←Back to the bookChapter 160 · One-Sample t-Test: Coffee Fill Weight

Chapter 160 · Capstone 1 · One-Sample t-Test

Is the Coffee Underfilled?

A full pass through the twelve-step framework on one messy quality-control dataset. We meet a day of fill-weight measurements, clean the impossible entries, check whether the data are normal enough for a t-test, and test the mean against the 340 g on the label, then turn the numbers into a decision.

import numpy as np, pandas as pd
import matplotlib.pyplot as plt
from scipy import stats
import pingouin as pg
plt.rcParams.update({"figure.dpi":110,"font.size":11,"axes.spines.top":False,"axes.spines.right":False,
    "axes.grid":True,"grid.alpha":0.22,"axes.titleweight":"bold","axes.titlesize":12.5,"axes.titlelocation":"left"})
GOLD, AMBER, DARK, RD, GR, MUT = "#b45309", "#d97706", "#78350f", "#dc2626", "#16a34a", "#94a3b8"
TARGET = 340.0   # grams printed on the label

BASE = "https://raw.githubusercontent.com/johnfisher-ai/Statistics-Data-Science-AI-Visual-Book/main/data/"
fn = "capstone-coffee-fill-weight.xlsx"
try: raw = pd.read_excel("../../data/" + fn, sheet_name="fills")
except FileNotFoundError: raw = pd.read_excel(BASE + fn, sheet_name="fills")
print("loaded", raw.shape[0], "rows")
raw.head()

loaded 78 rows

	bag_id	line	shift	fill_g
0	QC-001	Line 1	Night	342.2
1	QC-002	Line 2	Morning	343.4
2	QC-003	Line 2	Morning	342.2
3	QC-004	Line 2	Afternoon	337.9
4	QC-005	Line 1	Morning	343.7

Step 1 · Goal and hypotheses

Bay Roasters sells whole-bean bags labeled 340 g. Overfilling gives away coffee and money; underfilling shorts the customer and breaks net-content law. Quality control pulled a sample of bags across one production day. Our question is simple: is the mean fill weight on target, or is the line running light or heavy?

We write it as a two-sided one-sample test of the mean against the target value, and fix the significance level before looking at the result.

• H₀ (null): the mean fill weight equals 340 g.
• H₁ (alternative): the mean fill weight is not 340 g.
• α = 0.05, two-sided.

Step 2 · Know the data types

One measurement matters here, and it is continuous: fill_g, a weight in grams that we can average. The other columns (bag_id, line, shift) are nominal labels, useful for describing the sample but not part of this test. A continuous outcome compared to a single target value points to the one-sample t-test family.

print(raw.dtypes)
print()
print("a continuous outcome (fill_g) vs one target value  ->  one-sample t-test family")

bag_id        str
line          str
shift         str
fill_g    float64
dtype: object

a continuous outcome (fill_g) vs one target value  ->  one-sample t-test family

Step 3 · Describe the sample

Before any test, meet the data. Note the row count, the summary statistics, and how the sample spreads across lines and shifts.

s = raw.fill_g
print(f"rows {len(raw)}   unique bag_id {raw.bag_id.nunique()}   "
      f"fill_g recorded on {s.notna().sum()}")
print(f"fill_g, raw and uncleaned:  median {s.median():.2f} g   "
      f"quartiles {s.quantile(.25):.2f} to {s.quantile(.75):.2f}")
print(f"                            min {s.min():.2f} g   max {s.max():.2f} g   "
      f"mean {s.mean():.2f} g")
display(pd.crosstab(raw.line, raw["shift"], margins=True, margins_name="all"))

rows 78   unique bag_id 76   fill_g recorded on 75
fill_g, raw and uncleaned:  median 341.40 g   quartiles 340.00 to 342.75
                            min 0.00 g   max 3410.00 g   mean 373.56 g

shift	Afternoon	Morning	Night	all
line
Line 1	15	11	13	39
Line 2	12	15	12	39
all	27	26	25	78

The summary already looks suspicious: the minimum and maximum are nowhere near 340 g, and there are more rows than unique bag IDs. That is the raw-data mess we clean next, but first it helps to see it.

