R: اختبار الفرضيات R

# Two-Proportion Test
prop.test(c(80, 100), c(1000, 1000))
# 2-sample test for equality of proportions
# X-squared = 2.46, df = 1, p-value = 0.117
# Conclusion: p > 0.05, Differences **Not significant** (It might be a coincidence.)

# Example: Product Weight Standard 100g, test 10 products
weights <- c(98, 102, 99, 101, 100, 97, 103, 99, 101, 100)

t.test(weights, mu = 100)
# One Sample t-test
# t = 0, df = 9, p-value = 1
# 95% CI: [98.7, 101.3]
# mean of x = 100
# Conclusion: p = 1, **Don't reject H0**, the mean is equal to 100g

# Example: A/B Testing the conversion times for two groups
group_a <- c(12, 15, 14, 13, 16, 14, 15, 13, 14, 15)
group_b <- c(10, 11, 12, 13, 11, 10, 12, 11, 13, 12)

t.test(group_a, group_b)
# Welch Two Sample t-test
# t = 5.32, df = 14.6, p-value = 0.0001
# 95% CI: [1.36, 3.24]
# Conclusion: p < 0.05, **Reject H0**, Group A was significantly slower than Group B

# Example: Before medication vs after medication in the same group of patients
before <- c(180, 175, 190, 170, 185)
after <- c(165, 160, 175, 160, 170)

t.test(before, after, paired = TRUE)
# Paired t-test
# t = 8.5, df = 4, p-value = 0.001
# Conclusion: p < 0.05, **Significant decline** (The medication is effective)

# One-sided test (right-tailed)
t.test(x, mu = 100, alternative = "greater")

# One-sided test (left-tailed)
t.test(x, mu = 100, alternative = "less")

# Example: Historical Product Pass Rate 95%, now 92 out of 100 are Qualified
prop.test(92, 100, p = 0.95)
# 1-sample proportions test
# X-squared = 1.27, df = 1, p-value = 0.26
# Conclusion: p > 0.05, Pass Rate **No significant decrease**

# Example: Group A 1000 people, 80 conversions; Group B 1000 people, 100 conversions
prop.test(c(80, 100), c(1000, 1000))
# 2-sample test for equality of proportions
# X-squared = 2.46, df = 1, p-value = 0.117
# 95% CI: [-0.046, 0.006]
# Conclusion: p > 0.05, **The difference is not significant** (It might be a coincidence.)

# Example: Dice Fairness Test (toss 60 times)
observed <- c(8, 12, 10, 11, 9, 10)  # Number of times each face appears
expected <- rep(10, 6)  # Expect uniformity

chisq.test(observed, p = expected / sum(expected))
# Chi-squared test
# X-squared = 1.2, df = 5, p-value = 0.945
# Conclusion: p > 0.05, Dice **is fair** (Do not reject the uniformity assumption)

# Example: Gender vs Purchasing Behavior
gender <- c("M", "M", "F", "F", "M", "F", "M", "F")
purchase <- c("Buy", "Not buying", "Buy", "Buy", "Not buying", "Buy", "Buy", "Not buying")

tab <- table(gender, purchase)
print(tab)
#      Not buying Buy
#   M    1  3
#   F    1  3

chisq.test(tab)
# X-squared = 0, df = 1, p-value = 1
# Conclusion: p > 0.05, Gender and Purchasing **Independent**

# Note: Any 2x2 Table with Expected Frequencies < 5, use Fisher's Exact Test
fisher.test(tab)

# Chi-Square Test Requirements: Expected Frequency for Each Cell >= 5
# Use Fisher's Exact Test when not satisfied
result <- chisq.test(tab)
result$expected  # View Expected Frequency

# Example: Effect of 3 Fertilizer Types on Crop Yields
fertilizer <- c(rep("A", 10), rep("B", 10), rep("C", 10))
yield <- c(20, 22, 21, 19, 23, 20, 21, 22, 19, 20,   # A
           25, 26, 24, 27, 25, 26, 27, 25, 24, 26,  # B
           18, 19, 17, 18, 19, 20, 19, 18, 19, 20)  # C

df <- data.frame(fertilizer, yield)

# Single-factor ANOVA
model <- aov(yield ~ fertilizer, data = df)
summary(model)
#              Df Sum Sq Mean Sq F value Pr(>F)    
# fertilizer     2  220.0   110.0   91.7  <2e-16 ***
# Residuals     27   32.4     1.2
# Conclusion: p < 0.05, 3 Fertilizer Types **Significant difference**

# Post-hoc comparison: Tukey HSD
TukeyHSD(model)
#   diff        lwr        upr     p adj
# B-A   5.0  3.91       6.09    0.000
# C-A  -1.5 -2.59      -0.41    0.005
# C-B  -6.5 -7.59      -5.41    0.000

