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 10 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm? An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and × = 8. 02 cm? and s = 07 cm? Conduct the appropriate test at the a = 0.01 level of significance
State the null and altemative hypoteses
Answer
Final Answer: We reject the null hypothesis. The mean hippocampal volume in adolescents with alcohol use disorders is less than the normal volume. \(\boxed{H1: \mu < 9.02 \, cm^3}\)
Step 1 ：State the null and alternative hypotheses. The null hypothesis (H0) is that the mean hippocampal volume in adolescents with alcohol use disorders is equal to the normal volume. The alternative hypothesis (H1) is that the mean hippocampal volume in adolescents with alcohol use disorders is less than the normal volume. In mathematical terms, this can be represented as: \(H0: \mu = 9.02 \, cm^3\) and \(H1: \mu < 9.02 \, cm^3\), where \(\mu\) is the mean hippocampal volume in adolescents with alcohol use disorders.
Step 2 ：Conduct a one-sample t-test to test these hypotheses. The t-test is appropriate because we are comparing the mean of a sample to a known value (the normal volume), and we know the standard deviation of the sample. The level of significance (\(\alpha\)) is given as 0.01.
Step 3 ：Calculate the t-statistic and the p-value. The calculated t-statistic is -4.52 and the p-value is 0.00073.
Step 4 ：Compare the p-value with the level of significance. The p-value is less than the level of significance (\(\alpha = 0.01\)), so we reject the null hypothesis.
Step 5 ：This means that we have enough evidence to support the alternative hypothesis that the mean hippocampal volume in adolescents with alcohol use disorders is less than the normal volume.
Step 6 ：Final Answer: We reject the null hypothesis. The mean hippocampal volume in adolescents with alcohol use disorders is less than the normal volume. \(\boxed{H1: \mu < 9.02 \, cm^3}\)
