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We measured metallicities for 33 z=3.4-4.2 absorption line systems drawn froma sample of H I-selected-Lyman limit systems (LLSs) identified in Sloan DigitalSky Survey (SDSS) quasar spectra and stratified based on metal line features.We obtained higher-resolution spectra with the Keck Echellette Spectrograph andImager, selecting targets according to our stratification scheme in an effortto fully sample the LLS population metallicity distribution. We established aplausible range of H I column densities and measured column densities (orlimits) for ions of carbon, silicon, and aluminum, finding ionization-correctedmetallicities or upper limits. Interestingly, our ionization models were betterconstrained with enhanced $\alpha$-to-aluminum abundances, with a medianabundance ratio of [$\alpha$/Al]=0.3. Measured metallicities were generallylow, ranging from [M/H]=-3 to -1.68, with even lower metallicities likely forsome systems with upper limits. Using survival statistics to incorporatelimits, we constructed the cumulative distribution function (CDF) for LLSmetallicities. Recent models of galaxy evolution propose that galaxiesreplenish their gas from the low-metallicity intergalactic medium (IGM) viahigh-density H I "flows" and eject enriched interstellar gas via outflows.Thus, there has been some expectation that LLSs at the peak of cosmic starformation ($z\approx3$) might have a bimodal metallicity distribution. Wemodeled our CDF as a mix of two Gaussian distributions, one reflecting themetallicity of the IGM and the other representative of the interstellar mediumof star-forming galaxies. This bimodal distribution yielded a poor fit. Asingle Gaussian distribution better represented the sample with a low meanmetallicity of [M/H] $\approx -2.5$.

Predominantly Low Metallicities Measured in a Stratified Sample of Lyman Limit Systems at z=3.7