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