Question

the weekly salaries (in dollars) for 7 employees of a small business are given below. (note that these are already ordered from least to greatest.)
643, 747, 775, 840, 864, 922, 935
suppose that the $935 salary changes to $753. answer the following.
(a) what happens to the
mean?
\\(\circ\\) it decreases by \\$\square.
\\(\circ\\) it increases by \\$\square.
\\(\circ\\) it stays the same.
(b) what happens to the
median?
\\(\circ\\) it decreases by \\$\square.
\\(\circ\\) it increases by \\$\square.
\\(\circ\\) it stays the same.

Explanation:

Part (a) - Mean Calculation

Step 1: Find the original sum

The original salaries are \(643, 747, 775, 840, 864, 922, 935\). The sum of these values is:

$$
LATEXBLOCK0
$$

The original mean is \(\frac{5726}{7}\).

Step 2: Find the new sum

The salary \(935\) is changed to \(753\). The new sum is \(5726 - 935 + 753 = 5726 - 182 = 5544\).
The new mean is \(\frac{5544}{7}\).

Step 3: Find the difference in means

The original mean: \(\frac{5726}{7} = 818\) (exact value: \(5726\div7 = 818\)).
The new mean: \(\frac{5544}{7} = 792\) (exact value: \(5544\div7 = 792\)).
The difference is \(818 - 792 = 26\). So the mean decreases by \(26\).

Part (b) - Median Calculation

Step 1: Original median

For a set of \(7\) (odd number) ordered values, the median is the \(\frac{7 + 1}{2} = 4^{th}\) value. The original ordered set is \(643, 747, 775, 840, 864, 922, 935\). The \(4^{th}\) value is \(840\).

Step 2: New ordered set

After changing \(935\) to \(753\), the new ordered set (since we need to re - order) is \(643, 747, 753, 775, 840, 864, 922\). The \(4^{th}\) value is still \(840\). So the median stays the same.

Answer:

(a) It decreases by \(\$26\).
(b) It stays the same.