A sample of 12 measurements has a mean of 24 and a standard deviation of 4.5. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 24 each.
Find the standard deviation of the sample of 14 measurements?
The variance, which was (4.5)^2 = 20.25, is reduced by the ratio 12/14. That is because there are now 14 measurements and the two that were added equaled the previous mean value.
New variance = 17.357
New standard deviation = 4.17
(= sqrt of variance)
That answer is wrong
To find the standard deviation of the sample of 14 measurements, we can use the following formula:
Standard deviation (s) = √((∑(x - x̄)^2) / (n - 1))
Where:
- ∑ represents the sum of the values
- x represents each individual measurement
- x̄ represents the mean of the sample
- n represents the number of measurements in the sample
Given that the original sample has a mean (x̄) of 24, a standard deviation (s) of 4.5, and the sample is enlarged to 14 measurements by including two additional measurements having a common value of 24 each, we can calculate the standard deviation of the new sample.
Step 1: Calculate the sum of the original measurements
Since we know the mean and the number of measurements in the original sample, we can calculate the sum of the original measurements.
Sum of original measurements = x̄ * n
Sum of original measurements = 24 * 12
Sum of original measurements = 288
Step 2: Calculate the sum of the new measurements
The two additional measurements have a common value of 24 each, so we need to add 24 twice to the sum of the original measurements.
Sum of new measurements = Sum of original measurements + (24 * 2)
Sum of new measurements = 288 + 48
Sum of new measurements = 336
Step 3: Calculate the sum of squares of the new measurements
We need to calculate the sum of the squares of the new measurements.
Sum of squares of new measurements = (∑(x - x̄)^2) + 2 * (24 - 24)^2
Sum of squares of new measurements = (∑(x - x̄)^2) + 2 * 0
Sum of squares of new measurements = (∑(x - x̄)^2)
Step 4: Calculate the standard deviation of the new sample
Now we can plug the values into the formula for standard deviation.
Standard deviation (s) = √((∑(x - x̄)^2) / (n - 1))
Standard deviation (s) = √((Sum of squares of new measurements) / (n - 1))
Standard deviation (s) = √(∑(x - x̄)^2 / (n - 1))
Therefore, the standard deviation of the sample of 14 measurements is equal to the standard deviation of the original sample, which is 4.5. In this case, adding the two additional measurements does not affect the standard deviation.
To find the standard deviation of the sample of 14 measurements, we can use the formula for population standard deviation:
σ = √[ Σ(x - μ)² / N ]
Where:
- σ represents the standard deviation
- Σ is the sum
- x represents each individual measurement
- μ represents the mean of the measurements
- N represents the number of measurements
We are given that the mean of the original sample of 12 measurements is 24. Therefore, μ = 24.
We are also given that the standard deviation of the original sample of 12 measurements is 4.5. Therefore, σ = 4.5.
To calculate the standard deviation of the enlarged sample of 14 measurements, we need the individual measurements of the two additional values. Since both additional measurements have a common value of 24, we can simplify our calculation:
For the original 12 measurements:
Σ(x - μ)² = Σ(24 - 24)² = Σ(0)² = 0
For the two additional measurements:
Σ(x - μ)² = (24 - 24)² + (24 - 24)² = 0 + 0 = 0
Now we can calculate the standard deviation of the enlarged sample:
σ = √[ Σ(x - μ)² / N ] = √[ 0 / 14 ] = √0 = 0
Therefore, the standard deviation of the sample of 14 measurements is 0.