Question : A researcher is investigating a population that has a mean of 80 and a standard deviation of 12. The researcher has collected a sample of 25 participants and that sample has a mean of 72. Use the facts from the central limit theorem to answer the following questions about the sampling distribution of the mean.
a) What is the mean of the sampling distribution of the mean?
b) What is the standard error of the mean?
Answer : The central limit theorem only applies if the sample size is at least 30 or if the population distribution is normal. Here, the sample size is only 25, and there is no mention of a normal population distribution, so you can't use it.
IF the central limit theorem did apply, the mean of the sampling distribution would be the same as the mean of the population distribution, so the answer to part (a) would be 80.
IF the central limit theorem did apply, the standard error would be the population standard deviation divided by the square root of the sample size, so the answer to part (b) would be 12/sqrt(25) = 12/5 = 2.4.
Question : The diameters of a trees are normally distributed with a mean of 3feet and a standard deviation of 0.2feet. Random samples of 16 are drawn from this population and the mean of each sample is distributed. Find the mean and standard error of the mean of the sampling distribution then sketch the graph of the sampling distribution.
I do not know how to set this up at all, I dont know where to start off.
Answer : The mean of the means of the samples will be the same as the mean of the original distribution, namely 3 feet.
The standard deviation of the means of the samples will be the standard deviation of the original distribution divided by the square root of the sample size. That is, it's 0.2/sqrt(16) = 0.2/4 = 0.05 feet. The Central Limit Theorem tells you this.
The sampling distribution (that is, the distribution of the means of the samples) will be another normal distribution. Per the above, it will have the same mean as the original distribution, but it will have a narrower 'bump' in the middle because its standard deviation is smaller.
Question : I'm doing an experiment to demonstrate the equation R=V/I
I've measured V and I, and calculated the errors for both.
To find the mean of R, I divide the mean of V by the mean of I.
How do I calculate the error of R?
Answer : In quadrature.
For R = V/I, the error in R is calculated thus:
R/R (i.e. the fractional error) = ( ( V/V) + ( I/I) )
So square the fractional errors in V and I, add them together and then take the square root of the sum.
This method is also true when multiplying. If adding or subtracting numbers, do the same process but with the absolute error, not the fractional.