Reaction Time In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key on seeing a blue screen and reaction time (in seconds) to press the key is measured. The same person is then asked to press a key on seeing a red screen, again with reaction time measured. The results for six randomly sampled study participants are as follows:

c. Is the reaction time to the blue stimulus different from the reaction time to the red stimulus at the α=0.01 level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Verified step by step guidance

1. Step 1: Calculate the difference in reaction times for each participant by subtracting the red stimulus reaction time from the blue stimulus reaction time. That is, for each participant, compute \(D_i = Blue_i - Red_i\).
2. Step 2: Find the mean difference \(\bar{D}\) and the standard deviation of the differences \(s_D\) using the differences calculated in Step 1.
3. Step 3: Determine the sample size \(n\), which is the number of paired differences (in this case, 6 participants).
4. Step 4: Calculate the test statistic for the paired t-test using the formula: \[ t = \frac{\bar{D} - 0}{s_D / \sqrt{n}} \] Here, the null hypothesis assumes the mean difference is zero (no difference in reaction times).
5. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table with \(n-1\) degrees of freedom at the \(\alpha = 0.01\) significance level (two-tailed test). If the absolute value of the test statistic exceeds the critical value, reject the null hypothesis, indicating a significant difference in reaction times.

Key Concepts

Paired Sample t-Test

A paired sample t-test compares the means of two related groups to determine if there is a statistically significant difference between them. It is used when the same subjects are measured under two conditions, such as reaction times to blue and red stimuli for the same participants.

Normality Assumption

The paired t-test assumes that the differences between paired observations are approximately normally distributed. This assumption can be checked using normal probability plots or boxplots, which help validate the use of the t-test for small sample sizes.

Significance Level and Hypothesis Testing

The significance level (α) defines the threshold for rejecting the null hypothesis. At α=0.01, there is a 1% risk of concluding a difference exists when it does not. Hypothesis testing involves comparing the p-value from the t-test to α to decide if the reaction times differ significantly.