Mathematics Homework Help

1. a.Choose n=10$n=10$ random numbers from a Beta distribution with a=5$a=5$ and b=10$b=10$

b). Calculate the sample mean.

c. Repeat this process 1000 times and plot a histogram of sample means.

d.Now, repeat the steps from 1 to 3 for n=20$n=20$ and n=100$n=100$. How does the histogram changes as n$n$ increases from 10 to 20 to 100?

2. (15 points) The following data set is a simulated data containing information on credit card balance of 10,000 customers. The data is pre-loaded in R. We are going to consider this data set as the population, i.e., small world.The “population” mean is calculated for you.

Balance=Credit$Balance
pop_mean=mean(Balance)
pop_mean

## [1] 520.015

1. Take 1000 samples of size 30 and calculate the average of sample means, X¯30${\overline{X}}_{30}$. Is this number close pop_mean?
2. Take 1000 samples of size 50 and calculate the average of sample means, X¯50${\overline{X}}_{50}$. Is this number close pop_mean?
3. What does central Limit theorem telss us about the distribution of X¯$\overline{X}$ in parts a and b?