Question 1099720: Suppose a simple random sample of size n = 40 is taken from a population with μ=50 and σ=4.
(i)Does the population need to be normally distributed in order for the sampling distribution to be approximately normal? Why or why not?
(ii)What is the sampling distribution of the sample mean? Describe its shape and parameters, and provide a sketch of the distribution including labels/numbers for 3 standard deviations above and below the mean.
Can someone please help me do this problem. I do not know how to solve this :(
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! No, the population doesn't have to be exactly normally distributed, but it should not be too skewed for a sample size of 50 to be considered to be "approximately normal". With very large sample sizes, they can be considered normal, but that can often be abused by people. Short answer, yes, with qualifications.
This is what a sampling distribution is:
There are a huge number of samples on n=40 that can be taken from a population. The mean of all those samples, if one could compute it, would be the same as the population mean, which is 50. The standard deviation of this normal distribution of means would be 4/sqrt(40) or 0.6324. This is a normal curve compressed, which means that likelihood of getting a MEAN of say 45 is a lot less likely than getting a SINGLE OBSERVATION of 45, because you are grouping 40 numbers together, and they all aren't going to be fewer than 45. Sampling distributions of means have the same mean as the population, but the sd is the original sd DIVIDED by the sqrt (n) or the sample size. Draw both of these, the single observation and the means, and see how they look.
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