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Background: Previous research has shown that the slope of the electroencephalography (EEG) power spectrum mediates the difference between older and younger adults on a visuo-spatial working memory task. The present study sought to replicate and extend that work using a larger sample and a validated set of neuropsychological tests: The Repeatable Battery for the Assessment of Neuropsychological Status (RBANS). Methods: Forty-four participants (21 younger adults, 23 older adults) completed a battery of cognitive and motor tasks that included the RBANS. EEG data was collected both during rest and on-task. Excluding the alpha-band, RBANS scores were regressed onto the slope of the resting EEG power spectrum, controlling for age and using robust mediation analysis. Results: Older adults performed reliably lower on the composite RBANS and the Coding, List Recall, List Recognition, and Figure Recall subtests. However, boot-strapped mediation models only showed a mediating effect of the spectral slope on the RBANS composite and the Coding subtest. Conclusions: The resting slope of the EEG power spectrum mediated age-related differences in cognition in the current study, which replicates prior work and is consistent with the neural noise hypothesis of aging. In extending this work, it was shown that these effects are strongest in tasks requiring speeded processing and/or executive functions, whereas this effect was weaker (to absent) for delayed memory, even though age-related differences were present. This pilot study warrants further exploration of the EEG power spectrum in age-related cognitive decline.