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Chapter 10: Problem 15

Effects of Alcohol on the Brain In a study published in the American Journal of Psychiatry (157:737-744, May 2000), 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 12 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cubic centimeters \(\left(\mathrm{cm}^{3}\right)\). An analysis of the sample data revealed that the hippocampal volume is approximately normal with \(\bar{x}=8.10 \mathrm{~cm}^{3}\) and \(s=0.7 \mathrm{~cm}^{3}\). Conduct the appropriate test at the \(\alpha=0.01\) level of significance.

Short Answer

Expert verified
Reject the null hypothesis; mean hippocampal volume in alcoholic adolescents is less than 9.02 cm鲁.

Step by step solution

01

State the hypotheses

Define the null hypothesis (\text{H}_0) and the alternate hypothesis (\text{H}_a):\[\text{H}_0: \text{The mean hippocampal volume} = 9.02 \text{ cm}^3 \]\[\text{H}_a: \text{The mean hippocampal volume} < 9.02 \text{ cm}^3\]
02

Determine the test statistic

Use the t-test formula for a single sample:\[t = \frac{\bar{x} - \text{渭}}{s/\sqrt{n}} \]Given: \(\bar{x}=8.10\text{ cm}^3\), \(\text{渭}=9.02\text{ cm}^3\), \(s=0.7\text{ cm}^3\), and \(n=12\). Plug these values into the formula:\[t = \frac{8.10 - 9.02}{0.7/\sqrt{12}} \]
03

Calculate the value

Perform the calculations:\[t = \frac{8.10 - 9.02}{0.7/\sqrt{12}} = \frac{-0.92}{0.202} \approx -4.554 \]
04

Determine the critical value

At \(伪 = 0.01\) level of significance and \(n-1 = 11\) degrees of freedom, we look up the critical t-value for a one-tailed test. Using a t-table or calculator, the critical value \(t_{0.01, 11}\) is approximately \(-2.718\).
05

Compare the test statistic to the critical value

Compare the calculated t-value to the critical t-value:\[t = -4.554\text{ and } t_{0.01, 11} = -2.718 \]Since \( t < t_{0.01, 11} \), the null hypothesis is rejected.
06

State the conclusion

Since the test statistic is less than the critical value, there is sufficient evidence at the \(伪 = 0.01\) level of significance to conclude that the mean hippocampal volume in alcoholic adolescents is significantly less than 9.02 cubic centimeters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test
A t-test is a statistical test used to compare the means of two groups. It's especially useful when you have a small sample size. In this exercise, we used a one-sample t-test to determine if the mean hippocampal volume in adolescents with alcohol use disorders is less than the normal volume of 9.02 cubic centimeters. The formula for the t-test is: \[t = \frac{\bar{x} - 渭}{s/\sqrt{n}} \]Where:
  • \(\bar{x}\): the sample mean
  • \(渭\): the population mean
  • \(s\): the sample standard deviation
  • \(n\): the sample size.
By calculating the t-value, we can determine how far our sample mean is from the population mean, relative to the variability in our sample.
significance level
The significance level, often denoted as \( 伪 \), is the probability of rejecting the null hypothesis when it is actually true. It is also known as the Type I error rate. In this exercise, we used a significance level of \( 0.01 \), which means there is a 1% risk of concluding that the hippocampal volume is less when it is not. Choosing a lower significance level like 0.01 instead of 0.05 means we need stronger evidence to reject the null hypothesis.

This level of significance helps define the critical value that our test statistic (t-value) must exceed to justify rejecting the null hypothesis. At \( 伪 = 0.01 \) and with 11 degrees of freedom, the critical t-value for our one-tailed test is approximately -2.718. The calculated t-value must be less than this critical value to reject the null hypothesis.
null hypothesis
The null hypothesis, denoted as \(H_0\), is a statement that there is no effect or no difference. It serves as the default or starting assumption for any hypothesis test. In the context of this exercise, our null hypothesis (\(H_0\)) states that the mean hippocampal volume in alcoholic adolescents is equal to the normal volume of 9.02 cubic centimeters. The alternative hypothesis (\(H_a\)) challenges this assumption by stating that the hippocampal volume in alcoholic adolescents is less than 9.02 cubic centimeters.

The hypothesis testing process involves collecting data and calculating a test statistic to assess the evidence against the null hypothesis. If the test statistic falls within the rejection region defined by our significance level, we reject the null hypothesis in favor of the alternative hypothesis. In this exercise, since our calculated t-value (-4.554) is less than the critical value (-2.718), we reject the null hypothesis, concluding that the mean hippocampal volume in alcoholic adolescents is significantly less than the normal volume.

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Most popular questions from this chapter

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