Introduction

This report analyzes customer behavior at a fictional online retail store. The two datasets used here are synthetic: they were generated with an AI tool (Claude) and contain no real people or private information. Patterns were built into the data on purpose so that the statistical tests could be demonstrated.

Dataset 1 (between-subjects, N = 150) contains different customers in different groups. It is used for the chi-square test, the Pearson correlation, and the independent-samples t-test.

Dataset 2 (within-subjects, N = 60) contains the same customers measured twice, before and after a checkout redesign. It is used for the dependent (paired) t-test.

head(d1)

##   customer_id membership_type repeat_purchase monthly_visits monthly_spending
## 1        C001        Standard              No             15           153.80
## 2        C002        Standard             Yes             12           100.45
## 3        C003         Premium             Yes              7            59.78
## 4        C004         Premium             Yes             12           114.99
## 5        C005         Premium             Yes             13            87.81
## 6        C006        Standard             Yes             16            91.10
##   contacted_support satisfaction_score
## 1                No                 73
## 2                No                 78
## 3                No                 85
## 4                No                 66
## 5                No                 93
## 6                No                 79

head(d2)

##   customer_id satisfaction_before satisfaction_after
## 1        W001                  65                 80
## 2        W002                  37                 38
## 3        W003                  51                 54
## 4        W004                  55                 63
## 5        W005                  57                 72
## 6        W006                  64                 75

All tests use a significance level of α = .05.

Analysis 1: Chi-Square Test of Independence

Research Question

Is membership type associated with whether a customer made a repeat purchase?

Variables

membership_type (categorical): Free, Standard, Premium
repeat_purchase (categorical): Yes, No

Hypotheses

H₀: Membership type and repeat purchase are independent (not associated).
H₁: Membership type and repeat purchase are associated.

Statistical Test

A chi-square test of independence is appropriate because both variables are categorical. The assumption that all expected counts are at least 5 is checked below.

Results

chi_table <- table(d1$membership_type, d1$repeat_purchase)
chi_table

##           
##            No Yes
##   Free     39  21
##   Premium   4  31
##   Standard 28  27

# Row percentages
round(prop.table(chi_table, margin = 1) * 100, 1)

##           
##              No  Yes
##   Free     65.0 35.0
##   Premium  11.4 88.6
##   Standard 50.9 49.1

chi_result <- chisq.test(chi_table)
chi_result

## 
##  Pearson's Chi-squared test
## 
## data:  chi_table
## X-squared = 25.894, df = 2, p-value = 2.384e-06

# Assumption check: expected counts
chi_result$expected

##           
##                  No      Yes
##   Free     28.40000 31.60000
##   Premium  16.56667 18.43333
##   Standard 26.03333 28.96667

# Effect size: Cramer's V
n_chi <- sum(chi_table)
k_chi <- min(nrow(chi_table) - 1, ncol(chi_table) - 1)
cramers_v <- sqrt(as.numeric(chi_result$statistic) / (n_chi * k_chi))
cramers_v

## [1] 0.4154816

barplot(t(prop.table(chi_table, margin = 1)),
        beside = TRUE, legend.text = TRUE,
        main = "Repeat Purchase by Membership Type",
        xlab = "Membership Type", ylab = "Proportion",
        col = c("gray70", "steelblue"))

All expected counts were above 5 (the smallest was 16.57), so the assumption was met.

A chi-square test of independence showed a significant association between membership type and repeat purchase, χ²(2, N = 150) = 25.89, p < .001, Cramér’s V = .42.

Interpretation

Membership type and repeat purchasing are related. We reject the null hypothesis. Premium members were the most likely to buy again (88.6%), followed by Standard members (49.1%), while Free members were the least likely (35.0%). Cramér’s V of .42 indicates a large effect. In plain terms, customers with better memberships are much more likely to come back and buy again.

Analysis 2: Pearson Correlation

Research Question

Is the number of monthly website visits related to monthly spending?

Variables

monthly_visits (numeric): number of website visits per month
monthly_spending (numeric): dollars spent per month

Hypotheses

H₀: There is no linear relationship between monthly visits and monthly spending (ρ = 0).
H₁: There is a linear relationship between monthly visits and monthly spending (ρ ≠ 0).

Statistical Test

A Pearson correlation is appropriate because both variables are numeric. Assumptions are normality of each variable (Shapiro-Wilk test) and a linear relationship (scatterplot).

