Complete Question
Which of the following statements are true?
I. The sampling distribution of [tex]\= x[/tex] has standard deviation [tex]\frac{\sigma}{\sqrt{n} }[/tex] even if the population is not normally distributed.
II. The sampling distribution of [tex]\= x[/tex] is normal if the population has a normal distribution.
III. When n is large, the sampling distribution of [tex]\= x[/tex] is approximately normal even if the the population is not normally distributed.
A I and II
B I and III
C II and III
D I, II, and III
None of the above gives the complete set of true responses.
Answer:
The correct option is D
Step-by-step explanation:
Generally the mathematically equation for evaluating the standard deviation of the mean([tex]\= x[/tex]) of samples is [tex]\frac{\sigma}{\sqrt{n} }[/tex] hence the the first statement is correct
Generally the second statement is true, that is the sampling distribution of the mean ([tex]\= x[/tex]) is normal given that the population distribution is normal
Now according to central limiting theorem given that the sample size is large the distribution of the mean ([tex]\= x[/tex]) is approximately normal notwithstanding the distribution of the population