Step 4 · Visualize (first look, warts and all)

A histogram of the raw weights makes the problem obvious. A cluster sits near 340 g where real bags should be, and a few points are stranded far away, an empty-pan zero, a decimal typo, and a fat-finger extra digit.

fig, ax = plt.subplots(1, 2, figsize=(11, 3.4))
ax[0].hist(raw.fill_g.dropna(), bins=40, color=GOLD, alpha=0.85)
ax[0].set_title("Raw weights: impossible values")
ax[0].set_xlabel("fill_g (grams)"); ax[0].set_ylabel("bags")
plaus = raw.fill_g[(raw.fill_g > 300) & (raw.fill_g < 380)]
ax[1].hist(plaus, bins=20, color=AMBER, alpha=0.9)
ax[1].axvline(TARGET, color=RD, lw=2, ls="--", label="label 340 g")
ax[1].set_title("Zoomed in: the real distribution")
ax[1].set_xlabel("fill_g (grams)"); ax[1].legend()
plt.tight_layout(); plt.show()

Step 5 · Prepare and clean

Three problems to fix, in order: duplicate log rows, missing weights, and impossible values. A valid net weight for a 340 g bag sits comfortably inside 300 to 380 g, so anything outside that range is a recording error, not a light or heavy bag. We remove them and keep a clean analysis sample.

before = len(raw)
df = raw.drop_duplicates()
after_dupes = len(df)
df = df.dropna(subset=["fill_g"])
after_na = len(df)
bad = df[(df.fill_g <= 300) | (df.fill_g >= 380)]
df = df[(df.fill_g > 300) & (df.fill_g < 380)].copy()
print(f"start rows          : {before}")
print(f"after drop_duplicates: {after_dupes}   (removed {before-after_dupes} duplicate rows)")
print(f"after dropna fill_g : {after_na}   (removed {after_dupes-after_na} blank weights)")
print("impossible values removed:", sorted(bad.fill_g.tolist()))
print(f"clean analysis sample: n = {len(df)}")

start rows          : 78
after drop_duplicates: 76   (removed 2 duplicate rows)
after dropna fill_g : 73   (removed 3 blank weights)
impossible values removed: [0.0, 34.1, 3410.0]
clean analysis sample: n = 70

x = df.fill_g.values
n = len(x)
mean, sd = x.mean(), x.std(ddof=1)
se = sd/np.sqrt(n)
print(f"n = {n}    mean = {mean:.3f} g    sd = {sd:.3f} g    se = {se:.3f} g")
print(f"observed gap from target: {mean-TARGET:+.3f} g")

n = 70    mean = 341.229 g    sd = 2.105 g    se = 0.252 g
observed gap from target: +1.229 g

Step 6 · Name the design

This is about as simple as a study design gets: one sample of independent measurements, compared to a single fixed value (the 340 g label). There are no groups and no pairing. That design plus a continuous outcome is exactly what the one-sample t-test is built for, provided its assumption holds.

Step 7 · Check the assumption

The one-sample t-test assumes the data are roughly normal (or that the sample is large enough for the Central Limit Theorem to carry it). We check three ways: the Shapiro-Wilk test, the Kolmogorov-Smirnov test, and a Q-Q plot by eye. The plot matters as much as the p-values.

sw = stats.shapiro(x)
ks = stats.kstest(x, stats.norm(loc=mean, scale=sd).cdf)
print(f"Shapiro-Wilk : W = {sw.statistic:.3f}, p = {sw.pvalue:.3f}   -> {'normal enough' if sw.pvalue>0.05 else 'not normal'}")
print(f"Kolmogorov-Smirnov: D = {ks.statistic:.3f}, p = {ks.pvalue:.3f}   -> {'normal enough' if ks.pvalue>0.05 else 'not normal'}")
print(f"skew = {stats.skew(x):.2f}   kurtosis = {stats.kurtosis(x):.2f}")

fig, ax = plt.subplots(1, 2, figsize=(11, 3.4))
ax[0].hist(x, bins=18, color=AMBER, alpha=0.9, density=True)
xs = np.linspace(x.min(), x.max(), 200)
ax[0].plot(xs, stats.norm.pdf(xs, mean, sd), color=DARK, lw=2, label="normal fit")
ax[0].axvline(TARGET, color=RD, lw=2, ls="--", label="target 340 g")
ax[0].set_title("Clean weights with a normal curve overlaid"); ax[0].set_xlabel("fill_g (grams)"); ax[0].legend(fontsize=9)
stats.probplot(x, dist="norm", plot=ax[1])
ax[1].get_lines()[0].set(marker="o", markerfacecolor=GOLD, markeredgecolor="white", markersize=6)
ax[1].get_lines()[1].set(color=DARK, lw=2)
ax[1].set_title("Q-Q: the points hug the line")
plt.tight_layout(); plt.show()

Shapiro-Wilk : W = 0.973, p = 0.142   -> normal enough
Kolmogorov-Smirnov: D = 0.080, p = 0.735   -> normal enough
skew = -0.58   kurtosis = 1.01

Step 8 · Decide, and fix if needed

Shapiro-Wilk and Kolmogorov-Smirnov both fail to reject normality, and the Q-Q points sit close to the straight line, so the normality assumption is satisfied. No transformation is needed. We proceed with the parametric one-sample t-test, and as a safety check we will also run its rank-based twin, the Wilcoxon signed-rank test, which does not assume normality. If the two disagreed, we would trust the rank-based one; here they will agree.

