Questions: The weekly salaries (in dollars) for 10 employees of a small business are given below (note that these are already ordered from least to greatest.) 730, 761, 767, 777, 778, 795, 831, 880, 903, 948 Suppose that the 730 salary changes to 770. Answer the following. (a) What happens to the median? - It decreases by - It increases by - It stays the same. (b) What happens to the mean? - It decreases by - It increases by - It stays the same.

Solution Steps

The original weekly salaries are given as: \[ 730, 761, 767, 777, 778, 795, 831, 880, 903, 948 \] To find the median, we use the formula for the rank: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (10 + 1) = 5.5 \] Since the rank is 5.5, we take the average of the 5th and 6th values: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{778 + 795}{2} = 786.5 \] Thus, the original median is: \[ \text{Original Median} = 786.5 \]

After changing the salary from \(730\) to \(770\), the updated salaries are: \[ 761, 767, 770, 777, 778, 795, 831, 880, 903, 948 \] Using the same rank formula: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (10 + 1) = 5.5 \] Again, we take the average of the 5th and 6th values: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{778 + 795}{2} = 786.5 \] Thus, the updated median is: \[ \text{Updated Median} = 786.5 \]

To find the original mean: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{730 + 761 + 767 + 777 + 778 + 795 + 831 + 880 + 903 + 948}{10} = \frac{8170}{10} = 817.0 \] Thus, the original mean is: \[ \text{Original Mean} = 817.0 \]

For the updated mean: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{770 + 761 + 767 + 777 + 778 + 795 + 831 + 880 + 903 + 948}{10} = \frac{8210}{10} = 821.0 \] Thus, the updated mean is: \[ \text{Updated Mean} = 821.0 \]

The change in median is: \[ \text{Change in Median} = 786.5 - 786.5 = 0.0 \] The change in mean is: \[ \text{Change in Mean} = 821.0 - 817.0 = 4.0 \]

Final Answer