6.5: The Central Limit Theorem

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that μ = 1.2%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)

0.0000 This is very unlikely if μ = 1.2%. One would suspect that the European stock market may be heating up.

(a) If we have a distribution of x values that is more or less mound-shaped and somewhat symmetrical, what is the sample size n needed to claim that the distribution of sample means x from random samples of that size is approximately normal?

30

Consider two x distributions corresponding to the same x distribution. The first x distribution is based on samples of size n = 100 and the second is based on samples of size n = 225. Which x distribution has the smaller standard error?

The distribution with n = 225 will have a smaller standard error. -Since σx = σ/√n, dividing by the square root of 225 will result in a small standard error regardless of the value of σ.

(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the standard deviation decreases.

(b) If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means x taken from random samples of a given size is normal?

no

List two unbiased estimators and their corresponding parameters. (Select all that apply.)

p̂ is an unbiased estimator for p x is an unbiased estimator for μ

What is the standard error of a sampling distribution?

the standard deviation

Since σx = σ/√n, dividing by the square root of 225 will result in a small standard error regardless of the value of σ. (a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 225 stocks in the fund) has a distribution that is approximately normal? Explain.

yes, x is a mean of a sample of n = 225 stocks. By the central limit theorem, the x distribution is approximately normal