hi i need help calculating the standard error using the central limit theorem.
*a population of scores contains exactly 5 scores : 2, 3, 6, 8, 11, there are 5x5=25 samples of the two scores (n=2)*
i already calculated the standard deviation and mean. for the question my standard deviation was 3.6742 and my mean was 6.
i also need to calculate the standard deviation of the following sampling distribution of means. apparently this is the longhanded way of calculating standard error.
(2, 2) ... 2
(2, 3) ... 2.5
(2, 6) ... 4
(2, 8) ... 5
(2, 11) ... 6.5
(3, 2) ... 2.5
(3, 3) ... 3
(3, 6) ... 4.5
(3, 8) ... 5.5
(3, 11) ... 7
(6, 2) ... 4
(6, 3) ... 4,5
(6, 6) ... 6
(6, 8) ... 7
(6, 11) ... 8.5
(8, 2) ... 5
(8, 3) ... 5,5
(8, 6) ... 7
(8, 8) ... 8
(8, 11) ... 9.5
(11, 2) ... 6,5
(11, 3) ... 7
(11, 6) ... 8,5
(11, 8) ... 9,5
(11, 11) ... 11
thank you :)
To calculate the standard error using the Central Limit Theorem, you need to find the standard deviation of the sampling distribution of means.
1. Start by calculating the mean of the sampling distribution of means. This can be found by summing all the means and dividing by the number of samples.
For the given data, the mean of the sampling distribution of means is:
(2+2.5+4+5+6.5+2.5+3+4.5+5.5+7+4+4.5+6+7+8.5+5+5.5+7+8+9.5+6.5+7+8.5+9.5+11) / 25 = 6.44
2. Next, calculate the deviation of each sample mean from the mean of the sampling distribution of means.
Let's take the first sample mean (2, 2) as an example. The deviation from the mean is:
(2+2) / 2 - 6.44 = -2.44
Calculate the deviation for all 25 sample means.
3. Square each deviation and sum them up.
For example, (-2.44)^2 = 5.9536
Sum up all the squared deviations:
5.9536 + ... + (deviation of the last sample mean)^2
4. Divide the sum of squared deviations by the number of samples (n).
For example, if n=25, divide the sum of squared deviations by 25.
5. Finally, take the square root of the result to find the standard deviation of the sampling distribution of means.
Standard deviation of the sampling distribution of means = sqrt(result from step 4)
Note: The standard error will be the calculated standard deviation of the sampling distribution of means divided by the square root of the sample size. In this case, since n=2, you will divide the standard deviation by sqrt(2).
To calculate the standard error using the Central Limit Theorem, you first need to understand the concept of the sampling distribution of means.
The sampling distribution of means is a theoretical distribution that represents the distribution of sample means that would be obtained from an infinite number of samples, each of the same size, taken from a population. It is important to note that the Central Limit Theorem applies when the sample size is sufficiently large (typically n ≥ 30). However, since you are dealing with a population of exactly 5 scores (n=2), a modified version of the standard error calculation can be used.
The formula for calculating the standard error (SE) is as follows:
SE = σ/√n
Where:
SE = Standard Error
σ = Population Standard Deviation
n = Sample Size
In your case, you have the following information:
Population Standard Deviation (σ) = 3.6742
Sample Size (n) = 2
Therefore, you can calculate the standard error as follows:
SE = 3.6742 / √2
SE ≈ 2.597
So, the standard error for your sample is approximately 2.597.
Please note that this calculation assumes that your sample is representative of the population and that the scores in your population are independent and normally distributed.
I hope this helps! Let me know if you have any further questions.