91Ó°ÊÓ

In the study described in the paper "Exposure to Diesel Exhaust Induces Changes in EEG in Human Volunteers" (Particle and Fibre Toxicology [2007]), 10 healthy men were exposed to diesel exhaust for 1 hour. A measure of brain activity (called median power frequency, or MPF) was recorded at two different locations in the brain both before and after the diesel exhaust exposure. The resulting data are given in the accompanying table. For purposes of this exercise, assume that it is reasonable to regard the sample of 10 men as representative of healthy adult males. $$ \begin{array}{ccccc} &{\mathrm{MPF}(\operatorname{In} \mathrm{Hz})} \\ { 2 - 5 } \text { Subject } & \begin{array}{c} \text { Location 1 } \\ \text { Before } \end{array} & \begin{array}{c} \text { Location 1 } \\ \text { After } \end{array} & \begin{array}{c} \text { Location 2 } \\ \text { Before } \end{array} & \begin{array}{c} \text { Location 2 } \\ \text { After } \end{array} \\ \hline 1 & 6.4 & 8.0 & 6.9 & 9.4 \\ 2 & 8.7 & 12.6 & 9.5 & 11.2 \\ 3 & 7.4 & 8.4 & 6.7 & 10.2 \\ 4 & 8.7 & 9.0 & 9.0 & 9.6 \\ 5 & 9.8 & 8.4 & 9.7 & 9.2 \\ 6 & 8.9 & 11.0 & 9.0 & 11.9 \\ 7 & 9.3 & 14.4 & 7.9 & 9.1 \\ 8 & 7.4 & 11.3 & 8.3 & 9.3 \\ 9 & 6.6 & 7.1 & 7.2 & 8.0 \\ 10 & 8.9 & 11.2 & 7.4 & 9.1 \end{array} $$ Do the data provide convincing evidence that the mean MPF at brain location 1 is higher after diesel exposure than before diesel exposure? Test the relevant hypotheses using a significance level of 0.05 .

Step by step solution

01

Formulate the hypothesis

The null hypothesis (H0) is that there is no difference in the average MPF (Median Power Frequency) before and after diesel exposure at brain location 1. Or in other words, the mean difference is 0. The alternative hypothesis (HA) is that there is a difference in the average MPF before and after exposure, specifically that the mean MPF after exposure is higher than before.

02

Perform paired t-test

To test the hypothesis, use the paired t-test. This compares the means of two related groups to determine if there is statistical evidence that the associated population means are significantly different. When the sample size is smaller than 30, as in this case, a Student's t distribution is used in the test.

03

Calculate the mean difference

Calculate the mean difference between 'after' and 'before' values for each subject. Then, calculate the mean (often denoted as \(d\)) of these differences.

04

Calculate the standard deviation

Standard deviation (often denoted as \(sd\)) of these differences should be calculated. With \(d\) and \(sd\), calculate the test statistic \(t\) using the formula: \(t = \frac{d}{sd / \sqrt{n}}\), where \(n\) is the number of subjects.

05

Find the critical value

Using a t-distribution table, find the critical value for a one-tailed test with \(n - 1\) degrees of freedom at the 0.05 significance level.

06

Compare the t-statistic to the critical value

If the calculated t-statistic is greater than the critical value, reject the null hypothesis in favor of the alternative hypothesis.

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Key Concepts

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

Null Hypothesis

When conducting a paired t-test, as in the exercise involving the effect of diesel exhaust on median power frequency (MPF), we first need to define our null hypothesis (\(H_0\)). This hypothesis states that there is no effect—or more technically, no difference—in the variable we're measuring, before and after the treatment. In our case, the null hypothesis claims that the mean MPF at brain location 1 before and after diesel exposure is the same. Formulating a clear null hypothesis is critical because it gives us a specific claim to test against using statistical methods. It acts as a baseline assumption and the concept of 'innocence until proven guilty.' If our analysis shows that the data are highly unlikely if the null hypothesis were true, we then have grounds to reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

Conversely, the alternative hypothesis (\(H_A\)) proposes that there is a change, impact, or difference. For our exercise, the alternative hypothesis suggests that the mean MPF at brain location 1 after diesel exposure is higher than before exposure. This hypothesis is what researchers are often hoping to support, as it indicates a potential effect or relation that may be of interest. It’s essentially what you would accept if you find sufficient evidence to reject the null hypothesis. In hypothesis testing, evidence is gathered via data and analysis to assess whether the null hypothesis can be rejected, thereby lending support to the alternative hypothesis.

Median Power Frequency (MPF)

The median power frequency (MPF) is a measure used in electroencephalography (EEG) to quantify the central frequency of a power spectrum. MPF is useful in determining shifts in brain activity, and as seen in our study, it was measured before and after the subjects were exposed to diesel exhaust. This type of data is critical when performing a paired t-test as it reflects two matched samples from the same subjects under different conditions. When the paired t-test is applied to the MPF data, it evaluates whether the central tendency of the brain's electrical activity changes significantly following an intervention—in this case, diesel exposure.

Statistical Significance

To determine whether the observed differences in the MPF after diesel exposure are due to a real effect rather than random chance, we assess statistical significance. This concept involves calculating a p-value, which reflects the probability that the observed data would occur if the null hypothesis were true. A common threshold is a p-value of 0.05 or 5%. If our analysis yields a p-value lower than this threshold, we conclude the results are statistically significant, which means it's very unlikely that the observed difference in MPF happened by chance. In the given exercise, if the statistical analysis produces a p-value below 0.05, we reject the null hypothesis that there's no difference in MPF before and after diesel exposure. This leads us to believe that diesel exposure has a statistically significant effect on the MPF at brain location 1.