The following data set represents the monthly salaries of air hostesses from 10 different airlines in South Africa: 23.5; 33.1; 18.2; 56.2; 29.6; 44.3; 38.3; 26.3; 22.6; 27.8

Question 1: Calculate the mean salary of the air hostesses.

The formula for calculating the mean salary is as follows:

$$
\text{Mean} = \frac{23.5 + 33.1 + 18.2 + 56.2 + 29.6 + 44.3 + 38.3 + 26.3 + 22.6 + 27.8}{10}
$$

Therefore, the mean salary is calculated by summing all the values and dividing by the number of data points.

Question 2: Calculate the position of the median as well as the median value, and interpret your answer.

Arrange the data set in ascending order:

Position	1	2	3	4	5	6	7	8	9	10
Value	18.2	22.6	23.5	26.3	27.8	29.6	33.1	38.3	44.3	56.2

The formula for finding the median position is:

$$
\text{Median Position} = \frac{n + 1}{2}
$$

Since there are 10 values in the data set, the median position is:

$$
\text{Median Position} = \frac{10 + 1}{2} = 5.5
$$

Since the median position is 5.5 (an average of the 5th and 6th values), we find:

$$
\text{Median} = \frac{27.8 + 29.6}{2} = \frac{57.4}{2} = 28.7
$$

Therefore, the median salary is 28.7.

Question 3: Calculate the position of the first and the third quartile, and then also calculate the value of the first and third quartile, and interpret you answer.

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

Step 1: Determine the position of Q₁

$$
Q_1\ \text{Position} = \frac{n + 1}{4} = \frac{10 + 1}{4} = 2.75
$$

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

Lower value at position 2: (22.6)
Upper value at position 3: (23.5)

Step 3: Calculate the fractional leftover

$$
\text{Fractional Leftover} = 2.75 – 2 = 0.75
$$

Step 4: Find the fractional value

$$
\text{Fractional Value} = 0.75 \cdot (23.5 – 22.6) = 0.75 \cdot 0.9 = 0.675
$$

Step 5: Add the fractional value to the lower value

$$
Q_1 = 22.6 + 0.675 = 23.275
$$

Therefore, the first quartile (Q₁) value is:

$$
Q_1 = 23.275
$$

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

Step 1: Determine the position of Q₃

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

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

Lower value at position 8: (38.3)
Upper value at position 9: (44.3)

Step 3: Calculate the fractional leftover

$$
\text{Fractional Leftover} = 8.25 – 8 = 0.25
$$

Step 4: Find the fractional value

$$
\text{Fractional Value} = 0.25 \cdot (44.3 – 38.3) = 0.25 \cdot 6 = 1.5
$$

Step 5: Add the fractional value to the lower value

$$
Q_3 = 38.3 + 1.5 = 39.8
$$

Therefore, the third quartile (Q₃) value is:

$$
Q_3 = 39.8
$$

Interpretation: Approximately 25% of the air hostesses earn less than R23,280, while 75% earn less than R39,800. This indicates that the majority of air hostesses have salaries falling below R39,800, with a quarter earning below R23,280.

Question 4: Calculate the inter-quartile range (IQR).

$$
\text{IQR} = 39.8 – 23.275 = 16.525
$$

Question 5: Why is the mean not the best measure of central location in this case?

The mean is influenced by extreme values or outliers.

Question 6: What is the difference between the highest and lowest value in a data set?

Range.

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