In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set \[ 4,4,5,8,12 \] (a) Compute the mode, median, and mean. (Enter your answers to one (1) decimal places.) Mean value \( = \) \( \square \) Median \( = \) \( \square \) Mode \( = \) \( \square \) (b) Add 4 to each of the data values. Compute the mode, median, and mean. (Enter your answers to one (1) decimal places.) Mean value \( = \) \( \square \) Median \( = \) \( \square \) Mode \( = \) \( \square \) (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set? Adding the same constant \( c \) to each data value results in the mode, median, and mean remaining the same. There is no distinct pattern when the same constant is added to each data value in a set. Adding the same constant \( c \) to each data value results in the mode, median, and mean decreasing by \( c \) units. Adding the same constant \( c \) to each data value results in the mode, median, and mean increasing by \( c \) units.

Let's solve each part step-by-step. ### Part (a): Compute the Mode, Median, and Mean Given the data set: \[ 4, \, 4, \, 5, \, 8, \, 12 \] 1. **Mode**: The mode is the number that appears most frequently. - **Mode** = \(4\) (since 4 appears twice) 2. **Median**: The median is the middle number when the data is ordered. - Ordered data: \(4, \, 4, \, 5, \, 8, \, 12\) - **Median** = \(5\) (the third number) 3. **Mean**: The mean is the average of all the numbers. - Sum of data: \(4 + 4 + 5 + 8 + 12 = 33\) - Number of data points: \(5\) - **Mean** = \( \frac{33}{5} = 6.6 \) **Answers for Part (a):** - Mean value \( = \) **6.6** - Median \( = \) **5.0** - Mode \( = \) **4.0** --- ### Part (b): Add 4 to Each Data Value and Compute the Mode, Median, and Mean Add \(4\) to each data value: \[ 4 + 4 = 8, \quad 4 + 4 = 8, \quad 5 + 4 = 9, \quad 8 + 4 = 12, \quad 12 + 4 = 16 \] New data set: \[ 8, \, 8, \, 9, \, 12, \, 16 \] 1. **Mode**: - **Mode** = \(8\) (since 8 appears twice) 2. **Median**: - Ordered data: \(8, \, 8, \, 9, \, 12, \, 16\) - **Median** = \(9\) (the third number) 3. **Mean**: - Sum of data: \(8 + 8 + 9 + 12 + 16 = 53\) - Number of data points: \(5\) - **Mean** = \( \frac{53}{5} = 10.6 \) **Answers for Part (b):** - Mean value \( = \) **10.6** - Median \( = \) **9.0** - Mode \( = \) **8.0** --- ### Part (c): Comparison and General Effect **Comparison:** - **Mean** increased from \(6.6\) to \(10.6\) (\(+4\)) - **Median** increased from \(5\) to \(9\) (\(+4\)) - **Mode** increased from \(4\) to \(8\) (\(+4\)) **General Conclusion:** When the same constant \(c\) is added to each data value in a set: - The **mode**, **median**, and **mean** all **increase by \(c\) units**. **Correct Statement:** > **Adding the same constant \(c\) to each data value results in the mode, median, and mean increasing by \(c\) units.** ### Final Answers **(a)** - Mean value \( = \) 6.6 - Median \( = \) 5.0 - Mode \( = \) 4.0 **(b)** - Mean value \( = \) 10.6 - Median \( = \) 9.0 - Mode \( = \) 8.0 **(c)** - Adding the same constant \(c\) to each data value results in the mode, median, and mean increasing by \(c\) units.