Computer HW3
Emily Perez
2025-05-04
Exercise 1: Based on weighting the most animals in the population, the American Angus Association reported that mature Angus cows had a mean weight of 1350 pounds with a standard deviation of 163 pounds. A research randomly measures 100 mature Angus cows. Use R commands to answers the following questions.
(a) Find the probability that the average weight of the 100 mature Angus cows is less than 1325 pounds.
P(\(\bar{X} < 1325\)):
mu = 1350
sigma = 163
n = 100
se = sigma / sqrt(n)
ans = pnorm(1325, mean = mu, sd = se)
ans## [1] 0.06254653(b) Find the probability that the average weight of the 100 mature Angus cows is between 1325 and 1375
P(\(1325 < \bar{X} < 1375\)):
mu = 1350
sigma = 163
n = 100
se = sigma / sqrt(n)
ans = pnorm(1375, mean = mu, sd = se) - pnorm(1325, mean = mu, sd = se)
ans## [1] 0.8749069Exercise 2: The lifetime of a special type of battery is a random variable with mean 40 and standard deviation 20 hours. Assume a stockpile of 50 independent such batteries.Use R commands to answer the following questions.
(a) What is the probability that the total lifetime of the 50 batteries is less than 1900 hours?
P(\(\sum X_i < 1900\)):
n = 50
mu = 40
sigma = 20
mean_total = n * mu
sd_total = sqrt(n) * sigma
ans = pnorm(1900, mean = mean_total, sd = sd_total)
ans## [1] 0.2397501(b) What is the probability that the total lifetime of the 50 batteries is more than 2200 hours?
P(\(\sum X_i > 2200\)):
n = 50
mu = 40
sigma = 20
mean_total = n * mu
sd_total = sqrt(n) * sigma
ans = pnorm(2200, mean = mean_total, sd = sd_total, lower.tail = FALSE)
ans## [1] 0.0786496Exercise 3: The life time of a laser (in hours) is exponentially distributed with \(\lambda = 1/80\). Two such lasers are operating independently. Use R commands to answer the following questions.
(a) Use a simulated sample of size 1000 to estimate the probability that the sum of the two lifetimes is greater than 100 hours.
P(\(X_1 + X_2 > 100\)):
lambda = 1/80
n_sim = 1000
laser1_lifetimes = rexp(n_sim, lambda)
laser2_lifetimes = rexp(n_sim, lambda)
sum_lifetimes = laser1_lifetimes + laser2_lifetimes
prob = mean(sum_lifetimes > 100)
prob## [1] 0.628(b) Estimate the probability that the both lasers last less than 50 hours.
P(\(X_1 < 50\) and \(X_2 < 50\)):
lambda = 1/80
n_sim = 1000
laser1_lifetimes = rexp(n_sim, lambda)
laser2_lifetimes = rexp(n_sim, lambda)
prob = mean((laser1_lifetimes < 50) & (laser2_lifetimes < 50))
prob## [1] 0.239Exercise 4: Generate 500 samples, each of which is of size 5 and 50 (n=5, 50), from the following distributions.
(a) For sample size 5 and 50, what are the averages of the 500 sample means, respectively? What are the theoretical expected values of sample means, respectively?
(b) For sample size 5 and 50, what are the variances of the 500 sample means, respectively? What are the theoretical variances of sample means, respectively?
(c) Construct histogram for sample means for n =5 and n =50, respectively.
(d) Construct normal probability plot of sample means for n =5 and n =50, respectively.
(e) Summarize your findings and use central limit theorem to explain your findings.
