A sample of 12 measurements has a mean of 37 and a standard deviation of 4.25. Suppose that the sample is enlarged to 14 measurements, by including two additional measurements having a common value of 37 each.

A. Find the mean of the sample of 14 measurements.

Mean = 37

How I found it: 12 (the original sample) * 37 (mean) = 444 + (2*37, the two new scores) = 518/14 (the new sample size) = 37

WHAT I NEED HELP WITH (i.e. how can i calculate it?):

B. Find the standard deviation of the sample 14 measurements.

Standard Deviation =

To find the standard deviation of the sample of 14 measurements, you can use the formula for population variance (since you have all the data points):

Variance = (Σ(x - μ)^2) / n

where μ is the mean, which is 37, and n is the number of measurements, which is 14 for the enlarged sample.

But before that, let's find the sum of squared deviations (SSD) from the mean for the original 12 measurements:

SSD_12 = (standard deviation_12)^2 * (sample size_12)
SSD_12 = (4.25^2) * 12
SSD_12 = 216.75

Now let's find the SSD for the enlarged sample of 14 measurements:

SSD_14 = SSD_12 + (37 - 37)^2 + (37 - 37)^2
SSD_14 = 216.75 + 0 + 0
SSD_14 = 216.75

Now find the variance for the sample of 14 measurements:

Variance_14 = SSD_14 / 14
Variance_14 = 216.75 / 14
Variance_14 ≈ 15.48

Lastly, to find the standard deviation, take the square root of the variance:

Standard Deviation_14 = √Variance_14
Standard Deviation_14 = √15.48
Standard Deviation_14 ≈ 3.93

So, the standard deviation of the sample of 14 measurements is approximately 3.93.

To find the standard deviation of the sample of 14 measurements, we can use the formula for standard deviation.

The formula for standard deviation of a sample is:

s = √(Σ(x - x̄)² / (n - 1))

Where:
- s is the sample standard deviation
- Σ represents the sum of
- x is each individual measurement in the sample
- x̄ is the sample mean
- n is the sample size

In this case, we are given that the sample mean (x̄) is 37 and the sample size (n) is 14.

Let's calculate the standard deviation step-by-step:

1. Calculate the sum of squared differences:
For each measurement, subtract the sample mean (37) and square the result. Then, sum up all the squared differences.

For the original 12 measurements:
(37 - 37)² + (37 - 37)² + ... + (37 - 37)²
= 0² + 0² + ... + 0²
= 0

For the two new measurements:
(37 - 37)² + (37 - 37)²
= 0² + 0²
= 0

The sum of squared differences is 0.

2. Divide the sum of squared differences by (n - 1):
s² = 0 / (14 - 1)
s² = 0 / 13
s² = 0

3. Take the square root of the result:
s = √0
s = 0

Therefore, the standard deviation of the sample of 14 measurements is 0.

To find the standard deviation of the sample of 14 measurements, follow these steps:

1. Calculate the sum of the squared differences between each measurement and the mean of the original sample:
a. Square each difference between the measurement and the mean.
b. Sum up all the squared differences.

2. Add the squared differences of the two new measurements (which have a common value of 37) to the sum calculated in step 1.

3. Divide the sum of squared differences by the new sample size minus 1 (n-1). In this case, the new sample size is 14, so divide by 13.

4. Take the square root of the result from step 3 to get the standard deviation.

Let's calculate it step by step:

Step 1:
- For each observation in the original sample (12 measurements), calculate the squared difference between the observation and the mean of 37. Sum these squared differences.

Step 2:
- Add the squared differences of the two new measurements (which have a common value of 37) to the sum calculated in step 1.

Step 3:
- Divide the sum of squared differences by the new sample size minus 1 (13).

Step 4:
- Take the square root of the result obtained in step 3 to find the standard deviation.

By following these steps, you can calculate the standard deviation of the sample of 14 measurements.