# Non-parametric paired test (Wilcoxon signed-rank)
before <- c(180, 175, 190, 170, 185)
after <- c(165, 160, 175, 160, 170)

wilcox.test(before, after, paired = TRUE)
# V = 15, p-value = 0.0625
# Conclusion: p > 0.05 (Not significant), however, the sample size is small

# Independent non-parametric (Mann-Whitney U)
group_a <- c(12, 15, 14, 13, 16)
group_b <- c(10, 11, 12, 13, 11)
wilcox.test(group_a, group_b)
# W = 25, p-value = 0.028
# Conclusion: p < 0.05, the difference is significant

# Cohen's d Calculation
cohens_d <- function(x, y) {
  n1 <- length(x)
  n2 <- length(y)
  s_pooled <- sqrt(((n1-1)*var(x) + (n2-1)*var(y)) / (n1 + n2 - 2))
  (mean(x) - mean(y)) / s_pooled
}

# ============================================
# Comprehensive Evaluation of Marketing Campaign Performance
# Features: t-test / Proportionality Test / ANOVA combined
# ============================================

set.seed(42)

# 1. A/B Test Conversion Time
group_a <- rnorm(50, mean = 15, sd = 3)  # Original Page
group_b <- rnorm(50, mean = 13, sd = 3)  # New Page

cat("=== A/B Test: Conversion Time ===\n")
cat("A Group Mean:", round(mean(group_a), 2), "s\n")
cat("B Group Mean:", round(mean(group_b), 2), "s\n")
cat("Differences:", round(mean(group_a) - mean(group_b), 2), "s\n\n")

# 2. t-test
t_result <- t.test(group_a, group_b)
print(t_result)
cat("\n")

# 3. Cohen's d
n1 <- length(group_a)
n2 <- length(group_b)
s_pooled <- sqrt(((n1-1)*var(group_a) + (n2-1)*var(group_b)) / (n1+n2-2))
cohens_d <- (mean(group_a) - mean(group_b)) / s_pooled
cat("Cohen's d (Effect size):", round(cohens_d, 3), "\n")
cat("Effect Size:", ifelse(abs(cohens_d) > 0.8, "L",
                  ifelse(abs(cohens_d) > 0.5, "Mid", "S")), "\n\n")

# 4. A/B Testing Conversion Rates
convert_a <- 80
convert_b <- 100
visitors_a <- 1000
visitors_b <- 1000

cat("=== A/B Test: Conversion Rate ===\n")
cat("A Group Conversion Rate:", round(convert_a / visitors_a * 100, 2), "%\n")
cat("B Group Conversion Rate:", round(convert_b / visitors_b * 100, 2), "%\n\n")

# 5. Proportionality Test
prop_result <- prop.test(c(convert_a, convert_b),
                          c(visitors_a, visitors_b))
print(prop_result)
cat("\n")

# 6. Comparison of 3 Marketing Strategies (ANOVA)
strategy <- c(rep("Email", 20), rep("Text Message", 20), rep("Push", 20))
sales <- c(rnorm(20, 100, 15),
           rnorm(20, 110, 15),
           rnorm(20, 105, 15))
df_strategy <- data.frame(strategy, sales)

cat("=== Comparison of 3 Sales Strategies (ANOVA) ===\n")
model <- aov(sales ~ strategy, data = df_strategy)
summary(model)
cat("\nPost-hoc comparison:\n")
print(TukeyHSD(model))

# 7. Gender and Purchasing Behavior (Chi-square)
gender <- sample(c("M", "F"), 200, replace = TRUE)
purchase <- sample(c("Buy", "Not buying"), 200, replace = TRUE,
                   prob = c(0.3, 0.7))
tab <- table(gender, purchase)
cat("\n=== Gender vs Purchase Chi-Square Test ===\n")
print(tab)
cat("\n")
chisq_result <- chisq.test(tab)
print(chisq_result)

# 8. Comprehensive Report
cat("\n=== Summary of Decisions ===\n")
cat("1. Conversion Time: Group A is", round(cohens_d, 2), "standard deviations slower than Group B",
    ifelse(t_result$p.value < 0.05, " (Significant)", " (Not significant)"), "\n", sep = "")
cat("2. Conversion Rate: A vs B Difference",
    ifelse(prop_result$p.value < 0.05, "Significant", "Not significant"), "\n")
cat("3. 3 Strategies:",
    ifelse(summary(model)[[1]]$`Pr(>F)`[1] < 0.05, "Significant difference", "No difference"), "\n")

=== A/B Test: Conversion Time ===
A Group Mean: 14.87 s
B Group Mean: 12.94 s
Differences: 1.93 s

Welch Two Sample t-test
t = 3.18, df = 96.3, p-value = 0.002
Cohen's d (Effect size): 0.643
Effect Size: Mid

=== Summary of Decisions ===
1. Conversion Time: Group A is 0.64 standard deviations slower than Group B (Significant)
2. Conversion Rate: A vs B The difference is not significant
3. 3 Strategies: Significant difference