needed  <- c("dplyr", "ggplot2", "plotly", "pwr", "effsize", "knitr", "kableExtra")
new_pkg <- needed[!needed %in% installed.packages()[, "Package"]]
if (length(new_pkg) > 0) {
  install.packages(new_pkg, repos = "https://cloud.r-project.org", type = "binary")
}
library(dplyr); library(ggplot2); library(plotly)
library(pwr);   library(effsize); library(knitr); library(kableExtra)
df_raw <- read.csv("C:/Users/IU Student/Downloads/googleplaystore.csv - Copy/googleplaystore.csv", stringsAsFactors = FALSE)

df <- df_raw %>%
  mutate(Rating = suppressWarnings(as.numeric(Rating))) %>%
  filter(Type %in% c("Free", "Paid"), !is.na(Rating), Rating >= 1, Rating <= 5) %>%
  mutate(
    AudienceGroup = case_when(
      Content.Rating %in% c("Mature 17+", "Adults only 18+") ~ "Mature",
      Content.Rating == "Everyone"                            ~ "Everyone",
      TRUE                                                    ~ NA_character_
    )
  )
cat("Usable rows:", nrow(df))
## Usable rows: 9366

Background

The Google Play Store dataset covers ~10,000 Android apps with ratings, install counts, pricing, and content classifications. Two questions occured to me when going over this dataset:

  1. Do paid apps earn higher user ratings than free ones — does price signal quality?
  2. Is an app’s content rating (Everyone vs. Mature 17+) associated with whether it charges upfront?

Hypothesis 1 — Neyman–Pearson Framework

Null Hypothesis

\[H_0: \mu_{\text{Free}} = \mu_{\text{Paid}}\]

The mean star rating is the same for free and paid apps. The alternative is two-sided (\(H_1: \mu_{\text{Free}} \neq \mu_{\text{Paid}}\)) because there is no prior theoretical reason to expect one direction over the other.

Choosing \(\alpha\), Power, and Effect Size

  • \(\alpha = 0.05\) — A false rejection here means a product manager wastes resources adjusting recommendation algorithms based on a phantom effect. The cost is moderate, so 5% is still okay. I remember Leon said in the lecture, the justification for any alpha level should come from the real world cost of Type I error, not just convention alone.
  • Power = 0.80 — Standard for exploratory research; a 20% chance of missing a real effect is acceptable here.
  • Minimum effect \(d = 0.20\) (Cohen’s small) — Differences smaller than this (~0.05 stars) would not change any practical decision.
free_r <- df %>% filter(Type == "Free") %>% pull(Rating)
paid_r <- df %>% filter(Type == "Paid") %>% pull(Rating)

ss <- pwr.t.test(d = 0.20, sig.level = 0.05, power = 0.80,
                 type = "two.sample", alternative = "two.sided")

cat("Required n per group:", ceiling(ss$n), "\n")
## Required n per group: 394
cat("Available -- Free:", length(free_r), " | Paid:", length(paid_r), "\n")
## Available -- Free: 8719  | Paid: 647
cat("Sufficient data?", ifelse(length(paid_r) >= ceiling(ss$n), "YES" , "NO"))
## Sufficient data? YES

We have adequate data to do the test.

Test and Results

t_res <- t.test(free_r, paid_r, alternative = "two.sided", var.equal = FALSE)
cd    <- cohen.d(free_r, paid_r)

cat("t =", round(t_res$statistic, 3), "| p =", format(t_res$p.value, scientific = TRUE), "\n")
## t = -3.62 | p = 3.149307e-04
cat("Mean (Free):", round(mean(free_r), 3), "| Mean (Paid):", round(mean(paid_r), 3), "\n")
## Mean (Free): 4.186 | Mean (Paid): 4.267
cat("Cohen's D:", round(abs(cd$estimate), 3), "(", cd$magnitude, "effect)")
## Cohen's D: 0.156 ( 1 effect)

My interpretation: I think I can reject \(H_0\) — paid apps are rated slightly higher. However, with Cohen’s D ≈ 0.156 the effect is negligible in practice. A fraction of a star difference will not meaningfully shift a consumer’s decision. Pricing alone is a poor proxy for quality.

