Question

The reaction times are measured for 50 college students who are randomly assigned to two groups. The students in group 1 listen to loud music during the test and the students in group 2 perform the test in silence. There are 25 students in each group. The mean for the loud-music group is 2.64 seconds, and the mean for the silent

group is 3.22 seconds. The data for both groups is combined, 25 reaction times are selected at random, and the mean of these reaction times is compared to the mean
of the remaining 25 reaction times. This process is repeated 14 times. The means for
the groups are summarized in the table.
Is the difference in the mean reaction times from the original experimental groups likely due to chance, or is the evidence that the music has an effect on reaction time? Explain your reasoning.

Answer 1

See below.

Explanation:

To determine whether the difference in the mean reaction times between the two original experimental groups (loud music and silence) is likely due to chance or whether there's evidence that the music has an effect on reaction time, you can perform a statistical analysis, such as a t-test or analysis of variance (ANOVA). Here's a step-by-step explanation of the process:

Step 1: Define Hypotheses

Null Hypothesis (H0): There is no significant difference in the mean reaction times between the two groups; any observed difference is due to chance.

Alternative Hypothesis (Ha): There is a significant difference in the mean reaction times between the two groups, indicating that music has an effect on reaction time.

Step 2: Calculate the Test Statistic

You can use a t-test for independent samples to compare the means of the two groups. The test statistic is calculated as:

t = \frac{{\text{mean of Group 1} - \text{mean of Group 2}}{{\sqrt{\frac{{S1^2}}{{n1}} + \frac{{S2^2}}{{n2}}}}}

Where:

S1 and S2 are the sample standard deviations for each group.

n1 and n2 are the sample sizes for each group.

You have the means for both groups (2.64 seconds for loud music and 3.22 seconds for silence).

Step 3: Calculate the Degrees of Freedom

Degrees of freedom (df) is calculated as (n1 + n2 - 2), where n1 and n2 are the sample sizes for each group.

Step 4: Find the Critical Value

Based on your chosen level of significance (usually 0.05), find the critical value from a t-distribution table.

Step 5: Calculate the p-value

Use the calculated t-statistic and degrees of freedom to find the p-value associated with the test statistic.

Step 6: Make a Decision

If the p-value is less than your chosen level of significance, you reject the null hypothesis in favor of the alternative hypothesis. This indicates that there is evidence to suggest that music has an effect on reaction time.

If the p-value is greater than your chosen level of significance, you fail to reject the null hypothesis. This means there is no strong evidence that music has an effect on reaction time.

The conclusion you reach depends on the p-value. A small p-value (typically less than 0.05) suggests that the difference in mean reaction times is unlikely to have occurred due to random chance alone, supporting the alternative hypothesis. A large p-value suggests that the difference could have occurred by random sampling variability, and there's no strong evidence for an effect of music on reaction time.

You would need the sample standard deviations and sample sizes to perform the actual calculations and make a decision.