Question

To test the effect of alcohol in increasing the reaction time to respond to a given stimulus, the reaction times of seven people were measured. After consuming 3 ounces of \(40 \%\) alcohol, the reaction time for each of the seven people was measured again. Do the following data indicate that the mean reaction time after consuming alcohol was greater than the mean reaction time before consuming alcohol? Use \(\alpha=.05 .\) $$ \begin{array}{llllllll} \text { Person } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Before } & 4 & 5 & 5 & 4 & 3 & 6 & 2 \\ \text { After } & 7 & 8 & 3 & 5 & 4 & 5 & 5 \end{array} $$

Short answer

Answer: No, we cannot conclude that alcohol consumption significantly increases the mean reaction time, as there is not enough evidence to reject the null hypothesis that alcohol consumption does not significantly increase the mean reaction time, with a significance level of 0.05.

Step-by-step solution

Formulate the null and alternative hypothesis

Let \(\mu_d\) be the mean difference between reaction times before and after consuming alcohol. The null hypothesis (\(H_0\)) states that the mean difference is zero or negative (i.e., alcohol does not increase reaction time), and the alternative hypothesis (\(H_a\)) states that the mean difference is positive (i.e., alcohol increases reaction time): $$H_0 : \mu_d \leq 0$$ $$H_a : \mu_d > 0$$

Calculate the mean difference and standard deviation for the given data

Compute the difference between the "Before" and "After" reaction times for each person: $$ \begin{array}{|c|c|} \hline \text { Person } & \text { Difference } \\ \hline 1 & -3 \\ 2 & -3 \\ 3 & 2 \\ 4 & -1 \\ 5 & -1 \\ 6 & 1 \\ 7 & -3 \\ \hline \end{array} $$ Calculate the mean difference, denoted by \(\overline{d}\): $$\overline{d} = \frac{-3-3+2-1-1+1-3}{7} = -\frac{8}{7}$$ Calculate the standard deviation of the differences, denoted by \(s_d\): $$s_d = \sqrt{\frac{(-3-(-\frac{8}{7}) )^2 + (-3-(-\frac{8}{7}) )^2 + ... + (-3-(-\frac{8}{7}) )^2}{6}} = \sqrt{\frac{94}{7}}$$

Calculate the t-score and degrees of freedom

Calculate t-score, denoted by \(t\): $$t = \frac{\overline{d} - 0}{\frac{s_d}{\sqrt{n}}} = \frac{-\frac{8}{7}}{\frac{\sqrt{\frac{94}{7}}}{\sqrt{7}}}= -2.79$$ Calculate degrees of freedom, denoted by \(df\): $$df = n - 1 = 7 - 1 = 6$$

Determine the critical t-value

From t-distribution table (or online calculator), find the critical t-value using the significance level (\(\alpha = 0.05)\) and degrees of freedom (\(df=6\)), making sure it refers to the right-tailed test: $$t_{critical} = 1.943$$

Compare the t-score to the critical t-value

Compare the t-score with the critical t-value: $$t = -2.79 < t_{critical} = 1.943$$ Since the t-score is less than the critical t-value, we do not have enough evidence to reject the null hypothesis.

Conclusion

With a significance level of \(\alpha=0.05\), there is not enough evidence to conclude that consuming alcohol increases the mean reaction time of people. Therefore, based on the given data, we cannot reject the null hypothesis that alcohol consumption does not significantly increase the mean reaction time.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a fundamental concept. It represents a default position that there is no effect or difference between groups or variables. In our exercise, the null hypothesis (\(H_0\)) suggests that consuming alcohol does not increase reaction time. Specifically, it states that the mean difference between reaction times before and after alcohol consumption is zero or negative:

• \(H_0 : \mu_d \leq 0\)

The null hypothesis is always assumed to be true until there is sufficient evidence against it. Its role is to ensure that findings are not due to random chance but represent a true effect. When performing hypothesis testing, we aim to gather enough evidence to reject the null hypothesis if there's a significant effect.

Alternative Hypothesis

The alternative hypothesis (\(H_a\)) stands in contrast to the null hypothesis. It suggests that there is a significant effect or difference that we are testing for. In this context, the alternative hypothesis posits that alcohol consumption does lead to increased reaction times:

• \(H_a : \mu_d > 0\)

The alternative hypothesis is the statement we hope to find evidence for in our testing. By comparing our computed statistical values against critical values, we determine whether the data supports this new perspective, indicating a genuine effect of alcohol on reaction time.

t-test

The t-test is a statistical test used to compare the means of two groups to see if they are different. In this exercise, a paired t-test is appropriate because the same group of people is tested before and after alcohol consumption. Here's how it works:

• Calculate the mean difference between the two conditions for all subjects.
• Measure the variability with standard deviation of these differences.
• Compute the t-score using the formula:\[t = \frac{\overline{d} - 0}{\frac{s_d}{\sqrt{n}}}\]
• Consult a t-distribution table to find the critical t-value for the specific degree of freedom.

The t-test helps determine if any observed differences are significant or if they occurred by random chance. In this scenario, although the mean difference was calculated, it did not suffice to show a significant increase in reaction time.

Reaction Time Analysis

Reaction time analysis involves measuring how quickly someone responds to a stimulus, a crucial factor in many psychological and physiological studies. In this particular scenario, the researchers wanted to examine how alcohol influences this critical response time. Observing the changes in reaction time before and after alcohol consumption provided an empirical basis for hypothesis testing.
To analyze reaction time, we gathered data from individuals in both situations and calculated the difference for each person. Reaction time analysis in this context allows us to investigate whether a factor, like alcohol, significantly impacts the cognitive or neural processes responsible for quick responses.
Ultimately, it's not just about observing the differences but using statistical tools to determine the reliability and significance of these differences. That's why performing a thorough and accurate reaction time analysis alongside hypothesis testing is vital in behavioral research.