A paired design yields one observed difference D_s for each pair. The remaining question is inferential: what do the observed differences tell us about the average difference in the population represented by the design?
The paired t-test applies the one-sample t procedure to D_1,\ldots,D_S. It does not analyze the two condition columns separately because pairing has already converted the original measurements into the response relevant to the contrast. Thus the derivation below begins with the differences, not with two independent means.
Read Acoustic characteristics of American English vowels, focusing on the speaker sample, recording procedure, and formant measurements. The reading is included because each recorded speaker contributes measurements for multiple vowel categories. That repeated speaker design determines why the within-speaker F_1 difference, rather than an independent comparison of two vowel columns, bears on the Problem Set 1 contrast. The preceding page defines F_1 and distinguishes it from fundamental frequency.
Let
\mu_D = \mathbb{E}[D]
be the population mean of the within-pair difference D=Y_A-Y_B. The sample estimator is
\overline{D}=\frac{1}{S}\sum_{s=1}^S D_s.
The sign of \overline{D} inherits the subtraction order. A negative estimate means that condition A is lower on average than condition B.
The sample standard deviation of the differences is
s_D = \sqrt{ \frac{1}{S-1} \sum_{s=1}^S (D_s-\overline{D})^2 },
and the estimated standard error of the mean difference is
\widehat{\operatorname{SE}}(\overline{D}) = \frac{s_D}{\sqrt{S}}.
Note that S is the number of pairs. The calculation does not use 2S as though each original measurement supplied an independent condition contrast.
If the pair differences are independently sampled from a normal population, then
T = \frac{\overline{D}-\mu_D}{s_D/\sqrt{S}} \sim t_{S-1}.
The Student’s t distribution appears because the population scale of the differences is unknown and estimated by s_D. A central (1-\alpha) confidence interval for \mu_D is
\overline{D} \;\pm\; t_{1-\alpha/2,S-1} \frac{s_D}{\sqrt{S}},
where t_{1-\alpha/2,S-1} is the corresponding t quantile. For a 95% interval, \alpha=.05.
Consider the four synthetic acoustic differences from the preceding page:
(-69,-58,-82,-37).
Their mean is -61.5 Hz, and their standard deviation is approximately 19.05 Hz. Thus
\widehat{\operatorname{SE}}(\overline{D}) = \frac{19.05}{\sqrt{4}} \approx 9.53\text{ Hz}.
With three degrees of freedom, the .975 t quantile is approximately 3.182. Multiplying that reference quantile by the standard error gives the margin of error. Thus, the 95% interval is
-61.5 \pm 3.182(9.53) \approx [-91.82,-31.18]\text{ Hz}.
This small constructed sample yields a wide interval because four pairs supply little information about variation among pairs. The entire interval is negative, but its width still shows substantial uncertainty about the magnitude of the average contrast.
For the point null hypothesis H_0:\mu_D=0, the test statistic is
t_{\mathrm{obs}} = \frac{\overline{D}}{s_D/\sqrt{S}}.
A two-sided p value is the probability, under H_0 and the t reference distribution, of obtaining a statistic at least as far from zero as t_{\mathrm{obs}}:
p = 2\mathbb{P}\left(T\geq |t_{\mathrm{obs}}|\right), \qquad T\sim t_{S-1}.
The confidence interval and two-sided test encode the same comparison when they use the same model and \alpha: the .05 test rejects zero exactly when the 95% interval excludes zero. But the interval carries more information because it reports the effect direction, plausible magnitudes under the procedure, and uncertainty in the response’s original units.
The p value is not the probability that H_0 is true, nor is it the probability that a replication will reverse the result. It is a probability of test-statistic extremity conditional on the null model.
The manual calculation makes the unit visible.
d <- c(-69, -58, -82, -37)
n_pairs <- length(d)
mean_d <- mean(d)
sd_d <- sd(d)
se_d <- sd_d / sqrt(n_pairs)
t_critical <- qt(.975, df = n_pairs - 1)
ci <- mean_d + c(-1, 1) * t_critical * se_d
c(mean = mean_d, se = se_d, lower = ci[1], upper = ci[2])When the data remain in two aligned vectors, t.test() performs the same procedure.
y_a <- c(312, 347, 290, 401)
y_b <- c(381, 405, 372, 438)
t.test(y_a, y_b, paired = TRUE)The alignment is part of the input. R subtracts the first element of y_b from the first element of y_a, and so on. Sorting the two vectors independently would destroy the pairs while still returning a numerical result. A safer workflow joins observations by the pair identifier, checks that each pair has one value in each condition, and only then constructs the difference.
An analysis that ignores pairing estimates the variance of the difference between two sample means using separate marginal variances. The paired procedure instead uses the observed variance of D:
s_D^2=s_A^2+s_B^2-2s_{AB},
where s_{AB} is the sample covariance within pairs. Positive covariance makes the paired standard error smaller because stable pair baselines have been removed. This is not an automatic reward for choosing paired = TRUE; it is the consequence of using the covariance that the design supplies.
The two analyses also invoke different sampling stories: the paired procedure imagines sampling pairs and observing both conditions for each, whereas an independent procedure imagines sampling one collection for A and a separate collection for B. Only one story can match the actual design.
The normality assumption concerns the pair differences, not each condition separately. With many pairs, inference for the mean is often not very sensitive to moderate nonnormality, though strong skew, outliers, or a small number of pairs can matter. Thus, a plot of D_s is more relevant than two separate histograms of Y_A and Y_B.
The procedure also assumes that the pairs are independent under the sampling model. Two measurements within a pair are expected to be dependent; that is why they were paired. Dependence among different pairs, such as several lexical items from the same speaker, requires a richer representation.
Finally, the population must match the sample. A narrow interval for speakers recruited from one community does not create a probability sample of all English speakers. The interval quantifies uncertainty under the stated sampling model; the generalization claim still requires a defensible connection between observed speakers and the population of interest.
t.test(..., paired = TRUE) on data stored in long form?Paired t inference is one-sample inference on within-pair differences. The estimate is \overline{D}, its standard error is s_D/\sqrt{S}, and a t reference distribution accounts for estimating the scale of the differences. This procedure is the focal method in PS1, where the pairing follows from speakers producing both vowel categories.