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Identify the sequence of steps for testing whether two population means differ.

FRM® Part I · Quantitative Analysis · Hypothesis Testing · FRM1.QA.6.g

The idea

To test whether two population means differ, follow a fixed sequence. First, state H0: μ1 − μ2 = 0 against H1: μ1 − μ2 ≠ 0, or a one-sided version if justified in advance. Second, choose the significance level and critical value. Third, compute the difference in sample means and its standard error. For independent samples the variances of the two estimates add, so the standard error is: SE = √(s1²/n1 + s2²/n2). Here s1² and s2² are the sample variances and n1 and n2 the sample sizes. Fourth, form the test statistic: Statistic = (difference − 0) ÷ SE. Fifth, compare it with the critical value, or compute the p-value, and conclude. If the samples are paired, test the mean of the differences instead, because the covariance matters. Do not pool variances unless they are believed equal.

Formulas

Test statistic for a difference between two independent means

z = (x̄₁ − x̄₂) ÷ √(s₁² ÷ n₁ + s₂² ÷ n₂)

x̄₁, x̄₂
the two sample means
s₁, s₂
the two sample standard deviations
n₁, n₂
the two sample sizes

Independent large samples, testing H0 of no difference. For paired samples test the mean of the differences instead; the trap is pooling variances that are not believed equal.

A practice question

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

Which pair of hypotheses is the standard setup for testing whether two population means differ?

  1. H0: mu1 = mu2 versus H1: mu1 not equal to mu2 - correct
  2. H0: sample mean 1 = sample mean 2 versus H1: sample mean 1 not equal to sample mean 2Why not: Writes the hypotheses in terms of observed sample means; they must concern the unknown population means.
  3. H0: mu1 not equal to mu2 versus H1: mu1 = mu2Why not: Swaps the roles; the equality belongs in the null and the difference the test looks for in the alternative.
  4. H0: mu1 > mu2 versus H1: mu1 < mu2Why not: Imposes a direction and leaves out the equality, so the hypotheses do not cover mu1 = mu2.

To ask whether two means differ, the null states no difference (mu1 - mu2 = 0) and the alternative states a difference (not equal to zero).

A harder one

Sample 1 has n = 50, mean 8.4 and standard deviation 6. Sample 2 has n = 60, mean 6.9 and standard deviation 5. The samples are independent. For a two-sided 5 percent test of H0: mu1 = mu2 (critical value 1.96), which result is correct?

  1. z is about 3.33, so reject.Why not: Uses s rather than s² under the root: sqrt(6/50 + 5/60) instead of sqrt(36/50 + 25/60).
  2. z is about 1.00, so fail to reject.Why not: Adds the two standard errors (6/sqrt(50) + 5/sqrt(60)) instead of adding variances.
  3. z is about 1.41, so fail to reject. - correct
  4. z is about 2.01, so reject.Why not: Uses sqrt((s1² + s2²)/(n1 + n2)), adding the sample sizes instead of dividing each variance by its own n.

SE = sqrt(36/50 + 25/60) = sqrt(0.72 + 0.4167) = 1.066. z = (8.4 - 6.9)/1.066 = 1.41, which is below 1.96, so H0 is not rejected (two-sided p = 0.16).

The hardest tier

Over the same 36 months, strategy X has a mean monthly return 1.2 percentage points above strategy Y. The standard deviations of monthly returns are 4 percent for X and 5 percent for Y, and the correlation between them is 0.8. For a two-sided 5 percent test that the mean difference is zero, which is correct?

  1. Treat the samples as independent: SE = sqrt(41/36) = 1.12, z is about 1.07, so fail to reject.Why not: Ignores the positive correlation; the variance of the difference must subtract twice the covariance.
  2. Test the paired differences: variance of the difference = 16 + 25 - 2(0.8)(4)(5) = 9, SE = 3/6 = 0.5, z = 2.40, so reject. - correct
  3. Add the covariance term: variance = 16 + 25 + 2(0.8)(4)(5) = 73, z is about 0.84, so fail to reject.Why not: Uses the wrong sign on the covariance term; the variance of a difference subtracts it.
  4. Use the standard deviation of the differences as the denominator: z = 1.2/3 = 0.40, so fail to reject.Why not: Forgets to divide the standard deviation of the differences by the square root of n.

The returns are paired, so the difference series has variance 16 + 25 - 2 x 0.8 x 4 x 5 = 9 and standard deviation 3. SE = 3/sqrt(36) = 0.5 and z = 1.2/0.5 = 2.40, above 1.96.

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