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.
Asked, and answered. See previous post of same question by "Nao".
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To find the standard deviation of the sample of 14 measurements, we can use the formula for standard deviation.
The formula for standard deviation for a sample is:
s = sqrt(sum((x - x̄)^2) / (n - 1))
where:
- s is the standard deviation
- x is each individual measurement in the sample
- x̄ is the mean of the sample
- n is the number of measurements in the sample
We are given that the sample has a mean of 24 and a standard deviation of 4.5 for 12 measurements. We are also told that the sample is enlarged to 14 measurements by including two additional measurements with a common value of 24 each.
To find the standard deviation for the enlarged sample, we need to calculate the new mean and use the appropriate formula with the new sample size.
First, let's calculate the new mean:
The sum of the original 12 measurements is 12 * 24 = 288.
Adding the two additional measurements of 24 gives us a new sum of 288 + 24 + 24 = 336.
The new mean for the enlarged sample is 336 / 14 = 24.
Now we can calculate the standard deviation for the enlarged sample using the formula:
s = sqrt(sum((x - x̄)^2) / (n - 1))
For this formula, we use:
- n = 14 (the new sample size)
- x = each individual measurement in the 14-measurement sample
- x̄ = the new mean of 24
Since two measurements have a value of 24, let's consider them separately:
12 measurements have original values, so the sum expression would be (x - 24̄)^2.
Additionally, we need to consider the two measurements of 24, so the sum expression would be (24 - 24̄)^2 + (24 - 24̄)^2.
Plugging in the values, the formula becomes:
s = sqrt(((x_1 - x̄)^2 + (x_2 - x̄)^2 + ... + (x_12 - x̄)^2 + (24 - 24)^2 + (24 - 24)^2) / (14 - 1))
Simplifying further, we have:
s = sqrt(((x_1 - 24)^2 + (x_2 - 24)^2 + ... + (x_12 - 24)^2) / (13))
x_1, x_2, ..., x_12 are the original 12 measurements.
By substituting in the given values, we can calculate the standard deviation for the enlarged sample.