scipy.stats — distributions, hypothesis tests: Question bank

Parent: scipy.stats — distributions, hypothesis tests · builds on Normal Distribution, p-values and Significance, Central Limit Theorem, Chi-square Distribution, Maximum Likelihood Estimation.

The vocabulary these traps use (define before you answer)

Before the questions, pin down every symbol and function name so nothing below is a mystery. Look at the picture: it is ONE bell curve with the four functions all drawn on it.

Figure — scipy.stats — distributions, hypothesis tests

Definition The five names, anchored to the picture

PDF (probability density function) — the height of the curve at a point. In the figure it is the blue curve itself. Height is density, not probability; probability is area.
CDF (cumulative distribution function) $F (x) = P (X \leq x)$ — the shaded area to the LEFT of $x$ (yellow). Runs from $0$ up to $1$ .
SF (survival function) $S (x) = P (X > x) = 1 - F (x)$ — the area to the RIGHT (red). The yellow and red areas always sum to $1$ .
PPF (percent-point / quantile function) $= F^{- 1} (p)$ — read the picture backwards: give it a probability level $p$ (a shaded-area size, a number between $0$ and $1$ ) and it returns the $x$ that produces it.
loc / scale — the two knobs scipy gives every distribution: loc slides the curve left/right, scale stretches/squashes it. For norm only, loc $= μ$ (mean, the center) and scale $= σ$ (standard deviation, the width).

The next figure shows the CDF/SF trade-off directly: as $x$ moves right, yellow (left area) grows toward $1$ and red (right area) shrinks toward $0$ — they are two views of the same split.

The two-sided doubling of a tail is what makes a two-sided p-value; here it is drawn as both red tails of one curve.

And this last figure shows the far-tail precision trap: why 1 - cdf collapses to zero while .sf keeps every digit.

True or false — justify

A small p-value means the null hypothesis

H_{0}

is probably false.

False — the p-value is

P (data this extreme ∣ H_{0})

, the reverse conditioning of

P (H_{0} ∣ data)

; it says the data is surprising assuming

H_{0}

, not that

H_{0}

is unlikely. See p-values and Significance.

A p-value of

0.8

proves the two groups have equal means.

False — failing to reject is "no evidence of a difference," which can also happen with a tiny, low-power sample; absence of evidence is not evidence of absence.

For any distribution in scipy, loc is the mean and scale is the standard deviation.

False — that coincidence holds only for norm; in general loc shifts and scale stretches, e.g. for expon the mean equals scale

= 1/ λ

, not the sd formula you'd expect.

stats.norm.pdf(0) returns a probability.

False — the PDF returns a density (height of the bell in figure s01), not a probability; probability comes from area, so a single point has probability zero for a continuous variable.

.sf(x) and 1 - .cdf(x) always give identical numbers.

False as stated — they are mathematically equal but not numerically identical: far in the tail 1 - cdf subtracts two near-1 floats and loses precision (see figure s04), while .sf is computed directly and stays accurate.

A larger sample makes every true (nonzero) effect show up as statistically significant eventually.

True — the standard error shrinks like

1/ n

, so with enough data any real nonzero difference will eventually cross

p < α

; the catch is that statistical significance then no longer implies the effect is large or important.

The t-statistic follows a normal distribution when

n

is small.

False — because

σ

is estimated by

s

, the extra uncertainty gives heavier tails, so

t

follows Student-

t

with

n - 1

degrees of freedom; it only approaches normal as

n

grows.

Rejecting

H_{0}

α = 0.05

means there's a 5% chance you're wrong.

False —

α

is the false-positive rate when $H_{0}$ is actually true (a long-run property of the test), not the probability that this particular decision is a mistake.

The chi-square statistic

\sum (O - E)^{2} / E

can be negative.

False — every term is a squared difference divided by a positive expected count, so the sum is always

\geq 0

; it hits zero only when observed exactly equals expected. See Chi-square Distribution.

stats.norm.fit(sample) gives the true

μ

and

σ

of the population.

False — it returns maximum-likelihood estimates from the sample, which recover the truth only approximately and get better as the sample grows. See Maximum Likelihood Estimation.

Spot the error

p = stats.norm(100, 15).cdf(130) to find the fraction scoring above 130. Why is this wrong?

.cdf(130) gives the area to the left (fraction below, the yellow region in figure s02); for "above" you need the upper tail .sf(130)

= 1 - F (130)

, the red region.

stats.expon(scale=0.5) was written to model events arriving at rate

λ = 0.5

. Where's the trap?

For expon, scale

= 1/ λ

, so scale=0.5 means

λ = 2

; to get

λ = 0.5

you'd write scale=2.

Someone reads a p-value of

0.049

as "strong evidence" and

0.051

as "no evidence at all." What's the flaw?