I. Uniform distribution on interval (0,5)
(a)
num_samples = 500
n1 = 5
n2 = 50
samples_uniform_n5 = replicate(num_samples, runif(n1, min = 0, max = 5))
samples_uniform_n50 = replicate(num_samples, runif(n2, min = 0, max = 5))
sample_means_uniform_n5 = colMeans(samples_uniform_n5)
sample_means_uniform_n50 = colMeans(samples_uniform_n50)
avg_sample_means_uniform_n5 = mean(sample_means_uniform_n5)
avg_sample_means_uniform_n50 = mean(sample_means_uniform_n50)
theoretical_mean_uniform = (0 + 5) / 2
cat("Uniform Distribution:\n")## Uniform Distribution:cat("(a) Average of sample means (n=5):", avg_sample_means_uniform_n5, "\n")## (a) Average of sample means (n=5): 2.515572cat("(a) Average of sample means (n=50):", avg_sample_means_uniform_n50, "\n")## (a) Average of sample means (n=50): 2.512252cat("(a) Theoretical expected value:", theoretical_mean_uniform, "\n\n")## (a) Theoretical expected value: 2.5(b)
var_sample_means_uniform_n5 = var(sample_means_uniform_n5)
var_sample_means_uniform_n50 = var(sample_means_uniform_n50)
theoretical_var_uniform = (5 - 0)^2 / 12
theoretical_var_sample_means_uniform_n5 = theoretical_var_uniform / n1
theoretical_var_sample_means_uniform_n50 = theoretical_var_uniform / n2
cat("(b) Variance of sample means (n=5):", var_sample_means_uniform_n5, "\n")## (b) Variance of sample means (n=5): 0.426051cat("(b) Variance of sample means (n=50):", var_sample_means_uniform_n50, "\n")## (b) Variance of sample means (n=50): 0.04378969cat("(b) Theoretical variance of sample means (n=5):", theoretical_var_sample_means_uniform_n5, "\n")## (b) Theoretical variance of sample means (n=5): 0.4166667cat("(b) Theoretical variance of sample means (n=50):", theoretical_var_sample_means_uniform_n50, "\n\n")## (b) Theoretical variance of sample means (n=50): 0.04166667(c)
hist(sample_means_uniform_n5, main = "Histogram of Sample Means (Uniform, n=5)", xlab = "Sample Mean")
hist(sample_means_uniform_n50, main = "Histogram of Sample Means (Uniform, n=50)", xlab = "Sample Mean")
### (d)
qqnorm(sample_means_uniform_n5, main = "Normal Q-Q Plot (Uniform, n=5)")
qqline(sample_means_uniform_n5)
qqnorm(sample_means_uniform_n50, main = "Normal Q-Q Plot (Uniform, n=50)")
qqline(sample_means_uniform_n50)
### (e)
cat("(e) For the uniform distribution, the averages of the sample means for both n=5 and n=50 are very close to the theoretical expected value of 2.5. The variances of the sample means decrease as the sample size increases, approaching the theoretical values (population variance divided by n).\n\n")## (e) For the uniform distribution, the averages of the sample means for both n=5 and n=50 are very close to the theoretical expected value of 2.5. The variances of the sample means decrease as the sample size increases, approaching the theoretical values (population variance divided by n).cat("The histograms show that as the sample size increases from n=5 to n=50, the distribution of the sample means becomes more bell-shaped and centered around the theoretical mean. The normal Q-Q plots indicate that the distribution of sample means for n=50 is much closer to a normal distribution than for n=5.\n\n")## The histograms show that as the sample size increases from n=5 to n=50, the distribution of the sample means becomes more bell-shaped and centered around the theoretical mean. The normal Q-Q plots indicate that the distribution of sample means for n=50 is much closer to a normal distribution than for n=5.cat("This illustrates the Central Limit Theorem. Even though the original uniform distribution is not normal, the distribution of the sample means approaches a normal distribution as the sample size grows. The variability of these sample means also decreases with larger sample sizes.\n\n")## This illustrates the Central Limit Theorem. Even though the original uniform distribution is not normal, the distribution of the sample means approaches a normal distribution as the sample size grows. The variability of these sample means also decreases with larger sample sizes.II. Binomial distribution with n = 15, p = 0.
