Question

Word-Association Experiments A comparison of reaction times for two different stimuli in a word-association experiment produced the accompanying results when applied to a random sample of 16 people: Do the data present sufficient evidence to indicate a difference in mean reaction times for the two stimuli? Use the Wilcoxon rank sum test and explain your conclusions.

Short answer

Answer: Yes, if the p-value is less than the chosen significance level (typically 0.05), then we can conclude that there is a significant difference in the mean reaction times for the two stimuli.

Step-by-step solution

Organize the Data

Begin by organizing the reaction times for each stimulus in two separate columns. Label them as Stimulus A and Stimulus B. This will help us to compare them in the later steps.

Combine and Rank the Data

Combine the reaction times for both stimuli into a single list and assign ranks to each reaction time. If there are any ties in the data, assign the average rank to all the tied values.

Calculate the Rank Sum for Each Stimulus

Now, calculate the rank sum for each stimulus by adding up the ranks assigned to the reaction times for Stimulus A and the ranks assigned to the reaction times for Stimulus B.

Determine the Smaller Rank Sum

From the rank sums you just calculated, determine the smaller of the two rank sums (let's call this value W).

Calculate Expected Value and Standard Deviation of the Smaller Rank Sum

Calculate the expected value (E) and the standard deviation (SD) of the smaller rank sum W using the following formulas: E = n1 * (n1 + n2 + 1) / 2 SD = sqrt(n1 * n2 * (n1 + n2 + 1) / 12) where n1 and n2 are the sample sizes of Stimulus A and Stimulus B, respectively.

Calculate the Standardized Test Statistic

Calculate the standardized test statistic (Z) using the following formula: Z = (W - E) / SD

Determine the Wilcoxon Rank Sum Test Significance Level

Using a standard normal distribution table, find the p-value associated with the calculated Z value.

Interpret the Results

If the p-value found in Step 7 is less than the chosen significance level (typically 0.05), then we can conclude that there is sufficient evidence to indicate a difference in the mean reaction times for the two stimuli. If the p-value is greater than the chosen significance level, we fail to reject the null hypothesis and cannot conclude that there is a significant difference in mean reaction times between the two stimuli.

Key concepts

Word-Association Experiment

Word-association experiments are a staple in psychological research, measuring the connection between words and what they evoke in the mind. These experiments typically present subjects with a word, asking them to respond with the first word that comes to mind. Researchers are often interested in the reaction time, which is the time taken for a participant to respond after the stimulus word is presented.
The reaction time is telling – it can indicate the strength of association, cognitive processing speed, or even reveal connections and biases that are not immediately apparent. In educational settings, word-association experiments are used to study learning processes, memory, and language comprehension among students. For a more comprehensive analysis, comparisons of reaction times across different stimuli (like in the given exercise) are conducted. If differences are significant, it could imply varying levels of difficulty, emotional responses, or associations with the words presented.

Reaction Time Comparison

Comparing reaction times is a method to understand how different stimuli affect cognitive processing. It's not just the speed that's of interest; the consistency and variability of reaction times can be just as telling. For example, certain words may elicit faster reactions due to their commonality or emotional charge, while more complex or unfamiliar words might lead to slower reaction times.
In our exercise, reaction times were collected for two different stimuli in a word-association experiment, and the goal is to determine if there's a significant difference in these times between the two. This comparison can shed light on various psychological aspects, including cognitive load and associative strength. Furthermore, since reaction times are a type of continuous data, they are apt for statistical analysis – in particular, the use of nonparametric tests like the Wilcoxon rank sum test when the data don't necessarily follow a normal distribution.

Nonparametric Statistical Test

Nonparametric statistical tests are the tools of choice when data do not meet the assumptions necessary for parametric tests, such as normal distribution of the data points or homogeneity of variance. One such test is the Wilcoxon rank sum test, which is used to assess whether two independent samples come from the same distribution.
The test ranks all the data points and then sums these ranks for each group. It’s especially useful when dealing with skewed distributions or when sample sizes are small – conditions that are quite common in the psychological and biological sciences. Instead of comparing means like the t-test, the Wilcoxon rank sum test compares median reaction times, which can be more robust against outliers and non-normal data. This makes it an excellent choice for analyzing the reaction time data we encounter in the word-association experiment exercise.