A stock analyst claims to have devised a mathematical technique for selecting high-quality mutual funds and promises that a client’s portfolio will have higher average ten-year annualized returns and lower volatility; that is, a smaller standard deviation. After ten years, one of the analyst’s twenty-four-stock portfolios showed an average ten-year annualized return of 11.50% and a standard deviation of 10.17%. The benchmarks for the type of funds considered are a mean of 10.10% and a standard deviation of 15.67%.

(a) Let μ be the mean for a twenty-four-stock portfolio selected by the analyst’s method. Test at the 0.05 level that the portfolio beat the benchmark; that is, test H₀: μ = 10.1 versus H₁: μ > 10.1.

(b) Let σ be the standard deviation for a twenty-four stock portfolio selected by the analyst’s method. Test at the 0.05 level that the portfolio beat the benchmark; that is, test H₀: σ = 15.67 versus H₁: σ < 15.67.