Questions & Answers
Question 1
# Sample data
mu = 1.5 # Population mean
x_bar = 1.52 # Sample mean
sigma = 0.2 # Standard deviation
n = 100 # Sample size
alpha = 0.05 # Level of significance
# Z-test statistic
z_stat = (x_bar - mu) / (sigma / sqrt(n))
print(paste("Z-statistic:", z_stat))
# P-value for two-tailed test
p_value = 2 * pnorm(-abs(z_stat))
print(paste("P-value:", p_value))
# Decision
if (p_value < alpha) {
print("Reject Null Hypothesis: The machine is NOT working to the standards set.")
} else {
print("Fail to Reject Null Hypothesis: The machine is working to the Standards set.")
}
1] "Z-statistic: 1"
> > # P-value for two-tailed test
> p_value = 2 * pnorm(-abs(z_stat))
> print(paste("P-value:", p_value))
[1] "P-value: 0.317310507862914"
> > # Decision
> if (p_value < alpha) {
+ print("Reject Null Hypothesis: The machine is NOT working to the standards set.")
+ } else {
+ print("Fail to Reject Null Hypothesis: The machine is working to the Standards set.")
+ }
[1] "Fail to Reject Null Hypothesis: The machine is working to the Standards set."Question 2
# Sample data
x1 <- 72 # Mean of boys
s1 <- 8 # SD of boys
n1 <- 32 # Sample size of boys
x2 <- 70 # Mean of girls
s2 <- 6 # SD of girls
n2 <- 36 # Sample size of girls
alpha <- 0.01 # Level of significance
# t-test statistic
t_stat <- (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
print(paste("t-statistic:", t_stat))
# Degrees of freedom (Welch-Satterthwaite)
df <- ((s1^2 / n1 + s2^2 / n2)^2) /
((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
print(paste("Degrees of freedom:", df))
# P-value (two-tailed test)
p_value <- 2 * pt(-abs(t_stat), df)
print(paste("P-value:", p_value))
# Decision
if (p_value < alpha) {
print("Reject Null Hypothesis: Boys and girls perform differently.")
} else {
print("Fail to Reject Null Hypothesis: Boys and girls perform equally.")
}
Output
R version 4.4.2 (2024-10-31 ucrt) -- "Pile of Leaves"
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> # Sample data
> mu = 1.5 # Population mean
> x_bar = 1.52 # Sample mean
> sigma = 0.2 # Standard deviation
> n = 100 # Sample size
> alpha = 0.05 # Level of significance
> > # Z-test statistic
> z_stat = (x_bar - mu) / (sigma / sqrt(n))
> print(paste("Z-statistic:", z_stat))
[1] "Z-statistic: 1"
> > # P-value for two-tailed test
> p_value = 2 * pnorm(-abs(z_stat))
> print(paste("P-value:", p_value))
[1] "P-value: 0.317310507862914"
> > # Decision
> if (p_value < alpha) {
+ print("Reject Null Hypothesis: The machine is NOT working to the standards set.")
+ } else {
+ print("Fail to Reject Null Hypothesis: The machine is working to the Standards set.")
+ }
[1] "Fail to Reject Null Hypothesis: The machine is working to the Standards set."
> # Sample data
> x1 <- 72 # Mean of boys
> s1 <- 8 # SD of boys
> n1 <- 32 # Sample size of boys
> > x2 <- 70 # Mean of girls
> s2 <- 6 # SD of girls
> n2 <- 36 # Sample size of girls
> > alpha <- 0.01 # Level of significance
> > # t-test statistic
> t_stat <- (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
> print(paste("t-statistic:", t_stat))
[1] "t-statistic: 1.15470053837925"
> > # Degrees of freedom (Welch-Satterthwaite)
> df <- ((s1^2 / n1 + s2^2 / n2)^2) /
+ ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
> print(paste("Degrees of freedom:", df))
[1] "Degrees of freedom: 57.1052631578947"
> > # P-value (two-tailed test)
> p_value <- 2 * pt(-abs(t_stat), df)
> print(paste("P-value:", p_value))
[1] "P-value: 0.253022304616441"
> > # Decision
> if (p_value < alpha) {
+ print("Reject Null Hypothesis: Boys and girls perform differently.")