Step 9 · Choose the test

• Primary: one-sample t-test of the mean against 340 g (assumption met).
• Robustness check: Wilcoxon signed-rank test on the differences from 340 g.
• Reported alongside: Cohen's d (effect size) and the 95% confidence interval for the mean.

Step 10 · Run the test and report fully

We run the t-test with scipy and confirm it with pingouin, which reports the effect size, the confidence interval, and the achieved power in one table. Then the Wilcoxon cross-check.

t, p = stats.ttest_1samp(x, TARGET)
print(f"one-sample t-test:  t({n-1}) = {t:.3f},  p = {p:.2e}")

res = pg.ttest(x, TARGET)
print("\npingouin one-sample t-test:")
print(res[["T","dof","alternative","cohen_d","power"]].round(4).to_string(index=False))
ci = res["CI95"].iloc[0]
d  = float(res["cohen_d"].iloc[0])
print(f"\n95% CI for the mean fill weight: [{ci[0]:.2f}, {ci[1]:.2f}] g")
print(f"Cohen's d = {d:.3f}  ({'small' if d<0.5 else 'medium' if d<0.8 else 'large'} effect)")

w, pw = stats.wilcoxon(x - TARGET)
print(f"\nWilcoxon signed-rank (robustness): W = {w:.1f}, p = {pw:.2e}  -> same conclusion")

one-sample t-test:  t(69) = 4.884,  p = 6.47e-06

pingouin one-sample t-test:
     T  dof alternative  cohen_d  power
4.8841   69   two-sided   0.5838 0.9978

95% CI for the mean fill weight: [340.73, 341.73] g
Cohen's d = 0.584  (medium effect)

Wilcoxon signed-rank (robustness): W = 404.5, p = 7.50e-06  -> same conclusion

# The result in one picture: the sample mean and its 95% CI against the target.
tcrit = stats.t.ppf(0.975, n-1)
lo, hi = mean - tcrit*se, mean + tcrit*se
fig, ax = plt.subplots(figsize=(8.4, 3.4))
ax.hist(x, bins=18, color=AMBER, alpha=0.55)
ax.axvline(TARGET, color=RD, lw=2.4, ls="--", label=f"target {TARGET:.0f} g")
ax.axvline(mean, color=DARK, lw=2.4, label=f"sample mean {mean:.2f} g")
ax.axvspan(lo, hi, color=GOLD, alpha=0.20, label=f"95% CI [{lo:.2f}, {hi:.2f}]")
ax.set_title("The 95% CI sits entirely above 340 g: a real, if small, overfill")
ax.set_xlabel("fill_g (grams)"); ax.set_ylabel("bags"); ax.legend(fontsize=9)
plt.tight_layout(); plt.show()

Step 11 · Estimate, do not just test

The p-value answered a yes-or-no question: are the bags filled to 340 g? They are not. It said nothing about by how much, and that is the only number the plant can act on. An interval answers the question the p-value ducks.

# A percentile bootstrap: resample the data with replacement, recompute, repeat.
# Chapter 73 introduced this; it is how we get an interval when no formula exists.
def boot_ci(data, stat, B=10000, seed=11, ci=95):
    rng = np.random.default_rng(seed)
    data = np.asarray(data)
    reps = np.array([stat(rng.choice(data, size=len(data), replace=True)) for _ in range(B)])
    half = (100 - ci) / 2
    return np.percentile(reps, half), np.percentile(reps, 100 - half), reps

def boot_ci_pairs(a, b, stat, B=10000, seed=11, ci=95):
    """Bootstrap for two paired/related arrays: resample ROW indices together."""
    rng = np.random.default_rng(seed)
    a, b = np.asarray(a), np.asarray(b)
    idx = np.arange(len(a))
    reps = np.array([stat(a[i], b[i]) for i in
                     (rng.choice(idx, size=len(idx), replace=True) for _ in range(B))])
    reps = reps[np.isfinite(reps)]
    half = (100 - ci) / 2
    return np.percentile(reps, half), np.percentile(reps, 100 - half), reps

def boot_ci_groups(groups, stat, B=10000, seed=11, ci=95):
    """Bootstrap for k independent groups: resample within each group."""
    rng = np.random.default_rng(seed)
    groups = [np.asarray(g) for g in groups]
    reps = []
    for _ in range(B):
        rs = [rng.choice(g, size=len(g), replace=True) for g in groups]
        v = stat(rs)
        if np.isfinite(v):
            reps.append(v)
    reps = np.array(reps)
    half = (100 - ci) / 2
    return np.percentile(reps, half), np.percentile(reps, 100 - half), reps

