Question 880706: I am going through an assignment and my answers - I have worked out the answer to the question, but not really sure if I have got it right or not. So ...
The mean of salaries for a total of 350 staff is: µ = $50,500
The standard deviation for the 350 staff members is: σ = $18,800
(These numbers have been rounded to the nearest $100.00)
So the question is: If all possible samples of size 40 were taken from this population, what would you expect the mean and standard deviation of these samples means to be??
So I have calculated the means to be $49,200 and the standard deviation of $11,000.00, but don't ask me how I did this because after my computer shut down without warning, I couldn't retrieve my unsaved document, and each time I go to do it again to check my answer, I come up with different figures!
Am in right in ... coming up with a new figure - The sample standard deviation, is the standard deviation of $18800 divided by the square root of the sample figure of 40 to get $2970.00 (rounded to the nearest 100).
Then ... I am not sure how to go about getting the sample mean - part of me wants to take the mean total of $50500, and take $50500 away from this, and divide it by the original standard deviation of 18800, and then divide by the square root of 40, but this just doesn't seem to make sense - if I did that, my sample mean would be 50499??
Can someone please point me in the right direction, because the answer to this, is then used in a few other questions.
Thanks so much!!
Found 2 solutions by stanbon, rothauserc:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! The mean of salaries for a total of 350 staff is: µ = $50,500
The standard deviation for the 350 staff members is: σ = $18,800
(These numbers have been rounded to the nearest $100.00)
So the question is: If all possible samples of size 40 were taken from this population, what would you expect the mean and standard deviation of these samples means to be??
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Using the Central Limit Theorem::
mean of the sample means = the mean of the population = 50,500
std of the sample means = 18,800/sqrt(40)
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Cheers,
Stan H.
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Answer by rothauserc(4718)  (Show Source):
You can put this solution on YOUR website! 1)This problem is an application of a "Sampling Distribution of the Mean" using The Central Limit Theorem.
2)Our population size is 350 and we are sampling 40 members of that population.
3)The central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough.
As a rough rule of thumb, many statisticians say that a sample size of 30 is large enough. Our sample size of 40 is therefore large enough.
The central limit theorem tells us that the mean of the population will be the mean of the sample given that the sample is greater than or equal to 30.
Therefore,
i) sample mean is $50,500.
ii) sample standard deviation is given by the following formula
sample standard deviation = population standard deviation * square root (1/n - 1/N), where n is sample size and N is population size, note that for large populations 1/N approaches 0.
sample standard deviation = 18800 * square root( 1/40 - 1/350) = 2797.529522377 = $2,798.
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