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

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
Transcript text: 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
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Solution

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Solution Steps

Step 1: Calculate the Mean

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

μ=i=1NxiN=25112.27 \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{25}{11} \approx 2.27

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

Step 2: Calculate the Variance

The variance σ2 \sigma^2 is calculated using the formula:

σ2=(xiμ)2n=3745.83 \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} = 3745.83

Therefore, the variance of the measurements is σ2=3745.83 \sigma^2 = 3745.83 .

Step 3: Calculate the Standard Deviation

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

σ=3745.8361.2 \sigma = \sqrt{3745.83} \approx 61.2

Thus, the standard deviation of the measurements is σ61.2 \sigma \approx 61.2 .

Step 4: Calculate M Values

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

M=(x12)2 M = (x - 12)^2

The resulting M M values for each measurement are:

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

Final Answer

The results are summarized as follows:

  • Mean μ=2.27 \mu = 2.27
  • Variance σ2=3745.83 \sigma^2 = 3745.83
  • Standard Deviation σ61.2 \sigma \approx 61.2
  • M values: [256,900,5929,10000,2500,1600,1225,5041,12544,2025,225] [256, 900, 5929, 10000, 2500, 1600, 1225, 5041, 12544, 2025, 225]

Thus, the final answer is:

μ=2.27,σ2=3745.83,σ61.2 \boxed{\mu = 2.27, \sigma^2 = 3745.83, \sigma \approx 61.2}

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