Results

# Normality checks (p > .05 = normal)
shapiro.test(d1$monthly_visits)

## 
##  Shapiro-Wilk normality test
## 
## data:  d1$monthly_visits
## W = 0.99174, p-value = 0.5363

shapiro.test(d1$monthly_spending)

## 
##  Shapiro-Wilk normality test
## 
## data:  d1$monthly_spending
## W = 0.99278, p-value = 0.6539

cor_result <- cor.test(d1$monthly_visits, d1$monthly_spending,
                       method = "pearson")
cor_result

## 
##  Pearson's product-moment correlation
## 
## data:  d1$monthly_visits and d1$monthly_spending
## t = 11.26, df = 148, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.5823912 0.7570995
## sample estimates:
##       cor 
## 0.6792546

# Effect size: r squared
cor_result$estimate^2

##       cor 
## 0.4613868

plot(d1$monthly_visits, d1$monthly_spending,
     main = "Monthly Visits vs. Monthly Spending",
     xlab = "Monthly Visits", ylab = "Monthly Spending ($)",
     pch = 19, col = "steelblue")
abline(lm(monthly_spending ~ monthly_visits, data = d1), col = "red", lwd = 2)

Both variables met the normality assumption (visits: p = .536; spending: p = .654), and the scatterplot shows a linear pattern, so Pearson is appropriate.

A Pearson correlation showed a strong, positive, significant relationship between monthly website visits and monthly spending, r(148) = .68, p < .001, 95% CI [.58, .76]. Visits explained about 46% of the variance in spending (r² = .46).

Interpretation

We reject the null hypothesis. Customers who visit the website more often tend to spend more money. The relationship is strong and positive. Note that correlation does not prove that visiting causes spending; it only shows that the two go up together.

Analysis 3: Independent-Samples t-Test

Research Question

Do customers who contacted customer support differ in satisfaction from customers who did not?

Variables

contacted_support (categorical, two groups): Yes, No
satisfaction_score (numeric, 1 to 100)

Hypotheses

H₀: The mean satisfaction scores of the two groups are equal (μ_Yes = μ_No).
H₁: The mean satisfaction scores of the two groups are not equal (μ_Yes ≠ μ_No).

Statistical Test

An independent-samples t-test (Welch version) is appropriate because two separate groups of customers are compared on a numeric outcome. Assumptions checked are normality within each group (Shapiro-Wilk) and equal variances (F test).

Results

# Descriptive statistics
tapply(d1$satisfaction_score, d1$contacted_support, mean)

##    No   Yes 
## 74.73 65.22

tapply(d1$satisfaction_score, d1$contacted_support, sd)

##        No       Yes 
## 10.193933  8.748854

table(d1$contacted_support)

## 
##  No Yes 
## 100  50

# Normality within each group (p > .05 = normal)
shapiro.test(d1$satisfaction_score[d1$contacted_support == "Yes"])

## 
##  Shapiro-Wilk normality test
## 
## data:  d1$satisfaction_score[d1$contacted_support == "Yes"]
## W = 0.95709, p-value = 0.06712

shapiro.test(d1$satisfaction_score[d1$contacted_support == "No"])

## 
##  Shapiro-Wilk normality test
## 
## data:  d1$satisfaction_score[d1$contacted_support == "No"]
## W = 0.98859, p-value = 0.5525

# Equal variances (p > .05 = equal)
var.test(satisfaction_score ~ contacted_support, data = d1)

## 
##  F test to compare two variances
## 
## data:  satisfaction_score by contacted_support
## F = 1.3576, num df = 99, denom df = 49, p-value = 0.237
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.8159788 2.1686570
## sample estimates:
## ratio of variances 
##           1.357629

# Welch t-test
ind_t <- t.test(satisfaction_score ~ contacted_support, data = d1,
                var.equal = FALSE)
ind_t

## 
##  Welch Two Sample t-test
## 
## data:  satisfaction_score by contacted_support
## t = 5.9322, df = 112.46, p-value = 3.366e-08
## alternative hypothesis: true difference in means between group No and group Yes is not equal to 0
## 95 percent confidence interval:
##   6.333753 12.686247
## sample estimates:
##  mean in group No mean in group Yes 
##             74.73             65.22

# Effect size: Cohen's d
g_yes <- d1$satisfaction_score[d1$contacted_support == "Yes"]
g_no  <- d1$satisfaction_score[d1$contacted_support == "No"]
sd_pooled <- sqrt(((length(g_yes) - 1) * var(g_yes) +
                   (length(g_no)  - 1) * var(g_no)) /
                  (length(g_yes) + length(g_no) - 2))
cohens_d_ind <- abs(mean(g_yes) - mean(g_no)) / sd_pooled
cohens_d_ind

## [1] 0.9764596

boxplot(satisfaction_score ~ contacted_support, data = d1,
        main = "Satisfaction by Support Contact",
        xlab = "Contacted Support", ylab = "Satisfaction Score",
        col = c("lightblue", "lightgreen"))

Normality was met in both groups (No: p = .553; Yes: p = .067), and variances were equal (p = .237), so the assumptions were satisfied.