+ } else {
+ print("Fail to Reject Null Hypothesis: Boys and girls perform equally.")
+ }
[1] "Fail to Reject Null Hypothesis: Boys and girls perform equally."Question 3
# Quality Control Analysis for Plastic Golf Tees
# Hypothesis Testing to Determine if Production Process is Satisfactory
# Input the data
tee_weights <- c(0.253, 0.253, 0.249, 0.247, 0.252, 0.253, 0.250, 0.251, 0.253, 0.248,
0.254, 0.254, 0.256, 0.251, 0.251, 0.253, 0.254, 0.253, 0.251, 0.253,
0.256, 0.256, 0.255, 0.248, 0.252, 0.254, 0.251, 0.253, 0.251, 0.250,
0.253, 0.255, 0.253, 0.254, 0.252, 0.256, 0.252, 0.251, 0.255, 0.253)
# Calculate basic statistics
n <- length(tee_weights)
sample_mean <- mean(tee_weights)
sample_sd <- sd(tee_weights)
target_mean <- 0.256
# Display basic statistics
cat("Sample size:", n, "\n")
cat("Sample mean:", sample_mean, "\n")
cat("Sample standard deviation:", sample_sd, "\n")
cat("Target mean:", target_mean, "\n")
# a) Null and Alternative Hypotheses
cat("\na) Hypotheses:\n")
cat("Null Hypothesis (H0): μ = 0.256 (The production process is performing satisfactorily)\n")
cat("Alternative Hypothesis (Ha): μ ≠ 0.256 (The production process is performing unsatisfactorily)\n")
# b) Calculate the test statistic (t-score)
standard_error <- sample_sd / sqrt(n)
t_stat <- (sample_mean - target_mean) / standard_error
cat("\nb) Test statistic (t-score):", t_stat, "\n")
# c) Determine the rejection region
alpha <- 0.05
critical_value <- qt(1 - alpha/2, df = n - 1)
cat("\nc) Rejection Region:\n")
cat("Critical values at alpha =", alpha, ":", -critical_value, "and", critical_value, "\n")
cat("Rejection region: t <", -critical_value, "or t >", critical_value, "\n")
# Determine if we reject the null hypothesis
p_value <- 2 * pt(-abs(t_stat), df = n - 1)
cat("\nAdditional Analysis:\n")
cat("p-value:", p_value, "\n")
if(abs(t_stat) > critical_value) {
cat("Decision: Reject H0 (The production process is performing unsatisfactorily)\n")
} else {
cat("Decision: Fail to reject H0 (The production process may be performing satisfactorily)\n")
}
# Perform one-sample t-test using R's built-in function
t_test_result <- t.test(tee_weights, mu = target_mean)
cat("\nComplete t-test results:\n")
print(t_test_result)
# Create visualizations
par(mfrow=c(2,1)) # Set up a 2x1 plotting area
# Histogram
hist(tee_weights, breaks = 10, main = "Distribution of Golf Tee Weights",
xlab = "Weight (ounces)", col = "lightblue", border = "white")
abline(v = target_mean, col = "red", lwd = 2, lty = 2)
abline(v = sample_mean, col = "blue", lwd = 2)
legend("topright", legend = c("Target Mean (0.256)", "Sample Mean"),
col = c("red", "blue"), lty = c(2, 1), lwd = 2)
# Boxplot
boxplot(tee_weights, main = "Boxplot of Golf Tee Weights",
ylab = "Weight (ounces)", col = "lightgreen")
abline(h = target_mean, col = "red", lwd = 2, lty = 2)
text(1.3, target_mean, "Target Mean (0.256)", col = "red", pos = 4)
# Reset plotting area
par(mfrow=c(1,1))
Sample size: 40
> cat("Sample mean:", sample_mean, "\n")
Sample mean: 0.252475
