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.

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.
Transcript text: 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 datalset 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 .
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Solution

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Solution Steps

Step 1: Calculate the Original Mean

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

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

Step 2: Calculate the Mean After Increasing Each Data Point by 3

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

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

Step 3: Calculate the Original Standard Deviation

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:

\[ \sigma = \sqrt{5.2} \approx 2.28 \]

Step 4: Calculate the Standard Deviation After Increasing Each Data Point by 3

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:

\[ \sigma = 2.28 \]

Step 5: Conclusions
  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:

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

  1. 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

The answer to the change in the mean is B, and the change in the standard deviation is that it remains the same.

\(\boxed{\text{Mean: B, Standard Deviation: remains the same}}\)

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