Questions: Use the five numbers 12, 11, 16, 17, and 14 to complete parts a) through e) below. Draw a conclusion about the change in the mean. Choose the correct answer below. A. If each piece of data is increased, or decreased, by n, then the mean is multiplied, or divided, by n. B. If each piece of data is increased, or decreased, by n, then the mean is increased by n. That is, adding or subtracting a fixed value to a dataset will both increase the mean. C. If each piece of data is increased, or decreased, by n, then the mean is increased, or decreased, by n. D. The mean remains the same if each piece of data is increased, or decreased, by n.

Solution Steps

The original dataset consists of the values $12, 11, 16, 17, 14$. The mean $\mu$ is calculated as follows:

$$\mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{12 + 11 + 16 + 17 + 14}{5} = \frac{70}{5} = 14.0$$

When each data point is increased by $3$, the new dataset becomes $15, 14, 19, 20, 17$. The mean of this new dataset is:

$$\mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{15 + 14 + 19 + 20 + 17}{5} = \frac{85}{5} = 17.0$$

To find the standard deviation, we first calculate the variance $\sigma^2$:

$$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = \frac{(12 - 14)^2 + (11 - 14)^2 + (16 - 14)^2 + (17 - 14)^2 + (14 - 14)^2}{5} = \frac{4 + 9 + 4 + 9 + 0}{5} = \frac{26}{5} = 5.2$$

The standard deviation $\sigma$ is then:

$$\sigma = \sqrt{5.2} \approx 2.28$$

The standard deviation remains unchanged when each data point is increased by the same amount. Thus, the standard deviation after increasing each data point by $3$ is still:

$$\sigma = 2.28$$

1. The change in the mean indicates that if each piece of data is increased, or decreased, by $n$, then the mean is increased, or decreased, by $n$. Therefore, the conclusion is:

$$\text{If each piece of data is increased, or decreased, by } n, \text{ then the mean is increased, or decreased, by } n.$$

2. The change in the standard deviation shows that it remains the same when each piece of data is increased, or decreased, by $n$. Thus, the conclusion is:

$$\text{The standard deviation remains the same if each piece of data is increased, or decreased, by } n.$$

Final Answer