Visualization 1 — Rating the Distributions of Free vs. Paid

mean_free <- mean(free_r)
mean_paid <- mean(paid_r)

dens_free <- density(free_r, bw = "SJ", from = 1, to = 5)
dens_paid <- density(paid_r, bw = "SJ", from = 1, to = 5)

fig1 <- plot_ly() %>%
  add_trace(x = dens_free$x, y = dens_free$y, type = "scatter", mode = "lines",
            fill = "tozeroy", fillcolor = "rgba(0,180,216,0.35)",
            line = list(color = "#0096C7", width = 2.5), name = "Free Apps",
            hovertemplate = "Rating: %{x:.2f}<br>Density: %{y:.4f}<extra>Free</extra>") %>%
  add_trace(x = dens_paid$x, y = dens_paid$y, type = "scatter", mode = "lines",
            fill = "tozeroy", fillcolor = "rgba(230,57,70,0.35)",
            line = list(color = "#E63946", width = 2.5), name = "Paid Apps",
            hovertemplate = "Rating: %{x:.2f}<br>Density: %{y:.4f}<extra>Paid</extra>") %>%
  add_segments(x = mean_free, xend = mean_free, y = 0, yend = max(dens_free$y) * 1.05,
               line = list(color = "#023E8A", width = 2, dash = "dash"),
               name = paste0("Mean Free = ", round(mean_free, 2))) %>%
  add_segments(x = mean_paid, xend = mean_paid, y = 0, yend = max(dens_paid$y) * 1.05,
               line = list(color = "#9D0208", width = 2, dash = "dash"),
               name = paste0("Mean Paid = ", round(mean_paid, 2))) %>%
  add_annotations(
    x = 2.1, y = max(dens_free$y) * 0.85,
    text = paste0("<b>Welch t-test</b><br>p = ",
                  format(t_res$p.value, digits = 3, scientific = TRUE),
                  "<br>Cohen's D = ", round(abs(cd$estimate), 3),
                  " (", cd$magnitude, ")"),
    showarrow = FALSE, font = list(size = 11),
    bgcolor = "rgba(255,255,255,0.85)", bordercolor = "#aaa", borderwidth = 1
  ) %>%
  layout(
    title  = list(text = "<b>Rating Distribution: Free vs. Paid Apps</b>",
                  font = list(size = 17, color = "#1a1a2e")),
    xaxis  = list(title = "<b>User Rating (1-5 Stars)</b>", range = c(1, 5),
                  gridcolor = "#e0e0e0"),
    yaxis  = list(title = "<b>Density</b>", gridcolor = "#e0e0e0"),
    legend = list(orientation = "h", x = 0.01, y = 1.08),
    plot_bgcolor = "#FAFAFA", paper_bgcolor = "#FFFFFF"
  )

fig1

The teal (Free) and coral (Paid) density curves overlap almost entirely. The tiny mean difference is only visible by comparing the two dashed vertical lines. This confirms that while the difference is statistically real, it is practically invisible, statistical significance and practical significance are not the same thing.

Hypothesis 2 (Fisher’s Significance Testing Framework)

Null Hypothesis

\[H_0: P(\text{Paid} \mid \text{Mature}) = P(\text{Paid} \mid \text{Everyone})\]

Content rating and payment model are independent. Fisher’s Exact Test is purpose-built for exactly this computing the exact hypergeometric probability of a contingency table as extreme as what we observed, under the assumption that the two categorical variables share no relationship.

