Questions: The following is a list of measurements: -4.42, 89, -88, -38, 52, 47, -59, -100, 57, 27 Suppose that these 11 measurements are respectively labeled 1, 2, ..., 11. (Thus -4 is labeled 1, 42 is labeled 2, and so on.) Complete the following: M = (x - 12)^2

Solution Steps

The mean \( \mu \) of the measurements is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{25}{11} \approx 2.27 \]

Thus, the mean of the measurements is \( \mu = 2.27 \).

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = 3745.83 \]

Therefore, the variance of the measurements is \( \sigma^2 = 3745.83 \).

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{3745.83} \approx 61.2 \]

Thus, the standard deviation of the measurements is \( \sigma \approx 61.2 \).

The values of \( M \) for each measurement are calculated using the expression:

\[ M = (x - 12)^2 \]

The resulting \( M \) values for each measurement are:

\[ M = [256, 900, 5929, 10000, 2500, 1600, 1225, 5041, 12544, 2025, 225] \]

Final Answer