Homework 3

Due 10/11/2024 Friday before the class begins.

The purpose of this homework is to demonstate the celebrated Central Limit Theorem asserting that appropriately normalized independent samples converges to a normal distribution.

Problem 1. Generate 2000 samples from binomial distribution with parameter n=100 and p=0.6.

set.seed(2024);
population <-rbinom(2000, size=100, prob=0.6);

print(population)

##    [1] 71 67 64 60 60 53 60 66 64 66 64 70 67 64 66 61 66 62 58 60 56 59 60 55
##   [25] 58 58 64 64 59 55 60 60 61 58 61 63 53 59 73 65 53 57 52 66 63 57 64 61
##   [49] 63 56 47 55 57 64 64 73 64 57 53 65 60 63 57 60 58 58 58 62 64 67 61 59
##   [73] 62 62 59 61 54 53 56 59 62 64 61 61 57 57 57 60 64 52 57 58 57 60 62 59
##   [97] 53 62 62 62 62 53 63 51 64 52 55 62 63 58 68 57 62 59 55 60 59 58 56 57
##  [121] 57 62 69 56 65 56 68 60 59 53 58 63 62 62 62 69 59 48 65 64 65 58 58 48
##  [145] 55 67 58 65 63 65 56 63 46 60 61 64 50 56 60 66 65 59 60 60 62 59 54 57
##  [169] 70 53 63 57 62 63 63 44 61 58 56 63 56 54 51 58 55 65 63 60 58 52 66 59
##  [193] 65 62 61 65 70 58 58 59 61 67 62 65 59 59 42 58 64 50 62 64 60 62 64 54
##  [217] 49 62 57 65 58 54 54 58 62 61 54 53 63 54 57 61 56 62 57 58 60 66 67 58
##  [241] 62 57 50 59 59 66 54 53 62 45 56 66 60 67 54 60 73 62 55 50 63 53 54 54
##  [265] 60 58 63 62 52 57 54 49 63 54 52 49 69 60 70 68 61 61 57 61 51 67 59 57
##  [289] 65 56 58 61 53 62 64 58 54 58 55 57 63 65 59 54 67 55 65 67 65 60 57 62
##  [313] 60 57 57 68 67 62 60 70 64 54 61 54 60 62 63 63 58 60 56 61 59 57 67 61
##  [337] 63 56 60 66 60 52 64 54 62 59 58 56 66 63 61 64 58 64 57 52 57 54 58 67
##  [361] 61 60 49 56 51 61 66 60 65 62 65 62 64 64 62 63 61 53 67 61 54 62 51 60
##  [385] 61 53 56 56 56 60 63 66 62 65 61 59 58 51 57 53 63 53 60 57 59 67 53 53
##  [409] 63 66 62 60 64 54 63 63 55 50 58 57 64 60 59 62 71 63 58 56 68 60 58 56
##  [433] 55 53 60 59 53 65 54 57 66 54 56 60 62 65 56 61 56 53 56 65 55 60 53 54
##  [457] 58 58 52 65 59 61 60 61 65 57 56 58 59 62 59 49 58 67 53 62 59 65 61 68
##  [481] 54 68 53 66 65 69 64 63 60 59 70 59 62 53 61 59 67 54 68 56 67 65 69 54
##  [505] 66 64 59 63 64 62 58 62 54 60 58 64 59 63 61 55 52 62 62 65 58 59 51 58
##  [529] 63 71 56 68 55 65 62 59 61 51 57 68 60 65 63 66 62 59 60 58 65 56 62 58
##  [553] 62 61 59 67 53 61 59 60 59 63 63 60 75 49 64 64 58 60 58 48 60 56 57 63
##  [577] 61 55 56 56 64 67 61 62 61 59 53 64 55 61 58 49 68 61 52 61 59 72 62 58
##  [601] 61 61 69 58 55 57 61 56 59 64 56 72 63 55 61 59 75 49 56 70 65 53 47 61
##  [625] 61 58 59 59 59 65 61 56 52 60 56 59 63 67 70 62 66 61 60 54 62 65 57 57
##  [649] 57 61 61 62 64 63 55 63 64 62 60 58 59 56 64 56 63 57 60 58 62 54 59 63
##  [673] 60 61 57 65 58 64 59 54 59 61 65 56 66 57 67 61 55 61 61 61 60 59 49 65
##  [697] 69 65 61 60 70 56 59 63 58 56 57 57 60 54 61 63 67 64 56 61 62 58 47 60
##  [721] 58 58 52 54 58 52 54 62 64 51 56 67 68 60 70 58 58 54 60 61 61 57 63 60
##  [745] 62 68 54 62 62 64 58 68 55 62 58 52 56 65 61 47 56 68 57 52 60 55 68 63
##  [769] 59 66 61 62 66 62 49 61 56 57 47 67 58 56 55 55 64 60 65 67 66 58 51 58
##  [793] 59 70 56 65 70 56 57 57 63 51 63 68 55 55 52 65 65 63 65 59 54 63 60 59
##  [817] 64 57 61 56 65 61 62 59 67 55 63 66 63 60 60 58 62 64 58 65 58 61 58 70
##  [841] 67 64 66 67 61 57 64 60 59 65 61 56 54 71 72 63 51 54 56 58 60 60 56 51
##  [865] 60 60 55 64 61 61 62 57 61 70 69 56 58 69 66 52 65 58 54 65 63 54 63 65
##  [889] 63 63 71 63 61 69 67 60 61 60 65 55 55 51 56 55 56 56 56 60 58 60 68 60
##  [913] 59 66 60 65 60 60 60 52 62 62 56 66 60 59 61 56 60 56 63 62 63 68 61 60
##  [937] 48 