Must-Have Must-Ard Burger” promotion can be described as a/n Select the correct answer.

Median
Outlier
Coefficient
Measure of skewness

2. Outlier

Question 9: Descriptive Statistics: For each of the following measures, complete the table provided and state whether the “Must-Have Must-Ard Burger” promotion influenced the final answer. Use only the words “Yes”, “No” or “Possibly”.

Mean
Median
Mode

Mean “Yes”
Median “No”
Mode

Descriptive Statistics in Action: A Case Study of Bongi’s Burger Scenario

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