h2_df <- df %>%
  filter(AudienceGroup %in% c("Everyone", "Mature")) %>%
  mutate(Type_bin = ifelse(Type == "Paid", "Paid", "Free"))

ct <- table(h2_df$AudienceGroup, h2_df$Type_bin)
ct <- ct[c("Everyone", "Mature"), c("Free", "Paid")]

cat("-- Contingency Table --\n");      print(ct)
## -- Contingency Table --
##           
##            Free Paid
##   Everyone 6868  552
##   Mature    447   17
cat("\n-- Row Proportions --\n");      print(round(prop.table(ct, 1), 4))
## 
## -- Row Proportions --
##           
##              Free   Paid
##   Everyone 0.9256 0.0744
##   Mature   0.9634 0.0366
f_res <- fisher.test(ct, alternative = "two.sided")
cat("\nFisher p-value:", format(f_res$p.value, scientific = TRUE))
## 
## Fisher p-value: 1.550224e-03
cat("\nOdds Ratio:    ", round(f_res$estimate, 3))
## 
## Odds Ratio:     0.473
cat("\n95% CI (OR):   [", round(f_res$conf.int[1], 3), ",",
    round(f_res$conf.int[2], 3), "]")
## 
## 95% CI (OR):   [ 0.271 , 0.774 ]

My interpretation: The p-value is essentially zero. The odds ratio of 0.473 tells us Mature-rated apps have lower odds of being paid. This makes intuitive sense: adult-targeted apps tend to monetize through advertising and in-app purchases rather than upfront pricing, because a paywall discourages the casual browsers who make up most of that audience.

Visualization 2 — Free vs. Paid by Content Rating Group

h2_plot <- h2_df %>%
  group_by(AudienceGroup, Type_bin) %>%
  summarise(count = n(), .groups = "drop") %>%
  group_by(AudienceGroup) %>%
  mutate(pct = round(count / sum(count) * 100, 2)) %>%
  ungroup()

fig2 <- plot_ly(
  data      = h2_plot,
  x         = ~AudienceGroup,
  y         = ~pct,
  color     = ~Type_bin,
  colors    = c("Free" = "#00B4D8", "Paid" = "#F4A261"),
  type      = "bar",
  text      = ~paste0("<b>", AudienceGroup, " -- ", Type_bin, "</b><br>",
                      "Count: ", count, "<br>Proportion: ", pct, "%"),
  hoverinfo = "text",
  marker    = list(line = list(color = "white", width = 1.5))
) %>%
  layout(
    barmode = "group",
    title   = list(text = "<b>Proportion of Free vs. Paid Apps by Audience Group</b>",
                   font = list(size = 17, color = "#1a1a2e")),
    xaxis   = list(title = "<b>Content Rating Group</b>"),
    yaxis   = list(title = "<b>Percentage of Apps (%)</b>",
                   gridcolor = "#e0e0e0", range = c(0, 105)),
    legend  = list(title = list(text = "<b>App Type</b>"),
                   orientation = "h", x = 0.35, y = 1.08),
    annotations = list(list(
      x = 1.35, y = 90, xref = "x", yref = "y",
      text = paste0("<b>Fisher's Exact</b><br>p ~= 0<br>OR = ",
                    round(f_res$estimate, 3)),
      showarrow = FALSE, font = list(size = 11),
      bgcolor = "rgba(255,255,255,0.9)", bordercolor = "#aaa", borderwidth = 1
    )),
    plot_bgcolor = "#FAFAFA", paper_bgcolor = "#FFFFFF"
  )

fig2

The orange “Paid” bar nearly vanishes in the mature group compared to the everyone group. This visual asymmetry directly reflects the odds ratio below 1. Mature rated apps are overwhelmingly free. This pattern that aligns with the freemium economy of adult-targeted mobile content.

Summary

Hypothesis Framework Result Decision
H1: Free vs. Paid ratings Neyman-Pearson (Welch t-test) t = -3.62, Cohen’s D = 0.156 Reject H0 — but negligible practical effect
H2: Audience x Paid/Free Fisher’s Significance (Exact Test) p ~= 0, OR = 0.473 Reject H0 — Mature apps skew heavily free

Further questions: Does the Free/Paid rating gap hold within individual app categories, or does it wash out in the aggregate? And is the Mature-app freemium pattern driven purely by mobile gaming’s dominance of that segment, or does it persist across genres even after controlling for category?