Problem
In a study, researchers wanted to measure the effect of alcohol on the hippocampal region, the portion of the brain responsible for long-term memory storage, in adolescents. The researchers randomly selected 22 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 $\mathrm{cm}^{3}$. An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and $\bar{x}=8.16 \mathrm{~cm}^{3}$ and $s=0.7 \mathrm{~cm}^{3}$. Conduct Identify the t-statistic. $t_{0}=\square$ (Round to two decimal places as needed.)
Solution
Step 1 ：The t-statistic is calculated using the formula: t = (x̄ - μ) / (s/√n) where: x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. In this case, we have: x̄ = 8.16 cm³, μ = 9.02 cm³, s = 0.7 cm³, and n = 22. Substituting these values into the formula, we get: t = (8.16 - 9.02) / (0.7/√22) = -1.23/0.149 = -8.26 Therefore, the t-statistic, t₀, is -8.26 (rounded to two decimal places).
