Hypothesis Tests Concerning the Mean

Test Statistic for Hypothesis Tests of the Population Mean. H₀: μ = μ₀

The t-Test. If the population sampled has unknown variance and either:

1 the sample is large, or

2 the sample is small but the population sampled is normally distributed, or approximately normally distributed,

Tests Concerning Differences between Means.

H₀: μ₁ – μ₂ = 0

Independent samples, normally distributed populations with unknown variances, assume equal variances, t-test based on pooled estimator of the common variance.

Independent samples, normally distributed populations with unknown variances, cannot assume equal variances, t-test.

Dependent samples, arrange the data in paired observations and conduct a test of mean differences (a paired comparisons test).

A paired comparisons test is a statistical test for differences in dependent items.

H₀: μ_d = μ_d0

[Practice Problems] A fund manager reported a 2% mean quarterly return over the past ten years for its entire base of 250 client accounts that all follow the same investment strategy. A consultant employing the manager for 45 client accounts notes that their mean quarterly returns were 0.25% less over the same period.

The consultant tests the hypothesis that the return disparity between the returns of his clients and the reported returns of the fund manager’s 250 client accounts are significantly different from zero. Assuming normally distributed populations with unknown population variances, the most appropriate test statistic is:

A.       a paired comparisons t-test.

B.        a t-test of the difference between the two population means.

C.        an approximate t-test of mean differences between the two populations.

[Solutions] A

The sample sizes for both the fund manager and the consultant’s accounts consists of forty quarterly periods of returns. However, the consultant’s accounts are a subset of the fund manager’s entire account base. As such, they are not independent samples. When samples are dependent, a paired comparisons test is appropriate.