Question

If a population is normally distributed, the distribution of the sample means for a given sample size n will
A. be positively skewed
B. be negatively skewed
C. be uniform
D. be normal      
E. none of the above

If a population is not normally distributed, the distribution of the sample means for a given sample size n will
A. take the same shape as the population          
B. approach a normal distribution as n increases
C. be positively skewed
D. be negatively skewed
E. none of the above

Answer 1

Let x₁,x₂,.................,x_n follows normal distribution with mean and variance .

Then

  

which also follows normal distribution with mean and variance

Hence , if a population is normally distributed, the distribution of the sample means for a given sample size n will be normal.

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Clearly, for x₁,x₂,.................,x_n be random sample drawn from a distribution,then,

which tends to normality as n tends to

Hence,

If a population is not normally distributed, the distribution of the sample means for a given sample size n will  approach a normal distribution as n increases.

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