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Explain how the t-statistic, its p-value and the confidence interval are connected.

FRM® Part I · Quantitative Analysis · Linear Regression · FRM1.QA.7.g

The idea

The t-statistic, the p-value and the confidence interval are three views of the same test of H0: β = b0 against a two-sided alternative. The p-value is the probability, if the null were true, of getting an estimate at least as far from b0 as the one observed. For a t of 2.0 in a large sample it is 2 × (1 − Φ(2.0)), about 0.046. Reject at level α when the p-value is below α, equivalently when the absolute t exceeds the critical value, equivalently when b0 lies outside the (1 − α) confidence interval. The standard error can be recovered as the estimate divided by the t for a zero null, or as the interval half-width divided by 1.96. The usual mistake is using a one-tailed p-value for a two-sided test, or reading the p-value as the probability that the null is true.

A practice question

From the question bank, with the reason each wrong option is wrong.

The 95% confidence interval for a regression slope is 0.12 to 0.68. For a two-sided test of H0: slope = 0 at the 5% level, the conclusion is:

  1. fail to reject H0, because the interval is entirely positiveWhy not: A positive interval that excludes zero is exactly the reason to reject.
  2. reject H0, because zero lies outside the interval - correct
  3. fail to reject H0, because the interval is wideWhy not: Width alone does not decide the test; what matters is whether zero is inside.
  4. reject H0, because the interval's midpoint of 0.40 is above 0.05Why not: Compares the midpoint to the significance level, mixing an estimate with a probability.

A two-sided test at the 5% level rejects exactly when the hypothesised value lies outside the 95% confidence interval. Zero is outside 0.12 to 0.68.

A harder one

In a large sample, the t-statistic for a slope is 1.75 for H0: slope = 0. The two-sided p-value is closest to:

  1. 0.040Why not: Reports the one-tailed probability; a two-sided test doubles it.
  2. 0.920Why not: Reports one minus the p-value, the probability of a less extreme result.
  3. 0.080 - correct
  4. 0.160Why not: Doubles the tail probability twice.

Tail probability above 1.75 is 0.0401; two-sided p-value = 2 x 0.0401 = 0.080. At the 5% level this does not reject (0.080 > 0.05).

The hardest tier

A regression gives a slope estimate of 1.30 with a standard error of 0.20 in a large sample. A risk manager tests H0: slope = 1 against a two-sided alternative. Which statement is internally consistent?

  1. t = 1.5 and p is about 0.07; H0 is rejected at the 10% level because 0.07 is below 0.10Why not: Uses the one-tailed probability as if it were the two-sided p-value.
  2. t = 1.5 and p is about 0.87; H0 is not rejected because the p-value is largeWhy not: Uses one minus the two-sided p-value as the p-value.
  3. t = 1.5 and p is about 0.13; 1 lies outside the 90% interval, so H0 is rejected at 10%Why not: The 90% interval, 1.30 +/- 1.645 x 0.20, runs from 0.97 to 1.63 and does contain 1.
  4. t = 1.5, p about 0.13; 1 is inside the 90% interval (0.97 to 1.63), so H0 survives at 10% - correct

t = (1.30 - 1) / 0.20 = 1.5 and the two-sided p-value is 2 x 0.0668 = 0.134. Since 0.134 > 0.10, H0 is not rejected at 10%; the 90% interval 0.97 to 1.63 contains 1.

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