Paired t-Procedures

Unit 7 · Inference for Quantitative Data: Means

What AP Stats asks here

When data come in natural pairs — before/after, twins, matched-pair experiments — the right analysis collapses each pair into a single difference and runs one-sample t on those differences. The recurring AP trap: applying a two-sample t to paired data inflates the SE and destroys power.

Procedure on the differences

Differences

d_i = x_{1i} - x_{2i}

\bar{d} \pm t^* \dfrac{s_d}{\sqrt{n}}

Test statistic

t = \dfrac{\bar{d} - 0}{s_d/\sqrt{n}},\;\;\text{df} = n - 1

AP Tip: Paired analysis subtracts out between-subject variability. If most variation comes from people differing in their baseline, pairing reclaims an enormous amount of power.

Caution: Pairing requires a structural one-to-one link. Random assignment to two independent groups is NOT pairing — even if each group has the same n.

Type 1

Identify paired structure

Paired structure means a one-to-one link between an observation in group 1 and a specific observation in group 2.

Example 1

Which scenario is paired? (A) 30 students are randomly assigned to a tutoring program or no program; final exam scores are compared. (B) Identical-twin pairs are split: in each pair one twin gets the new sleep aid, the other a placebo; hours of sleep are measured. (C) 50 patients are measured on a control day; 50 different patients are measured on a treatment day. (D) 25 senior students' GPAs are compared with 25 junior students' GPAs.

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Type 2

Paired CI or test on the differences

Compute the column of differences $d_i$ , then $\bar{d}, s_d$ , then apply the one-sample t to those differences.

Example 2

Six participants take a typing-speed test before and after a one-week course. The After − Before differences (wpm) are: 6, 3, 4, 3, −1, 5 with

\bar{d} = 3.333

and

s_d \approx 2.42

. For a one-sided paired t-test of

H_0: \mu_d = 0

H_a: \mu_d > 0

: (A)

t \approx 3.37

, df = 5, p-value ≈ 0.010, reject — convincing evidence of an increase. (B)

t \approx 1.16

, df = 5, p-value ≈ 0.149, fail to reject. (C)

t \approx 3.37

, df = 10, p-value ≈ 0.001, reject. (D)

t \approx 0.50

, df = 5, p-value ≈ 0.30, fail to reject.

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Type 3

Paired analysis vs (wrong) two-sample

Ignoring the pairing inflates the SE because between-subject variability is no longer subtracted. Power evaporates and real effects go undetected.

Example 3

On the typing-speed data, paired t gives

t \approx 3.37

, p-value ≈ 0.010 (reject). Running an unpooled two-sample t on Before vs After gives

t \approx 1.16

, p-value ≈ 0.149 (fail to reject). Which is correct? (A) The two-sample test is more rigorous and should be used. (B) The paired test is correct here. Ignoring pairing inflates the SE because within-subject variability is no longer subtracted, masking a real effect. (C) Both tests are equivalent. (D) The two tests cannot disagree if the data are the same.

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