56 62 66 71 51 63 62 61 61 62 60 51 54 63 56 65 60 69 61 66 60 63 62
##  [961] 63 49 66 60 63 59 73 58 54 69 55 63 59 58 57 55 55 62 59 68 56 63 66 68
##  [985] 58 59 61 54 54 58 65 60 63 66 61 62 60 62 56 64 59 51 51 66 59 67 60 51
## [1009] 61 53 62 66 51 52 64 63 60 60 57 52 58 56 60 65 63 63 60 66 64 60 63 62
## [1033] 63 65 66 57 56 60 60 53 63 60 54 54 58 56 54 64 59 58 58 51 58 61 64 63
## [1057] 55 60 51 63 67 53 66 59 58 57 52 60 61 61 59 58 65 62 46 45 56 60 59 54
## [1081] 56 64 55 65 58 56 68 61 54 56 56 70 47 61 66 61 69 55 56 57 63 55 56 58
## [1105] 60 63 66 55 60 63 65 55 57 63 55 71 61 60 63 62 56 59 60 51 57 60 58 62
## [1129] 56 65 59 65 66 55 61 61 57 65 58 45 64 71 61 53 58 56 61 60 63 53 54 56
## [1153] 63 61 63 57 57 57 64 62 61 55 59 54 59 65 49 67 57 58 64 57 57 66 63 57
## [1177] 59 66 58 67 66 57 56 54 56 61 55 58 58 65 61 65 65 55 62 59 60 56 63 58
## [1201] 58 63 57 66 56 64 57 51 52 59 54 63 69 58 63 74 60 62 58 64 63 49 58 54
## [1225] 56 64 52 53 53 70 66 63 55 66 60 62 56 53 71 69 64 52 64 62 56 65 65 61
## [1249] 58 67 54 68 67 62 61 55 64 64 58 60 57 57 66 62 61 53 53 72 57 55 56 58
## [1273] 57 50 56 64 61 55 61 66 60 60 68 55 66 61 58 67 58 52 60 62 62 66 68 66
## [1297] 53 61 61 59 62 66 54 66 58 65 60 59 56 55 59 71 65 64 59 69 57 62 62 65
## [1321] 63 56 59 60 59 57 66 57 65 70 59 58 66 60 50 59 66 56 56 64 64 56 61 57
## [1345] 58 71 54 59 67 65 52 53 68 64 60 61 67 57 60 72 55 57 54 61 55 61 64 56
## [1369] 60 62 59 61 57 59 52 66 67 53 59 58 59 70 61 44 59 56 65 60 53 65 60 48
## [1393] 59 59 62 63 56 60 55 59 72 62 53 66 65 62 72 66 61 59 53 54 60 62 55 68
## [1417] 65 46 72 60 56 64 60 67 56 59 59 63 58 60 61 65 60 51 53 54 49 57 63 58
## [1441] 63 71 49 53 58 60 66 62 69 61 61 62 62 59 54 59 56 60 64 63 71 60 58 59
## [1465] 63 58 54 60 59 65 57 54 68 62 55 52 56 68 64 62 61 61 58 58 60 63 50 70
## [1489] 58 60 55 62 70 59 66 58 53 57 64 58 54 64 58 63 56 55 57 61 55 67 63 63
## [1513] 66 53 63 66 62 52 58 56 54 61 57 58 61 61 58 65 56 61 58 51 59 70 60 59
## [1537] 64 62 62 55 64 56 59 60 56 60 58 51 58 65 54 53 56 63 52 54 63 61 63 60
## [1561] 63 61 61 65 58 61 63 64 62 57 67 54 61 64 59 53 62 49 62 61 60 71 66 61
## [1585] 58 57 64 61 53 58 62 58 72 62 55 64 57 55 52 60 56 60 56 57 60 66 64 65
## [1609] 57 54 63 58 58 69 62 58 67 63 61 52 55 59 60 62 65 55 62 60 53 59 60 56
## [1633] 65 60 62 54 59 67 69 51 67 53 57 57 60 53 60 67 53 54 68 61 60 56 68 59
## [1657] 64 54 55 60 56 62 72 56 61 62 58 64 59 59 61 59 51 56 55 61 50 53 57 53
## [1681] 67 61 65 61 59 59 52 60 55 62 55 64 67 61 55 60 54 59 64 67 61 60 65 57
## [1705] 63 60 61 65 57 62 68 54 64 60 57 65 55 63 62 62 61 56 59 57 63 43 46 57
## [1729] 62 52 70 60 59 66 66 68 65 59 54 57 56 58 64 61 62 58 72 64 60 66 67 62
## [1753] 63 58 57 61 58 63 61 62 62 67 60 62 65 68 61 55 54 51 58 63 55 63 60 65
## [1777] 63 66 52 60 64 54 65 56 70 60 60 66 59 59 58 61 68 52 52 55 61 67 58 62
## [1801] 50 55 60 66 60 62 65 57 54 61 58 62 65 56 53 59 63 63 67 63 61 63 71 57
## [1825] 55 56 65 65 61 62 66 64 65 66 50 66 57 57 62 66 54 55 57 66 66 58 68 60
## [1849] 54 62 63 62 64 62 62 58 65 55 58 64 59 60 59 66 68 58 57 56 59 66 64 61
## [1873] 59 58 59 57 70 61 66 58 61 62 59 57 64 66 65 52 53 57 57 48 59 60 56 69
## [1897] 52 61 54 55 48 62 57 63 51 59 63 62 65 66 49 63 60 65 55 53 59 65 60 63
## [1921] 55 60 64 63 59 64 63 60 64 62 60 63 54 67 59 62 66 62 60 63 68 64 64 53
## [1945] 54 68 51 57 63 64 68 58 67 61 55 58 66 60 53 65 63 61 53 56 62 62 55 56
## [1969] 57 59 67 57 68 61 57 67 62 66 51 68 66 54 64 62 53 61 65 60 63 60 63 63
## [1993] 60 62 55 57 59 66 64 66