> cat("Sample standard deviation:", sample_sd, "\n")
Sample standard deviation: 0.002230183
> cat("Target mean:", target_mean, "\n")
Target mean: 0.256
> > # a) Null and Alternative Hypotheses
> cat("\na) Hypotheses:\n")
a) Hypotheses:
> cat("Null Hypothesis (H0): μ = 0.256 (The production process is performing satisfactorily)\n")
Null Hypothesis (H0): μ = 0.256 (The production process is performing satisfactorily)
> cat("Alternative Hypothesis (Ha): μ ≠ 0.256 (The production process is performing unsatisfactorily)\n")
Alternative Hypothesis (Ha): μ ≠ 0.256 (The production process is performing unsatisfactorily)
> > # b) Calculate the test statistic (t-score)
> standard_error <- sample_sd / sqrt(n)
> t_stat <- (sample_mean - target_mean) / standard_error
> cat("\nb) Test statistic (t-score):", t_stat, "\n")
b) Test statistic (t-score): -9.996513
> > # c) Determine the rejection region
> alpha <- 0.05
> critical_value <- qt(1 - alpha/2, df = n - 1)
> cat("\nc) Rejection Region:\n")
c) Rejection Region:
> cat("Critical values at alpha =", alpha, ":", -critical_value, "and", critical_value, "\n")
Critical values at alpha = 0.05 : -2.022691 and 2.022691
> cat("Rejection region: t <", -critical_value, "or t >", critical_value, "\n")
Rejection region: t < -2.022691 or t > 2.022691
> > # Determine if we reject the null hypothesis
> p_value <- 2 * pt(-abs(t_stat), df = n - 1)
> cat("\nAdditional Analysis:\n")
Additional Analysis:
> cat("p-value:", p_value, "\n")
p-value: 2.58434e-12
> > if(abs(t_stat) > critical_value) {
+ cat("Decision: Reject H0 (The production process is performing unsatisfactorily)\n")
+ } else {
+ cat("Decision: Fail to reject H0 (The production process may be performing satisfactorily)\n")
+ }
Decision: Reject H0 (The production process is performing unsatisfactorily)
> > # Perform one-sample t-test using R's built-in function
> t_test_result <- t.test(tee_weights, mu = target_mean)
> cat("\nComplete t-test results:\n")
Complete t-test results:
> print(t_test_result)
One Sample t-test
data: tee_weights
t = -9.9965, df = 39, p-value = 2.584e-12
alternative hypothesis: true mean is not equal to 0.256
95 percent confidence interval:
0.2517618 0.2531882
sample estimates:
mean of x
0.252475
> > # Create visualizations
> par(mfrow=c(2,1)) # Set up a 2x1 plotting area
> > # Histogram
> hist(tee_weights, breaks = 10, main = "Distribution of Golf Tee Weights",
+ xlab = "Weight (ounces)", col = "lightblue", border = "white")
> abline(v = target_mean, col = "red", lwd = 2, lty = 2)
> abline(v = sample_mean, col = "blue", lwd = 2)
> legend("topright", legend = c("Target Mean (0.256)", "Sample Mean"),
+ col = c("red", "blue"), lty = c(2, 1), lwd = 2)
> > # Boxplot
> boxplot(tee_weights, main = "Boxplot of Golf Tee Weights",
+ ylab = "Weight (ounces)", col = "lightgreen")
> abline(h = target_mean, col = "red", lwd = 2, lty = 2)
> text(1.3, target_mean, "Target Mean (0.256)", col = "red", pos = 4)
> > # Reset plotting area
> par(mfrow=c(1,1))
References
Information
- date: 2025.03.25
- time: 12:27