(a)
n_binom = 15
p_binom = 0.3
samples_binom_n5 = replicate(num_samples, rbinom(n1, size = n_binom, prob = p_binom))
samples_binom_n50 = replicate(num_samples, rbinom(n2, size = n_binom, prob = p_binom))
sample_means_binom_n5 = colMeans(samples_binom_n5)
sample_means_binom_n50 = colMeans(samples_binom_n50)
avg_sample_means_binom_n5 = mean(sample_means_binom_n5)
avg_sample_means_binom_n50 = mean(sample_means_binom_n50)
theoretical_mean_binom = n_binom * p_binom
cat("Binomial Distribution (n=15, p=0.3):\n")## Binomial Distribution (n=15, p=0.3):cat("(a) Average of sample means (n=5):", avg_sample_means_binom_n5, "\n")## (a) Average of sample means (n=5): 4.4812cat("(a) Average of sample means (n=50):", avg_sample_means_binom_n50, "\n")## (a) Average of sample means (n=50): 4.49192cat("(a) Theoretical expected value:", theoretical_mean_binom, "\n\n")## (a) Theoretical expected value: 4.5(b)
var_sample_means_binom_n5 = var(sample_means_binom_n5)
var_sample_means_binom_n50 = var(sample_means_binom_n50)
theoretical_var_binom = n_binom * p_binom * (1 - p_binom)
theoretical_var_sample_means_binom_n5 = theoretical_var_binom / n1
theoretical_var_sample_means_binom_n50 = theoretical_var_binom / n2
cat("(b) Variance of sample means (n=5):", var_sample_means_binom_n5, "\n")## (b) Variance of sample means (n=5): 0.5889044cat("(b) Variance of sample means (n=50):", var_sample_means_binom_n50, "\n")## (b) Variance of sample means (n=50): 0.05906885cat("(b) Theoretical variance of sample means (n=5):", theoretical_var_sample_means_binom_n5, "\n")## (b) Theoretical variance of sample means (n=5): 0.63cat("(b) Theoretical variance of sample means (n=50):", theoretical_var_sample_means_binom_n50, "\n\n")## (b) Theoretical variance of sample means (n=50): 0.063(c)
hist(sample_means_binom_n5, main = "Histogram of Sample Means (Binomial, n=5)", xlab = "Sample Mean")
hist(sample_means_binom_n50, main = "Histogram of Sample Means (Binomial, n=50)", xlab = "Sample Mean")
### (d)
qqnorm(sample_means_binom_n5, main = "Normal Q-Q Plot (Binomial, n=5)")
qqline(sample_means_binom_n5)
qqnorm(sample_means_binom_n50, main = "Normal Q-Q Plot (Binomial, n=50)")
qqline(sample_means_binom_n50)
### (e)
cat("(e) For the binomial distribution, the averages of the sample means are close to the theoretical expected value of 4.5. The variances of the sample means decrease with the larger sample size, consistent with theory.\n\n")## (e) For the binomial distribution, the averages of the sample means are close to the theoretical expected value of 4.5. The variances of the sample means decrease with the larger sample size, consistent with theory.cat("The histograms show a trend towards a more symmetric, bell-shaped distribution as the sample size increases. The normal Q-Q plot for n=50 shows points lying closer to the straight line than for n=5, indicating a better approximation to a normal distribution.\n\n")## The histograms show a trend towards a more symmetric, bell-shaped distribution as the sample size increases. The normal Q-Q plot for n=50 shows points lying closer to the straight line than for n=5, indicating a better approximation to a normal distribution.cat("The Central Limit Theorem applies here as well. Even though the binomial distribution is discrete, the distribution of its sample means becomes approximately normal for larger sample sizes.\n\n")## The Central Limit Theorem applies here as well. Even though the binomial distribution is discrete, the distribution of its sample means becomes approximately normal for larger sample sizes.III. Exponential distribution with λ = 5
(a)
lambda_exp = 5
