Answer :
Answer:
The sampling distribution of the sample proportion of adults who have credit card debts of more than $2000 is approximately normally distributed with mean [tex]\mu = 0.39[/tex] and standard deviation [tex]s = 0.0488[/tex]
Step-by-step explanation:
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this question:
[tex]p = 0.39, n = 100[/tex]
Then
[tex]s = \sqrt{\frac{0.39*0.61}{100}} = 0.0488[/tex]
By the Central Limit Theorem:
The sampling distribution of the sample proportion of adults who have credit card debts of more than $2000 is approximately normally distributed with mean [tex]\mu = 0.39[/tex] and standard deviation [tex]s = 0.0488[/tex]