Claude Binomial Analysis

Author

Robert W. Walker

Published

February 24, 2026

Prompt

This dataset represents Admissions decisions by two traits: M.F. captures Male/Female election and Dept captures the department, labeled A through F. Please provide exact binomial confidence intervals, 95%, for the probability of Admission for Males, for Females, and then for each of Males and Females by department. Next, reanalyse the data using the normal approximation and provide differences overall and for each department. Collect your responses and code into a quarto markdown document with python code chunks that I can download as text.

Overview

This document analyses the UC Berkeley admissions dataset. Each row represents one applicant with columns M.F (Male/Female), Admit (Yes/No), and Dept (A–F). We compute:

Exact (Clopper–Pearson) 95% binomial confidence intervals for the probability of admission for Males overall, Females overall, and each gender × department cell.
Normal-approximation (Wald) 95% CIs for the same quantities, and the Male − Female difference in admission probability, overall and by department.

Setup

How’s that done?

import pandas as pd
import numpy as np
from scipy.stats import norm, beta as beta_dist

# ── load data ────────────────────────────────────────────────────────────────
df = pd.read_csv("./data/UCB-Admit.csv")
df.columns = df.columns.str.strip()          # remove BOM / whitespace
df["admitted"] = (df["Admit"] == "Yes").astype(int)

print(df.shape)

(4526, 4)

How’s that done?

print(df.head())

    M.F Admit Dept  admitted
0  Male   Yes    A         1
1  Male   Yes    A         1
2  Male   Yes    A         1
3  Male   Yes    A         1
4  Male   Yes    A         1

How’s that done?

print(df["M.F"].value_counts())

M.F
Male      2691
Female    1835
Name: count, dtype: int64

How’s that done?

print(df["Dept"].value_counts().sort_index())

Dept
A    933
B    585
C    918
D    792
E    584
F    714
Name: count, dtype: int64

Part 1 — Exact (Clopper–Pearson) Binomial CIs

The exact 95% CI for a binomial proportion with \(k\) successes out of \(n\) trials is

\[ \left[\, B\!\left(\tfrac{\alpha}{2};\, k,\, n-k+1\right),\; B\!\left(1-\tfrac{\alpha}{2};\, k+1,\, n-k\right) \right] \]

where \(B(p; a, b)\) is the \(p\)-th quantile of a Beta distribution.
scipy.stats.binom.proportion_confint (method 'exact') implements this directly.

How’s that done?

from scipy.stats import beta as beta_dist

def exact_ci(k, n, alpha=0.05):
    """Clopper-Pearson exact 95% CI via the Beta distribution."""
    lo = beta_dist.ppf(alpha / 2,     k,     n - k + 1) if k > 0 else 0.0
    hi = beta_dist.ppf(1 - alpha / 2, k + 1, n - k)     if k < n else 1.0
    return k / n, lo, hi

def summarise(sub):
    n   = len(sub)
    k   = sub["admitted"].sum()
    p, lo, hi = exact_ci(k, n)
    return pd.Series({"n": n, "admits": k, "p_hat": p,
                      "CI_low": lo, "CI_high": hi})

1a. Overall by gender

How’s that done?

overall_exact = (
    df.groupby("M.F")
      .apply(summarise)
      .reset_index()
)
overall_exact.columns = ["Gender", "n", "Admits", "p_hat", "CI_low", "CI_high"]

# Format for display
disp = overall_exact.copy()
for c in ["p_hat", "CI_low", "CI_high"]:
    disp[c] = disp[c].map("{:.4f}".format)
print(disp.to_string(index=False))

Gender      n  Admits  p_hat CI_low CI_high
Female 1835.0   557.0 0.3035 0.2826  0.3252
  Male 2691.0  1198.0 0.4452 0.4263  0.4642

1b. By gender × department

How’s that done?

dept_exact = (
    df.groupby(["M.F", "Dept"])
      .apply(summarise)
      .reset_index()
)
dept_exact.columns = ["Gender", "Dept", "n", "Admits", "p_hat", "CI_low", "CI_high"]

disp2 = dept_exact.copy()
for c in ["p_hat", "CI_low", "CI_high"]:
    disp2[c] = disp2[c].map("{:.4f}".format)
print(disp2.to_string(index=False))

Gender Dept     n  Admits  p_hat CI_low CI_high
Female    A 108.0    89.0 0.8241 0.7390  0.8906
Female    B  25.0    17.0 0.6800 0.4650  0.8505
Female    C 593.0   202.0 0.3406 0.3025  0.3803
Female    D 375.0   131.0 0.3493 0.3011  0.4000
Female    E 393.0    94.0 0.2392 0.1978  0.2845
Female    F 341.0    24.0 0.0704 0.0456  0.1029
  Male    A 825.0   512.0 0.6206 0.5865  0.6538
  Male    B 560.0   353.0 0.6304 0.5889  0.6705
  Male    C 325.0   120.0 0.3692 0.3166  0.4242
  Male    D 417.0   138.0 0.3309 0.2859  0.3784
  Male    E 191.0    53.0 0.2775 0.2153  0.3467
  Male    F 373.0    22.0 0.0590 0.0373  0.0879

