Calculating \(p\)-values in R: A Quick and Dirty Guide

• Critical Region (\(\alpha = .05\)) is shown in light red.
• Solid vertical black line is the test-statistic.
• The probability of a test-statistic (i.e. \(p\)-value) is shown in brown.
• Degrees of freedom are set to 4 in all examples.
• Additional notes pertaining to one- and two-sided confidence interval calculations are also provided.

Two-Tailed Tests

• For two-tailed tests, what is matters is whether the test-statistic lands in the left or right tail; i.e., whether it is negative or positive.
• Note that the probability (i.e., \(p\)-value’s area under the curve) is mirrored in the case of two-tailed tests.

1. If your test-statistic is negative (e.g., \(T = -1.5\)), then you are going to use . . .

pt(-1.5, df = 4) * 2

## [1] 0.208

Confidence Interval Information

\(\begin{aligned} \text{Confidence Interval} &= \bar{x} \pm \text{(critical $t$-score)} \cdot \text{(standard error)} \\ &= [\text{ lower bound}, \text{ upper bound }] \end{aligned}\)

# Critical t-calculation
alpha <- 0.05
abs(qt(alpha / 2, df = 4))

## [1] 2.776445

• The absolute value (abs()) is used because the confidence interval equation already accounts for the direction of each boundary through addition (+) and subtraction (-). This eliminates the need to compute separate critical \(t\)-scores for each side.

2. If your test-statistic is positive (e.g., \(T = 1.5\)), then you are going to use . . .

pt(1.5, df = 4, lower.tail = FALSE) * 2

## [1] 0.208

Confidence Interval Information

\(\begin{aligned} \text{Confidence Interval} &= \bar{x} \pm \text{(critical $t$-score)} \cdot \text{(standard error)} \\ &= [\text{ lower bound}, \text{ upper bound }] \end{aligned}\)

# Critical t-calculation
alpha <- 0.05
abs(qt(alpha / 2, df = 4))

## [1] 2.776445

• The absolute value (abs()) is used because the confidence interval equation already accounts for the direction of each boundary through addition (+) and subtraction (-). This eliminates the need to compute separate critical \(t\)-scores for each side.

One-tailed tests

• For one-tailed tests, what is matters is where the critical region (red) is located (left or right tail). This is determined by the null hypothesis.
• Whether the test-statistic is positive or negative has no bearing on how the \(p\)-value is calculated.

3. If your critical region is in the left tail (e.g., \(H_0 : \mu \geq 1.5\)) then you are going to use . . .

pt(1.5, df = 4)

## [1] 0.896

Confidence Interval Information

\(\begin{aligned} \text{Confidence Interval} &= \bar{x} + \text{(critical $t$-score)} \cdot \text{(standard error)} \\ &= (-\infty, \text{ upper bound }] \end{aligned}\)

# Critical t-calculation
alpha <- 0.05
abs(qt(alpha, df = 4))

## [1] 2.131847

• The confidence interval is expressed in terms of the alternative hypothesis \(H_1: \mu < 1.5\).
• For one-tailed tests, the critical \(t\)-score is calculated using the entire significance level \(\alpha\).
• The absolute value (abs()) is used because the direction of the boundary is already specified by the addition (+) in the confidence interval equation.

4. If your critical region is in the right tail (e.g., \(H_0 : \mu \leq 1.5\)) then you are going to use . . .

pt(1.5, df = 4, lower.tail = FALSE)

## [1] 0.104

Confidence Interval Information

\(\begin{aligned} \text{Confidence Interval} &= \bar{x} - \text{(critical $t$-score)} \cdot \text{(standard error)} \\ &= [\text{ lower bound}, \infty) \end{aligned}\)

# Critical t-calculation
alpha <- 0.05
abs(qt(alpha, df = 4))

## [1] 2.131847

• The confidence interval is expressed in terms of the alternative hypothesis \(H_1: \mu > 1.5\).
• For one-tailed tests, the critical \(t\)-score is calculated using the entire significance level \(\alpha\).
• The absolute value (abs()) is used because the direction of the boundary is already specified by the subtraction (-) in the confidence interval equation.