Question

From a population which is according to a normal distribution with mean 10 units and
standard deviation 8 units, a sample of size n=25 is taken. What would the distribution of
the sum of the sample values be? Provide the name of the distribution and its parameters.
Are any assumptions made regarding n?

Answer 1

distribution of
the sum of the sample will be normal .with its mean being the sum of the two means, and its variance being the sum of the two variances

• X1, X2, .., Xx are normal ‘iid' rv's, then, the rv Z=(X,+ X,+ ..+Xx) is also normal with, M: = < w; and o? = Ï o? п i= 1

given mean= 10

and s.d. 8, variance =64

then for n= 25

mean of sum of samples= 25×10=250

and variance of sum of samples= 25×64=160

Thus as the sample size increases, the standard deviation of sum of samples decreases; and as the sample size decreases, the standard deviation of the sample summation increases.

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