Part 2 — Normal (Wald) Approximation CIs and Differences

The Wald CI for a proportion is

\[\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

For the difference \(\delta = \hat{p}_M - \hat{p}_F\) the standard error is

\[\text{SE}_\delta = \sqrt{\frac{\hat{p}_M(1-\hat{p}_M)}{n_M} + \frac{\hat{p}_F(1-\hat{p}_F)}{n_F}}\]

How’s that done?

z = norm.ppf(0.975)   # ≈ 1.96

def wald_ci(k, n):
    p = k / n
    se = np.sqrt(p * (1 - p) / n)
    return p, p - z * se, p + z * se

def diff_wald(km, nm, kf, nf):
    pm, lom, him = wald_ci(km, nm)
    pf, lof, hif = wald_ci(kf, nf)
    delta = pm - pf
    se_d  = np.sqrt(pm*(1-pm)/nm + pf*(1-pf)/nf)
    return delta, delta - z*se_d, delta + z*se_d

2a. Overall Wald CIs

How’s that done?

def wald_summarise(sub):
    n = len(sub)
    k = sub["admitted"].sum()
    p, lo, hi = wald_ci(k, n)
    return pd.Series({"n": n, "admits": k,
                      "p_hat": p, "Wald_low": lo, "Wald_high": hi})

overall_wald = (
    df.groupby("M.F")
      .apply(wald_summarise)
      .reset_index()
)
overall_wald.columns = ["Gender", "n", "Admits", "p_hat", "Wald_low", "Wald_high"]

disp3 = overall_wald.copy()
for c in ["p_hat", "Wald_low", "Wald_high"]:
    disp3[c] = disp3[c].map("{:.4f}".format)
print(disp3.to_string(index=False))

Gender      n  Admits  p_hat Wald_low Wald_high
Female 1835.0   557.0 0.3035   0.2825    0.3246
  Male 2691.0  1198.0 0.4452   0.4264    0.4640

2b. Department-level Wald CIs

How’s that done?

dept_wald = (
    df.groupby(["M.F", "Dept"])
      .apply(wald_summarise)
      .reset_index()
)
dept_wald.columns = ["Gender", "Dept", "n", "Admits", "p_hat", "Wald_low", "Wald_high"]

disp4 = dept_wald.copy()
for c in ["p_hat", "Wald_low", "Wald_high"]:
    disp4[c] = disp4[c].map("{:.4f}".format)
print(disp4.to_string(index=False))

Gender Dept     n  Admits  p_hat Wald_low Wald_high
Female    A 108.0    89.0 0.8241   0.7523    0.8959
Female    B  25.0    17.0 0.6800   0.4971    0.8629
Female    C 593.0   202.0 0.3406   0.3025    0.3788
Female    D 375.0   131.0 0.3493   0.3011    0.3976
Female    E 393.0    94.0 0.2392   0.1970    0.2814
Female    F 341.0    24.0 0.0704   0.0432    0.0975
  Male    A 825.0   512.0 0.6206   0.5875    0.6537
  Male    B 560.0   353.0 0.6304   0.5904    0.6703
  Male    C 325.0   120.0 0.3692   0.3168    0.4217
  Male    D 417.0   138.0 0.3309   0.2858    0.3761
  Male    E 191.0    53.0 0.2775   0.2140    0.3410
  Male    F 373.0    22.0 0.0590   0.0351    0.0829

2c. Male − Female differences (overall and by department)

How’s that done?

def make_diff_table(wald_df):
    male   = wald_df[wald_df["Gender"] == "Male"].copy()
    female = wald_df[wald_df["Gender"] == "Female"].copy()

    grp_col = [c for c in wald_df.columns if c not in
               ("Gender", "n", "Admits", "p_hat", "Wald_low", "Wald_high")]

    rows = []
    if grp_col:
        keys = male[grp_col[0]].unique()
        for k in sorted(keys):
            rm = male  [male  [grp_col[0]] == k].iloc[0]
            rf = female[female[grp_col[0]] == k].iloc[0]
            delta, dlo, dhi = diff_wald(rm["Admits"], rm["n"],
                                        rf["Admits"], rf["n"])
            rows.append({grp_col[0]: k,
                         "p_Male": rm["p_hat"], "p_Female": rf["p_hat"],
                         "Diff (M-F)": delta,
                         "Diff_CI_low": dlo, "Diff_CI_high": dhi})
    else:
        rm = male.iloc[0]; rf = female.iloc[0]
        delta, dlo, dhi = diff_wald(rm["Admits"], rm["n"],
                                    rf["Admits"], rf["n"])
        rows.append({"p_Male": rm["p_hat"], "p_Female": rf["p_hat"],
                     "Diff (M-F)": delta,
                     "Diff_CI_low": dlo, "Diff_CI_high": dhi})

    return pd.DataFrame(rows)

# Overall difference
overall_diff = make_diff_table(overall_wald)
disp5 = overall_diff.copy()
for c in ["p_Male", "p_Female", "Diff (M-F)", "Diff_CI_low", "Diff_CI_high"]:
    disp5[c] = disp5[c].map("{:.4f}".format)
print("=== Overall Male − Female difference ===")

=== Overall Male − Female difference ===

How’s that done?

print(disp5.to_string(index=False))

p_Male p_Female Diff (M-F) Diff_CI_low Diff_CI_high
0.4452   0.3035     0.1416      0.1134       0.1698

How’s that done?