samples_exp_n5 = replicate(num_samples, rexp(n1, rate = lambda_exp))
samples_exp_n50 = replicate(num_samples, rexp(n2, rate = lambda_exp))
sample_means_exp_n5 = colMeans(samples_exp_n5)
sample_means_exp_n50 = colMeans(samples_exp_n50)
avg_sample_means_exp_n5 = mean(sample_means_exp_n5)
avg_sample_means_exp_n50 = mean(sample_means_exp_n50)
theoretical_mean_exp = 1 / lambda_exp
cat("Exponential Distribution (λ = 5):\n")## Exponential Distribution (λ = 5):cat("(a) Average of sample means (n=5):", avg_sample_means_exp_n5, "\n")## (a) Average of sample means (n=5): 0.195069cat("(a) Average of sample means (n=50):", avg_sample_means_exp_n50, "\n")## (a) Average of sample means (n=50): 0.2008036cat("(a) Theoretical expected value:", theoretical_mean_exp, "\n\n")## (a) Theoretical expected value: 0.2(b)
var_sample_means_exp_n5 = var(sample_means_exp_n5)
var_sample_means_exp_n50 = var(sample_means_exp_n50)
theoretical_var_exp = 1 / lambda_exp^2
theoretical_var_sample_means_exp_n5 = theoretical_var_exp / n1
theoretical_var_sample_means_exp_n50 = theoretical_var_exp / n2
cat("(b) Variance of sample means (n=5):", var_sample_means_exp_n5, "\n")## (b) Variance of sample means (n=5): 0.008409833cat("(b) Variance of sample means (n=50):", var_sample_means_exp_n50, "\n")## (b) Variance of sample means (n=50): 0.0007785209cat("(b) Theoretical variance of sample means (n=5):", theoretical_var_sample_means_exp_n5, "\n")## (b) Theoretical variance of sample means (n=5): 0.008cat("(b) Theoretical variance of sample means (n=50):", theoretical_var_sample_means_exp_n50, "\n\n")## (b) Theoretical variance of sample means (n=50): 8e-04(c)
hist(sample_means_exp_n5, main = "Histogram of Sample Means (Exponential, n=5)", xlab = "Sample Mean")
hist(sample_means_exp_n50, main = "Histogram of Sample Means (Exponential, n=50)", xlab = "Sample Mean")
### (d)
qqnorm(sample_means_exp_n5, main = "Normal Q-Q Plot (Exponential, n=5)")
qqline(sample_means_exp_n5)
qqnorm(sample_means_exp_n50, main = "Normal Q-Q Plot (Exponential, n=50)")
qqline(sample_means_exp_n50)
### (e)
cat("(e) For the exponential distribution, even though the original distribution is heavily skewed, the averages of the sample means are close to the theoretical expected value of 0.2. The variances of the sample means decrease significantly with the larger sample size.\n\n")## (e) For the exponential distribution, even though the original distribution is heavily skewed, the averages of the sample means are close to the theoretical expected value of 0.2. The variances of the sample means decrease significantly with the larger sample size.cat("The histograms clearly show the transformation from a skewed distribution (for n=5) to a more symmetric, bell-shaped distribution (for n=50). The normal Q-Q plot for n=50 is much closer to a straight line than for n=5, indicating a better approximation to a normal distribution.\n\n")## The histograms clearly show the transformation from a skewed distribution (for n=5) to a more symmetric, bell-shaped distribution (for n=50). The normal Q-Q plot for n=50 is much closer to a straight line than for n=5, indicating a better approximation to a normal distribution.cat("The Central Limit Theorem is powerfully demonstrated here. It shows that even with a non-symmetric original distribution, the distribution of the sample mean tends towards normality as the sample size grows.\n\n")## The Central Limit Theorem is powerfully demonstrated here. It shows that even with a non-symmetric original distribution, the distribution of the sample mean tends towards normality as the sample size grows.V. Poisson distribution with parameter μ = 2