The

α = 0.05

cutoff is an arbitrary convention, not a physical wall; two p-values straddling it carry almost identical strength of evidence, so treating them as opposite worlds is a false dichotomy.

To test if the mean is

5

, someone uses ttest_ind(data, [5,5,5,5,5]). Why is that the wrong test?

ttest_ind compares two independent samples; testing one sample against a fixed number

μ_{0}

is a one-sample test, ttest_1samp(data, popmean=5).

stats.chisquare(obs, exp) is called where several expected counts are

1

2

. What's unsound?

The chi-square approximation assumes expected counts are reasonably large (rule of thumb

\geq 5

); tiny expected values make the p-value untrustworthy.

z = (x - mu) / sigma**2 was written to standardize a value. Spot the bug.

You divide by

σ

, not

σ^{2}

— dividing by the variance would give the wrong units and wrong scale; the z-score is

(x - μ) / σ

A one-sided p-value was doubled and then compared to

α /2

. Why is this a double penalty?

You either double the one-sided p-value (for a two-sided test) and compare to

α

, or keep it one-sided and compare to

α /2

— doing both applies the correction twice.

Why questions

Why does a two-sided t-test multiply the tail probability by 2 (see figure s03)?

"At least as extreme" means as far from

μ_{0}

in either direction, so you add both symmetric red tails

P (T > ∣ t ∣)

and

P (T < - ∣ t ∣)

, which by symmetry equals

2 P (T > ∣ t ∣)

Why does the standard error of the mean shrink by

n

and not by

n

Variance of a sum of

n

independent values grows like

n

, so the mean's variance is

σ^{2} / n

; taking the square root to get the sd gives

σ / n

. See Central Limit Theorem.

Why can one stats.norm object answer questions about every normal distribution?

The z-transform

Z = (X - μ) / σ

slides the curve to center

0

(that's loc) and squashes width to

1

(that's scale), mapping any

N (μ, σ)

onto the standard

N (0, 1)

. See Normal Distribution.

Why is the PDF defined as the derivative of the CDF?

The CDF accumulates probability (the growing yellow area in figure s02); the density is how fast it accumulates, so probability in a sliver

[x, x + d x]

f (x) d x = F (x + d x) - F (x)

Why use

t

(Student-t) rather than

z

when the population sd is unknown?

Replacing

σ

with the sample estimate

s

injects extra randomness; the heavier-tailed

t_{n - 1}

widens the critical region to honestly account for that uncertainty.

Why does the sampling distribution of the mean look bell-shaped even when the raw data isn't?

The Central Limit Theorem says sums (and hence averages) of many independent contributions converge to a normal shape regardless of the original distribution. See Central Limit Theorem.

Why does the chi-square statistic divide each squared mismatch by the expected count

E

Dividing by

E

turns a raw squared deviation into a relative one — a gap of 5 matters far more when you expected 10 than when you expected 1000 — so the terms are comparably scaled. See Chi-square Distribution.

Edge cases

What is stats.norm(0,1).cdf(0) and why?

Exactly

0.5

— the standard normal is symmetric about

0

(figure s01), so half the area lies to the left of the center.

What does the PDF give for a point outside the support, e.g. stats.expon.pdf(-1)?

Zero — the exponential is only defined for

x \geq 0

, so density is zero everywhere the variable can't occur.

What p-value do you get from a one-sample t-test where the sample mean exactly equals

μ_{0}

t = 0

, so the p-value is

1

— the data is perfectly consistent with

H_{0}

, giving zero evidence against it.

What breaks in a one-sample t-test with

n = 1

Degrees of freedom

n - 1 = 0

, so there is no sample sd

s

to estimate spread (you'd divide by zero) and the

t_{0}

distribution is undefined — a single point carries zero information about variability, so the test simply cannot run.

As sample size

n \to \infty

with

σ

unknown, what happens to the

t_{n - 1}

distribution?

Its tails thin out and it converges to the standard normal

N (0, 1)

, because

s

estimates

σ

almost perfectly and the extra uncertainty vanishes.

What does stats.chisquare return when observed counts equal expected exactly?

Statistic

= 0

and p-value

= 1

— no mismatch at all, so there's no evidence against the "expected" model.

What is .ppf(0) and .ppf(1) for a normal?

- \infty

and

+ \infty

— the value below which

0%

of mass lies is unbounded below, and

100%

is unbounded above, since the normal has infinite support.

Why does 1 - cdf fail in the far tail but .sf survive (figure s04)?

Far out, cdf is like

0.99999999

; subtracting it from

1

leaves only the last few digits, so catastrophic cancellation throws away precision, while .sf computes the tiny tail area directly and keeps every digit.

Recall One-line self-test

Cover every answer above. If you can justify at least 20 of them in a sentence — not guess yes/no — you own the concepts. The traps that trip people most: p-value direction, scale vs $σ$ , .cdf vs .sf, and "failed to reject $\neq =$ proved."