An independent-samples t-test showed that customers who did not contact support (M = 74.73, SD = 10.19) reported significantly higher satisfaction than customers who did contact support (M = 65.22, SD = 8.75), t(112.46) = 5.93, p < .001, 95% CI [6.33, 12.69], Cohen’s d = 0.98.

Interpretation

We reject the null hypothesis. Customers who contacted support were, on average, about 9.5 points less satisfied than those who did not. Cohen’s d of 0.98 is a large effect. This may suggest that customers who need support are having problems, which lowers their satisfaction.

Analysis 4: Dependent (Paired) t-Test

Research Question

Did customer satisfaction change from before to after the checkout redesign?

Variables

satisfaction_before (numeric): satisfaction score before the redesign
satisfaction_after (numeric): satisfaction score after the redesign

The same 60 customers were measured both times.

Hypotheses

H₀: The mean difference between after and before scores is zero (μ_diff = 0).
H₁: The mean difference between after and before scores is not zero (μ_diff ≠ 0).

Statistical Test

A dependent (paired) t-test is appropriate because the same customers were measured twice. The assumption is that the difference scores are normally distributed (Shapiro-Wilk test and histogram).

Results

# Descriptive statistics
mean(d2$satisfaction_before)

## [1] 63.83333

sd(d2$satisfaction_before)

## [1] 11.50117

mean(d2$satisfaction_after)

## [1] 69.76667

sd(d2$satisfaction_after)

## [1] 14.35549

# Difference scores (after minus before)
d2$difference <- d2$satisfaction_after - d2$satisfaction_before
mean(d2$difference)

## [1] 5.933333

sd(d2$difference)

## [1] 5.871121

# Normality of the difference scores (p > .05 = normal)
shapiro.test(d2$difference)

## 
##  Shapiro-Wilk normality test
## 
## data:  d2$difference
## W = 0.97678, p-value = 0.3078

# Paired t-test
paired_t <- t.test(d2$satisfaction_after, d2$satisfaction_before,
                   paired = TRUE)
paired_t

## 
##  Paired t-test
## 
## data:  d2$satisfaction_after and d2$satisfaction_before
## t = 7.828, df = 59, p-value = 1.069e-10
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  4.416662 7.450005
## sample estimates:
## mean difference 
##        5.933333

# Effect size: Cohen's d for paired data
cohens_d_paired <- mean(d2$difference) / sd(d2$difference)
cohens_d_paired

## [1] 1.010596

hist(d2$difference,
     main = "Change in Satisfaction (After - Before)",
     xlab = "Difference Score", col = "steelblue", border = "white")

The difference scores were normally distributed (Shapiro-Wilk p = .308), so the assumption was met.

A paired-samples t-test showed that satisfaction was significantly higher after the redesign (M = 69.77, SD = 14.36) than before the redesign (M = 63.83, SD = 11.50), t(59) = 7.83, p < .001, mean difference = 5.93, 95% CI [4.42, 7.45], Cohen’s d = 1.01.

Interpretation

We reject the null hypothesis. On average, customers’ satisfaction increased by about 6 points after the checkout redesign. Cohen’s d of 1.01 is a large effect. Because the same customers were measured both times, this suggests the redesign was followed by higher satisfaction. Without a control group, we cannot be certain the redesign itself caused the increase.

Conclusion

All four analyses produced statistically significant results:

Analysis	Test	Key Result
Membership type and repeat purchase	Chi-square	χ²(2, N = 150) = 25.89, p < .001, V = .42
Visits and spending	Pearson correlation	r(148) = .68, p < .001
Support contact and satisfaction	Independent t-test	t(112.46) = 5.93, p < .001, d = 0.98
Satisfaction before and after redesign	Paired t-test	t(59) = 7.83, p < .001, d = 1.01

Higher membership levels were linked to more repeat purchases, more visits were linked to more spending, customers who contacted support were less satisfied, and satisfaction rose after the checkout redesign.

Limitations: The data are synthetic, and the effects were built in on purpose, so real-world data would likely show weaker and messier patterns. These results also show associations and differences, not proof of cause and effect.

Customer Satisfaction and Purchasing Behavior: Final Project

Shruti Gajbe

October 8, 2026

Introduction

Analysis 1: Chi-Square Test of Independence

Research Question

Variables

Hypotheses

Statistical Test

Results

Interpretation

Analysis 2: Pearson Correlation

Research Question

Variables

Hypotheses

Statistical Test

Results

Interpretation

Analysis 3: Independent-Samples t-Test

Research Question

Variables

Hypotheses

Statistical Test

Results

Interpretation

Analysis 4: Dependent (Paired) t-Test

Research Question

Variables

Hypotheses

Statistical Test

Results

Interpretation

Conclusion