r-function |
Description |
|---|---|
rnorm(n,mean,sd) |
Generate n random values of standard normal distribution with the given mean and sd. |
hist(x) |
Plot the histogram of the data vector x, pass probability=TRUE to use density estimate. Pass breaks argument to specify edges of bins. Eg.: breaks = seq(0,1, by=0.1). breaks="FD" is a method based on data variability. |
seq(start, end, by=step) |
Generate a sequence. |
density(x) |
Estimate the density of the data vector x |
lines(x,y) |
Add a line to an existing plot. y may be omitted depending on x |
pnorm(q,mean=0, sd=1) |
Calculate the cumulative probability P(X\le q) for a normal distributed random variable X with a given mean and sd. |
diff(x) |
Calculate the first difference of a vector x. |
qnorm(p, mean=0, sd=1) |
Calculate the quantile of the normal distribution corresponding to the probability p (from left-tail). |
scale(x,center, scale) |
Scale data x to z-score using a given mean (center) and standard deviation (scale). E.g.: scale(x, center=5, scale=2) |
rbinom(s,size=n, prob=p) |
Generate s random binomial-distributed values with n trials and success probability p |
replicate(n, expr) |
Perform the Monte-Carlo simulation by replicating the experiment given by the expression expr n times. |
set.seed(123) # Set the seed for reproducibility
x <- rnorm(1000, mean = 0, sd = 1) # Generate data for a standard normal distribution
# Plot the data with density curve
hist(x, probability = TRUE, col = "lightblue", main = "Standard Normal Distribution")
lines(density(x), col = "red", lwd = 2)
# Find the area under the curve to the left of a certain value: P(z<1)
pnorm(1, mean = 0, sd = 1)[1] 0.8413447# Find the area under the curve to the right of a certain value: P(z>1)
1-pnorm(1, mean = 0, sd = 1)[1] 0.1586553# Find the area under the curve between two values: P(-1<z<1)
diff(pnorm(c(-1, 1), mean = 0, sd = 1))[1] 0.6826895# Find the value with a certain area under the curve to its left: critical value
alpha <- 0.05
qnorm(1-alpha, mean = 0, sd = 1) # find the critical Z score.[1] 1.644854x <- 80 # the individual value
mu <- 75 # the mean of the distribution
sigma <- 10 # the standard deviation of the distribution
# Calculate z-scores for the individual value using scale()
z_scores <- scale(x, center = mu, scale = sigma)
cat("Z-score:", z_scores, "\n") # print the z-scoreZ-score: 0.5 z <- (x - mu) / sigma # find the z-score by using the formula
cat("Z =", z, "\n") # print the z-scoreZ = 0.5 x1 <- 60
x2 <- 80
mu <- 69.6
sigma <- 11.3
# Find the probability that X is less than 60: P(X<60)
pnorm(x1, mean = mu, sd = sigma)[1] 0.1977856# Find the probability that X is great than 80: P(X>80)
1-pnorm(x2, mean = mu, sd = sigma)[1] 0.1786939# Find the probability between two values: P(60<X<80)
diff(pnorm(c(x1, x2), mean = mu, sd = sigma))[1] 0.6235205z <- 1.96 # the z-score
mu <- 100 # the mean of the distribution
sigma <- 15 # the standard deviation of the distribution
x <- z * sigma + mu # convert the z score to individual x value using formula
cat("X =", x, "\n") # print the individual x valueX = 129.4 # Set the seed for reproducibility
set.seed (123)
# Generate data
n <- 10 # sample size
p <- 0.5 # population proportion
samples <- replicate(50000, rbinom(1, size = n, prob = p))
# Calculate sample proportions of successes
sample_props <- samples / n
# Plot the histogram
hist(sample_props, breaks = seq( 0, 1, by = 0.1 ), col = "lightblue",
main = "Sampling Distribution of Sample Proportion")
#input the parameter values
mu <- 3.5
sigma <- 1.7
n <- 5
# Simulate sampling distribution
sample_means <- replicate(10000, mean(rnorm(n, mu, sigma)))
# Create a histogram of the sampling distribution of the sample mean
hist(sample_means, breaks ="FD", main = "Sampling Distribution of Sample Mean",
xlab = "Sample Mean", ylab = "Frequency", col = "lightblue",
border = "black")
mu <- 4 # True population mean
sigma <- 8 # Population standard deviation
sample_size <- 10 # Sample size
num_samples <- 10000 # Number of samples
# Function to calculate sample variance
sample_variance <- function(sample) {
n <- length(sample)
mean_sample <- mean(sample)
sum_squared_deviations <- sum((sample - mean_sample)^2)
return(sum_squared_deviations / (n - 1))
}
# Simulate sampling distribution
sample_variances <- replicate(num_samples, sample_variance(rnorm(sample_size,
mu, sigma)))
# Create a histogram of the sampling distribution of sample variance
hist(sample_variances, breaks = "FD", freq = FALSE,
main = "Sampling Distribution of Sample Variance",
xlab = "Sample Variance", ylab = "Frequency", col = "lightblue",
border = "black")
mu <- 171 # population mean
sigma <- 46 # population standard deviation
n <- 25 # sample size
x_lower <- 140
x_upper <- 211
# Find the probability between two X values
probability_range <- diff(pnorm(c(x_lower, x_upper), mean = mu, sd = sigma))
probability_range[1] 0.5575477# Find the probability between two mean values $x/bar$ (CLT)
standard_error <- sigma / sqrt(n) # Calculate the standard error of the sample mean
probability_range <- diff(pnorm(c(x_lower, x_upper), mean = mu,
sd = standard_error))# Find the probability
probability_range[1] 0.9996167