# 1. The overfill itself. The CI for the mean shifts straight onto the overfill scale.
overfill = mean - TARGET
over_lo, over_hi = lo - TARGET, hi - TARGET
print(f"overfill per bag       {overfill:+.2f} g    95% CI [{over_lo:+.2f}, {over_hi:+.2f}] g")

# 2. The same interval expressed as the thing a plant manager budgets for.
print(f"per million bags       {overfill*1000:,.0f} kg   95% CI [{over_lo*1000:,.0f}, {over_hi*1000:,.0f}] kg")

# 3. An interval for the effect size, which is almost always reported bare.
d_lo, d_hi, _ = boot_ci(x, lambda s: (s.mean() - TARGET) / s.std(ddof=1))
print(f"Cohen's d              {d:.2f}         95% CI [{d_lo:.2f}, {d_hi:.2f}]")

# 4. Margin of error: half the width of the interval for the mean.
print(f"\nmargin of error on the mean fill: +/- {(hi-lo)/2:.2f} g at 95% confidence")
print(f"to halve it we would need roughly {n*4} bags, since precision improves with the square root of n")

overfill per bag       +1.23 g    95% CI [+0.73, +1.73] g
per million bags       1,229 kg   95% CI [727, 1,730] kg

Cohen's d              0.58         95% CI [0.32, 0.94]

margin of error on the mean fill: +/- 0.50 g at 95% confidence
to halve it we would need roughly 280 bags, since precision improves with the square root of n

What the intervals add. The overfill is not "1.23 g". It is somewhere between about 0.7 and 1.7 g per bag, and every downstream figure inherits that width: the giveaway is between roughly 700 and 1,700 kg per million bags, not a precise 1,229. Quoting the point estimate alone would let a planner build a budget on a number the data cannot support to that precision.

And the effect size gets an interval too. Cohen's d of 0.58 comes with a 95% interval of roughly 0.32 to 0.94, which runs from "small" all the way to "large" in the conventional labels. That is worth knowing before anyone writes "a medium-sized effect" as though the category had been settled by the data.

Step 12 · Interpret in plain language

The mean fill weight is about 341.2 g, and the test says that gap from 340 g is far too large to be chance (p is tiny). The 95% confidence interval, roughly [340.7, 341.7] g, lies entirely above the label, so we are confident the line runs heavy, not light. The effect size (Cohen's d near 0.6) is medium: the overfill is modest per bag, a bit over a gram, but it is consistent.

overfill = mean - TARGET
print(f"average overfill: {overfill:.2f} g per bag")
print(f"over a 1,000,000-bag run that is about {overfill*1_000_000/1000:,.0f} kg of extra coffee given away")
print("\nStatistically significant AND practically meaningful at scale,")
print("but every bag is still ABOVE the 340 g label, so no customer is shorted.")

average overfill: 1.23 g per bag
over a 1,000,000-bag run that is about 1,229 kg of extra coffee given away

Statistically significant AND practically meaningful at scale,
but every bag is still ABOVE the 340 g label, so no customer is shorted.

This is the honest, three-part read the framework asks for: a tiny p-value (chance is ruled out), a medium effect size (the overfill is real but modest per bag), and a confidence interval that says how precisely we know it. Significant and meaningful are not the same word, and here both happen to be true.

Step 13 · Ethics, bias, and limits

A clean result still deserves a skeptical last look.

• Sampling: the bags were pulled across lines and shifts in one day. If QC tends to sample when the machine is freshly calibrated, the true overfill could be understated. A conclusion about "the process" needs sampling that spans machine states, not one convenient window.
• Measurement: everything rides on the scale. A scale reading 0.5 g high would manufacture an overfill that is not there. Calibration against a certified weight is part of trusting this number.
• Generalization: this is one day on two lines. It supports a claim about that run, not a permanent property of the brand, until more days confirm it.
• Use: the safe and honest action is to nudge the target down toward roughly 340.5 g, saving coffee while staying comfortably above the legal minimum. Cutting straight to 340 g would push half of future bags below the label, which is exactly what the law forbids.

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From analysis to report. The notebook produced the evidence: the cleaning trail, the assumption checks, the test, the effect size, and the interval. The companion written report turns that into a one-page memo a plant manager can act on, leading with the recommendation and keeping the statistics in plain language. Automate the evidence; author the argument.

←Back to the bookChapter 160 · One-Sample t-Test: Coffee Fill Weight