Say we have two sets of data that are not independent of each other and come from a matched-pairs experiment, and we want to construct a confidence interval for the mean difference between these two samples. How do we make this confidence interval? Let’s assume we’ve chosen a confidence level of $\alpha$ = 0.05.
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We have two samples of data, $x_1, x_2, x_3, \ldots, x_k$ and $x’_1, x’_2, x’_3, \ldots, x’_k$. We’re going to use some fake data below just as an example; replace it with your real data.
1 2 sample.1 <- c(15, 10, 7, 22, 17, 14) sample.2 <- c(9, 1, 11, 13, 3, 6)
The shortest way to create the confidence interval is with R’s
It’s just one line of code (after we choose $\alpha$).
1 2 alpha <- 0.05 # replace with your chosen alpha (here, a 95% confidence level) t.test(sample.1, sample.2, paired = TRUE, conf.level = 1-alpha)
1 2 3 4 5 6 7 8 9 10 Paired t-test data: sample.1 and sample.2 t = 2.8577, df = 5, p-value = 0.0355 alternative hypothesis: true mean difference is not equal to 0 95 percent confidence interval: 0.7033862 13.2966138 sample estimates: mean difference 7
If you need the lower and upper bounds later, you can save them as variables as follows.
1 2 3 conf.interval <- t.test(sample.1, sample.2, paired = TRUE, conf.level = 1-alpha) lower.bound <- conf.interval$conf.int upper.bound <- conf.interval$conf.int
It’s also possible to do the computation manually, using the code below.
1 2 3 4 5 6 7 8 diff.samples <- sample.1 - sample.2 # differences between the samples n = length(sample.1) # number of observations per sample diff.mean <- mean(diff.samples) # mean of the differences diff.variance <- var( diff.samples ) # variance of the differences critical.val <- qt(p = alpha/2, df = n - 1, lower.tail=FALSE) # critical value radius <- critical.val*sqrt(diff.variance)/sqrt(n) # radius of confidence interval c( diff.mean - radius, diff.mean + radius ) # confidence interval
1  0.7033862 13.2966138
Either method gives the same result. Our 95% confidence interval is $[0.70338, 13.2966]$.
Content last modified on 24 July 2023.
Contributed by Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)