# Department-level differences
dept_diff = make_diff_table(dept_wald)
disp6 = dept_diff.copy()
for c in ["p_Male", "p_Female", "Diff (M-F)", "Diff_CI_low", "Diff_CI_high"]:
    disp6[c] = disp6[c].map("{:.4f}".format)
print("\n=== Department-level Male − Female differences ===")


=== Department-level Male − Female differences ===

How’s that done?

print(disp6.to_string(index=False))

Dept p_Male p_Female Diff (M-F) Diff_CI_low Diff_CI_high
   A 0.6206   0.8241    -0.2035     -0.2825      -0.1244
   B 0.6304   0.6800    -0.0496     -0.2368       0.1375
   C 0.3692   0.3406     0.0286     -0.0363       0.0935
   D 0.3309   0.3493    -0.0184     -0.0845       0.0477
   E 0.2775   0.2392     0.0383     -0.0379       0.1145
   F 0.0590   0.0704    -0.0114     -0.0476       0.0248

Part 3 — Side-by-side comparison: Exact vs. Wald

How’s that done?

# Merge exact and wald for overall
cmp_overall = overall_exact.merge(
    overall_wald[["Gender", "Wald_low", "Wald_high"]],
    on="Gender"
)
print("=== Overall: Exact vs Wald CI bounds ===")

=== Overall: Exact vs Wald CI bounds ===

How’s that done?

disp7 = cmp_overall.copy()
for c in ["p_hat", "CI_low", "CI_high", "Wald_low", "Wald_high"]:
    disp7[c] = disp7[c].map("{:.4f}".format)
print(disp7.to_string(index=False))

Gender      n  Admits  p_hat CI_low CI_high Wald_low Wald_high
Female 1835.0   557.0 0.3035 0.2826  0.3252   0.2825    0.3246
  Male 2691.0  1198.0 0.4452 0.4263  0.4642   0.4264    0.4640

How’s that done?

# By department
cmp_dept = dept_exact.merge(
    dept_wald[["Gender", "Dept", "Wald_low", "Wald_high"]],
    on=["Gender", "Dept"]
)
print("\n=== By Department: Exact vs Wald CI bounds ===")


=== By Department: Exact vs Wald CI bounds ===

How’s that done?

disp8 = cmp_dept.copy()
for c in ["p_hat", "CI_low", "CI_high", "Wald_low", "Wald_high"]:
    disp8[c] = disp8[c].map("{:.4f}".format)
print(disp8.to_string(index=False))

Gender Dept     n  Admits  p_hat CI_low CI_high Wald_low Wald_high
Female    A 108.0    89.0 0.8241 0.7390  0.8906   0.7523    0.8959
Female    B  25.0    17.0 0.6800 0.4650  0.8505   0.4971    0.8629
Female    C 593.0   202.0 0.3406 0.3025  0.3803   0.3025    0.3788
Female    D 375.0   131.0 0.3493 0.3011  0.4000   0.3011    0.3976
Female    E 393.0    94.0 0.2392 0.1978  0.2845   0.1970    0.2814
Female    F 341.0    24.0 0.0704 0.0456  0.1029   0.0432    0.0975
  Male    A 825.0   512.0 0.6206 0.5865  0.6538   0.5875    0.6537
  Male    B 560.0   353.0 0.6304 0.5889  0.6705   0.5904    0.6703
  Male    C 325.0   120.0 0.3692 0.3166  0.4242   0.3168    0.4217
  Male    D 417.0   138.0 0.3309 0.2859  0.3784   0.2858    0.3761
  Male    E 191.0    53.0 0.2775 0.2153  0.3467   0.2140    0.3410
  Male    F 373.0    22.0 0.0590 0.0373  0.0879   0.0351    0.0829

Discussion

Simpson’s Paradox

The aggregate data (Part 1a / 2a) suggest Males are admitted at a higher rate than Females overall. However, when we examine each department separately (Parts 1b / 2b / 2c), the Male − Female difference is small and often negative (females admitted at equal or higher rates within most departments).

This reversal is a classic demonstration of Simpson’s Paradox: females disproportionately applied to departments with low overall admission rates (C–F), dragging down the female aggregate even though they fared similarly or better within each department.

Exact vs. Normal approximation

At the large sample sizes present here the Wald and Clopper–Pearson intervals agree closely overall. Some department-level cells (especially small female counts in Dept A) show modest discrepancies, which is expected: the Wald approximation degrades when \(n\) is small or \(\hat p\) is close to 0 or 1. The exact interval is preferred in those situations.