Question

In the experiment 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 example, assume that it is reasonable to regard the sample of 10 men as representative of healthy adult males. $$ \begin{array}{ccrcr} \hline & \text { Location 1 } & \text { Location 1 } & \text { Location 2 } & \text { Location 2 } \\ \text { Subject } & \text { Before } & \text { After } & \text { Before } & \text { After } \\ \hline 1 & 6.4 & 8.0 & 6.9 & 9.4 \\ 2 & 8.7 & 12.6 & 9.5 & 11.2 \\ 3 & 7.4 & 8.4 & 6.7 & 10.2 \\ 4 & 8.7 & 9.0 & 9.0 & 9.6 \\ 5 & 9.8 & 8.4 & 9.7 & 9.2 \\ 6 & 8.9 & 11.0 & 9.0 & 11.9 \\ 7 & 9.3 & 14.4 & 7.9 & 9.1 \\ 8 & 7.4 & 11.3 & 8.3 & 9.3 \\ 9 & 6.6 & 7.1 & 7.2 & 8.0 \\ 10 & 8.9 & 11.2 & 7.4 & 9.1 \\ \hline \end{array} $$ a. Do the data provide convincing evidence that the mean MPF at brain location 1 is higher after diesel exposure? Test the relevant hypotheses using a significance level of \(.05 .\) b. Construct and interpret a \(90 \%\) confidence interval estimate for the difference in mean MPF at brain location 2 before and after exposure to diesel exhaust.

Short answer

For part a, the conclusion depends on the p-value from the computation in Step 3. If the p-value is less than 0.05, then there is convincing evidence to suggest that the mean MPF at brain location 1 is higher after diesel exposure. For part b, the interpretation will depend on the constructed 90% confidence interval in Step 4. If the interval includes 0, there's no significant effect of diesel exposure on the mean MBF at brain location 2, but if 0 is not in the interval, it suggests a significant effect.

Step-by-step solution

State the hypotheses

The null hypothesis (H0) is that the mean MPF is the same before and after diesel exposure. The alternative hypothesis (Ha) is that the mean MPF is higher after diesel exposure. H0: µ_before = µ_after, Ha: µ_before < µ_after. Remember, µ_before represents the mean MBF at a specific brain location before diesel exposure and µ_after represents the mean MBF after diesel exposure.

Compute the test statistic

First, compute the mean and standard deviation of the differences in MBF before and after diesel exposure for both locations. Use these to compute the t test statistic: t = (Xbar - µ) / (s/√n), where Xbar is the sample mean, µ is the population mean under H0, s is the standard deviation of the sample, and n is the sample size.

Compute the p-value and reach a decision

Use a t-distribution to compute the p-value corresponding to the absolute value of the t statistic. If the p-value is less than 0.05, the significance level, then reject the null hypothesis.

Compute the confidence interval

For part b, to prepare the 90% confidence interval for the difference in mean MBF at brain location 2 before and after exposure, use this formula: (Xbar_diff - t*s/√n , Xbar_diff + t*s/√n), where Xbar_diff is the mean of the differences, s is the standard deviation of the differences, and n is the sample size. 't' is the t critical value for a 90% confidence level, which can be found in a t-table.

Interpret the confidence interval

The 90% confidence interval provides a range of plausible values for the mean difference in MBF before and after diesel exposure at brain location 2. If the interval includes 0, that suggests no significant effect of diesel exposure. If 0 is not in the interval, it suggests a significant effect.

Key concepts

Mean Power Frequency

When studying brain activity, one important measure researchers use is the Mean Power Frequency (MPF). MPF represents the average frequency at which the power of the electrical signals generated by the brain is highest. In simpler terms, it is a method to quantify the activity in different parts of the brain under various conditions.

MPF is particularly useful in neuroscience and psychology research, as changes in MPF may indicate alterations in neuronal activity, possibly due to exposure to certain stimuli or substances, such as diesel exhaust in the given exercise. By comparing MPF measurements taken at different times, such as before and after exposure, scientists can infer whether an external factor has a significant effect on brain function.

Confidence Interval Estimation

Understanding confidence interval estimation is crucial when dealing with statistics. A confidence interval gives a range within which we are fairly sure the true parameter (like the mean power frequency difference) lies. It's calculated from the data and has a corresponding confidence level, like 90%, which in plain language means 'we're 90% confident that the true difference between MPFs before and after exposure is within this interval'.

When constructing confidence intervals, you use the mean of your sample measurements and adjust for variability and sample size. The interval adds and subtracts a margin of error from the sample mean, which includes the t critical value from the t-distribution reflecting the level of confidence you're aiming for.

t-test

The t-test is a statistical method used to determine if there is a significant difference between the means of two groups, under the assumption that the populations follow a normal distribution. In the context of our exercise, a t-test helps us test the hypothesis that the mean power frequency at specific brain locations has changed after exposure to diesel exhaust.

The t-test calculates a 't statistic', which, in essence, shows how much the sample mean deviates from the null hypothesis' expected mean, scaled by the variability of the data. A p-value is then derived from this t statistic, which gives the probability of observing such a difference if the null hypothesis were true. A p-value lower than the chosen significance level, like .05, indicates strong evidence against the null hypothesis, suggesting that the diesel exposure had a significant effect on MPF.

Brain Activity Measurement

Assessing brain activity involves various techniques, and measuring the mean power frequency (MPF) is one of them. MPF captures the electrical activity in the brain, which can be recorded using electroencephalography (EEG). During EEG, electrodes placed on the scalp detect the brain's electrical signals, and these readings allow researchers to explore how different factors, like environmental pollutants, may affect mental states and cognitive functions.

This type of measurement is highly relevant in studies like the one mentioned in the exercise because it provides direct insight into the physiological effects that substances, such as diesel exhaust, can have on the brain. By evaluating the MPF changes, scientists gather evidence on how certain exposures can influence neuronal dynamics, potentially leading to recommendations for public health or workplace safety.