Question

If we wish to use a normal distribution as a model for xbar, how many years will we need to consider?

a. between 15 and 30 years should be considered.
b. at least 15 years should be considered.
c. since the annual yield is skewed, at least 31 years should be considered.
d. since the annual yield is right-skewed, any number of years will work.

Answer 1

The correct answer is c. since the annual yield is skewed, at least 31 years should be considered, in line with the central limit theorem indicating that larger samples better approximate a normal distribution for the sample mean even when the underlying distribution is not normal.

Step-by-step explanation:

The correct option is c:

since the annual yield is skewed, at least 31 years should be considered. According to the central limit theorem, to approximate a right-skewed distribution with a normal distribution for the sample mean (xbar), a larger sample size is needed. If the sample size is sufficiently large (typically $n \ge 30$), the distribution of the sample means will be approximately normal even if the underlying distribution is not. This is a key principle in inferential statistics, ensuring that the sample mean can be effectively used for hypothesis testing and confidence intervals.

As the degrees of freedom increase with larger sample sizes, the Student's t distribution, which should be used when the population standard deviation is unknown and the sample size is small (typically $n < 30$), becomes closer in shape to the normal distribution. However, for skewed distributions, even the t-distribution requires a larger sample size to approximate normality effectively.