Problem 2. Draw a histogram of your data (population).

hist(population, breaks = 50, main = "Histogram with Density Curve")

Problem 3. Take 500 random samples of size 20 from the data (population). Draw plot and histogram of these samples.

# Set the sample size and number of samples
sample_size <- 20;
num_samples <- 500;

# Draw random samples
samples <- replicate(num_samples, sample(population, size = sample_size,replace = TRUE));


plot(samples)

hist(samples)

Problem 4. Compute the sample mean, sample standard deviation, population mean, and population standard deviaion.

# Calculate sample means
sample_means <- colMeans(samples);

# For sample
x_bar <- mean(sample_means);
std <- sd(sample_means);

print('Sample Mean and Variance')

## [1] "Sample Mean and Variance"

print(x_bar)

## [1] 60.0473

print(std**2) # a**x is the same as a^x

## [1] 1.128565

# For Population
mu <- mean(population);
sigma <- sd(population);

print('Population Mean and Variance')

## [1] "Population Mean and Variance"

print(mu)

## [1] 60.001

print((sigma**2)/sample_size)

## [1] 1.226063

Problem 5. Visualize those sample means taked from the population. Also, draw a normal curve with matching mean and variance and check that they are very close to each other.

# Visualize the sample means
hist(sample_means, breaks = 15, prob = TRUE, main = "Distribution of Sample Means",
     xlab = "Sample Mean")

curve(dnorm(x,mean=x_bar, sd=std), col="red", lwd=5, add=TRUE, from=x_bar-3*std, to=x_bar+3*std)