Question

for students who borrow money to pay for college, the amount of debt for college graduates is normally distributed with mean $37,172 and standard deviation $11,536. assume that samples of 10 college graduates are randomly selected. is the distribution of sample means normal?

no, because ( n < 30 ).

yes, because the original distribution was already normal.

yes, because ( n geq 30 ).

no, since we dont know if lifespans of televisions are normally distributed.

Explanation:

Step1: Recall the Central Limit Theorem

If the population is normally distributed, then the sampling distribution of the sample mean \(\bar{X}\) is normally distributed for any sample size \(n\).

Step2: Analyze the given situation

The population (amount of debt for college graduates) is normally distributed. The sample size \(n = 10\). Since the population is normal, the distribution of sample means is normal regardless of the sample size.

Answer:

Yes, because the original distribution was already normal.