A sample of 12 measurements has a mean of 20 and a standard deviation of 2. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 20 each.
Find the standard deviation of the sample of 14 measurements.
To find the standard deviation of the sample of 14 measurements, you can use the formula for the standard deviation.
The formula for the standard deviation of a sample is:
s = sqrt((∑(x - x̄)²) / (n - 1))
where s is the standard deviation, ∑ represents the sum, x is each individual measurement, x̄ is the mean, and n is the number of measurements in the sample.
In this case, we have a sample of 12 measurements with a mean of 20 and a standard deviation of 2. We want to calculate the standard deviation of the sample when it is enlarged to 14 measurements by including two additional measurements with a common value of 20 each.
First, let's calculate the sum of the squares of the differences between each measurement and the mean:
For the original 12 measurements:
(x1 - x̄)² + (x2 - x̄)² + ... + (x12 - x̄)²
Since the common value of the two additional measurements is 20, the squares of the differences for these two measurements will be 0, as (20 - x̄)² = (20 - 20)² = 0.
So the sum of the squares of the differences for the original 12 measurements will be the same as the sum of the squares of the differences for the 14 measurements.
Next, let's find the sum of the squares of the differences for the original 12 measurements:
(20 - x̄)² + (20 - x̄)²
Since both differences are the same (20 - x̄), we can simplify the equation as:
2 * (20 - x̄)²
Now, we can substitute the given values into the formula for the standard deviation of the sample:
s = sqrt((∑(x - x̄)²) / (n - 1))
s = sqrt((2 * (20 - x̄)²) / (12 - 1))
Since the mean of the original 12 measurements is 20, we can substitute that into the equation:
s = sqrt((2 * (20 - 20)²) / (12 - 1))
s = sqrt((2 * 0) / 11)
s = sqrt(0)
s = 0
Therefore, the standard deviation of the sample of 14 measurements, including the two additional measurements with a common value of 20 each, is 0.