(a)
mu_poisson = 2
samples_poisson_n5 = replicate(num_samples, rpois(n1, lambda = mu_poisson))
samples_poisson_n50 = replicate(num_samples, rpois(n2, lambda = mu_poisson))
sample_means_poisson_n5 = colMeans(samples_poisson_n5)
sample_means_poisson_n50 = colMeans(samples_poisson_n50)
avg_sample_means_poisson_n5 = mean(sample_means_poisson_n5)
avg_sample_means_poisson_n50 = mean(sample_means_poisson_n50)
theoretical_mean_poisson = mu_poisson
cat("Poisson Distribution (μ = 2):\n")## Poisson Distribution (μ = 2):cat("(a) Average of sample means (n=5):", avg_sample_means_poisson_n5, "\n")## (a) Average of sample means (n=5): 2.052cat("(a) Average of sample means (n=50):", avg_sample_means_poisson_n50, "\n")## (a) Average of sample means (n=50): 1.99484cat("(a) Theoretical expected value:", theoretical_mean_poisson, "\n\n")## (a) Theoretical expected value: 2(b)
var_sample_means_poisson_n5 = var(sample_means_poisson_n5)
var_sample_means_poisson_n50 = var(sample_means_poisson_n50)
theoretical_var_poisson = mu_poisson
theoretical_var_sample_means_poisson_n5 = theoretical_var_poisson / n1
theoretical_var_sample_means_poisson_n50 = theoretical_var_poisson / n2
cat("(b) Variance of sample means (n=5):", var_sample_means_poisson_n5, "\n")## (b) Variance of sample means (n=5): 0.3987335cat("(b) Variance of sample means (n=50):", var_sample_means_poisson_n50, "\n")## (b) Variance of sample means (n=50): 0.04157573cat("(b) Theoretical variance of sample means (n=5):", theoretical_var_sample_means_poisson_n5, "\n")## (b) Theoretical variance of sample means (n=5): 0.4cat("(b) Theoretical variance of sample means (n=50):", theoretical_var_sample_means_poisson_n50, "\n\n")## (b) Theoretical variance of sample means (n=50): 0.04(c)
hist(sample_means_poisson_n5, main = "Histogram of Sample Means (Poisson, n=5)", xlab = "Sample Mean")
hist(sample_means_poisson_n50, main = "Histogram of Sample Means (Poisson, n=50)", xlab = "Sample Mean")
### (d)
qqnorm(sample_means_poisson_n5, main = "Normal Q-Q Plot (Poisson, n=5)")
qqline(sample_means_poisson_n5)
qqnorm(sample_means_poisson_n50, main = "Normal Q-Q Plot (Poisson, n=50)")
qqline(sample_means_poisson_n50)
### (e)
cat("(e) For the Poisson distribution, the averages of the sample means for both sample sizes are very close to the theoretical mean of 2. The variances of the sample means decrease significantly from n=5 to n=50.\n\n")## (e) For the Poisson distribution, the averages of the sample means for both sample sizes are very close to the theoretical mean of 2. The variances of the sample means decrease significantly from n=5 to n=50.cat("The histograms show a clear progression from a slightly skewed discrete distribution (for n=5) towards a more continuous and symmetric bell shape (for n=50). The normal Q-Q plot for n=50 shows a much better fit to the normal line compared to n=5.\n\n")## The histograms show a clear progression from a slightly skewed discrete distribution (for n=5) towards a more continuous and symmetric bell shape (for n=50). The normal Q-Q plot for n=50 shows a much better fit to the normal line compared to n=5.cat("The Central Limit Theorem again explains these observations. As the sample size increases, the distribution of the sample means of a Poisson distribution approximates a normal distribution, and the spread of these means decreases.\n")## The Central Limit Theorem again explains these observations. As the sample size increases, the distribution of the sample means of a Poisson distribution approximates a normal distribution, and the spread of these means decreases.