Question

1. Draw the PDF for random variable X ~ N (ux, o3), marking clearly the location of jix and the approximate locations of ux tox. 2. Draw the CDF for random variable X N (ux,0), marking clearly the location of uix and the approximate locations of hx tox. 3. What are the mean and variance of a standard normally distributed variable? 4. Give a brief summary of the central limit theorem (CLT). Address, specifically (a) What limit does the CLT put on the distribution function of the underlying variable? (b) Explain in words (only) what the CLT says about the relationship of the sample mean X to the population mean ix. (c) Give the equation for what the CLT says about the retiationship of the sample mean X to the population mean ix. 5. Student's t distribution relates an observable mean to a standard normal distribution based on the CLT making what important substitution? 6. What is meant by degrees of freedom in Student's t distribution? 7. Changing what aspect of Student's t distribution causes it to converge to a standard normal distribution? 8. What is the relationship of the chi-square distribution to the standard normal distribution? 9. Sketch two chi-square distributions on the same axes: one with df=2 and one with df=10.

Answer 1

1.)

) ) ) (M)(87) ,

2.)

3.)

CDF for the Normal Distribution 4ox lx. Matty

Mean of variable X = ux Variance of the variable X = 0%

4.)

a.)

i.) the samples should be drawn from the population randomly.

ii.) The size of the randomly drawn samples from the population should be equal to or greater than 30.

b.)

According to the Central limit Theorem(CLT ), the mean of the sampling distribution will be approximately equal to the mean of the population.

c.)

X = μχ