5 10 points a If a simple random sample is right skewed with

#5 (10 points) a) If a simple random sample is right skewed with mean=25 and standard deviation=5, what would most likely be the shape of the population? What is the approximate value for mean and standard deviation of the population?

b) Suppose we take 10000 samples of size 5 from the same population in (b), what should be the sampling distribution shape? Is it appropriate to conclude that the sampling distribution follows normal distribution? (Explain what theorem (if any) you use to come up with the distribution for x-bar)

c) Suppose we take 10000 samples of size 50 from the same population in (b), what should be the sampling distribution shape? Does it follow normal distribution? What are the mean and standard deviation of the sampling distribution? (Explain what theorem (if any) you use to come up with the distribution for x-bar)

Solution

If a simple random sample is right skewed with mean=25 and standard deviation=5,then the shape of the population will also be right skewed with mean=25 and standard deviation=5 because samples follow the distribution of the population. Suppose we take 10000 samples of size 5 from the same population in (b), according to central limit theorem, the sampling distribution will also be right skewed because for small samples (n<30), the sampling distribution does not follow normal distribution. Suppose we take 10000 samples of size 50 from the same population in (b), according to central limit theorem, the sampling distribution will be normally distributed because for large samples (n>30), the sampling distribution follow normal distribution with mean and standard deviation /n.