91Ó°ÊÓ

Published reports indicate that a brain region called the nucleus accumbens (NAC) is involved in interval timing, which is the perception of time in the seconds-to-minutes range. To test this, researchers investigated whether removing the NAC interferes with rats' ability to time the presentation of a liquid reward. Using a conditioning procedure, the researchers had rats press a lever for a reward that was delivered after 16 seconds. The time that eight rats responded the most (peak responding) was recorded before and after a surgery to remove the NAC. The peak responding times are given in the following table. $$ \begin{array}{|l|l|} \hline \multicolumn{2}{|l|}{\text { Peak Interval Timing }} \\ \hline \text { Before NAC Surgery } & \text { After NAC Surgery } \\ \hline 15 & 20 \\ \hline 14 & 26 \\ \hline 16 & 20 \\ \hline 16 & 18 \\ \hline 17 & 25 \\ \hline 18 & 21 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} 15 & 18 \\ \hline 16 & 23 \\ \hline \end{array} $$ a. Test whether or not the difference in peak responding changed at a \(.05\) level of significance (two-tailed test). State the value of the test statistic and the decision to retain or reject the null hypothesis. b. Compute effect size using estimated Cohen's \(d\).

Step by step solution

01

State the Hypotheses

We need to determine if there is a significant difference between peak responding times before and after the NAC surgery. The null hypothesis \(H_0\) states there is no difference in the mean peak responding times: \(\mu_{\text{before}} = \mu_{\text{after}}\). The alternative hypothesis \(H_a\) states there is a difference: \(\mu_{\text{before}} eq \mu_{\text{after}}\).

02

Calculate the Differences

Calculate the difference between the peak responding times before and after the surgery for each rat. The differences are: \(-5, -12, -4, -2, -8, -3, -3, -7\).

03

Calculate the Mean and Standard Deviation of Differences

Compute the mean and standard deviation of the differences. The mean difference \(\bar{x}\) is \(-5.5\). Calculate the standard deviation \(s_d\) of these differences.

04

Perform a t-Test

Use the differences calculated to perform a paired t-test. The t-statistic is given by \(t = \frac{\bar{x}}{s_d / \sqrt{n}}\), where \(n\) is the number of pairs. With \(n = 8\) and given level of significance \(\alpha = 0.05\), look up the critical t-value for \(df = n - 1 = 7\).

05

Decision Making

Compare the calculated t-statistic with the critical t-value. If the absolute value of the calculated t-statistic exceeds the critical value, reject the null hypothesis \(H_0\); otherwise, do not reject \(H_0\).

06

Calculate Cohen's d for Effect Size

Compute Cohen's \(d\) using the formula \(d = \frac{\bar{x}}{s_d}\). Interpret the effect size to determine if the change in peak responding times is practically significant.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test

A t-test is a statistical method used to determine if there is a significant difference between the means of two groups. In the context of this exercise, we are exploring the paired samples t-test, which is ideal for comparing two related groups, such as the peak responding times of rats before and after surgery. The paired t-test considers each pair of observations from the same subject: one before the surgery, one after.

The formula for the t-statistic in a paired t-test is:

• \( t = \frac{\bar{x}}{s_d / \sqrt{n}} \)

where \(\bar{x}\) is the mean of the differences, \(s_d\) is the standard deviation of the differences, and \(n\) is the number of paired observations.

The calculated t-statistic is then compared to a critical t-value from the t-distribution table, determined by the chosen level of significance (\(\alpha = 0.05\) in this case) and the degrees of freedom (\(df = n - 1\)). If the t-statistic exceeds the critical value, it indicates a significant difference, allowing us to reject the null hypothesis.

Cohen's d

Cohen's d is a measure of effect size used to indicate the standardized difference between two means. It provides a sense of how meaningful the difference is, beyond mere statistical significance. In the exercise you're working on, Cohen's d helps determine if the change in peak responding times for the rats is practically significant, after acknowledging it as statistically significant through the t-test.

The formula for calculating Cohen's d is:

• \( d = \frac{\bar{x}}{s_d} \)

where \(\bar{x}\) is the mean of the differences between paired observations, and \(s_d\) is the standard deviation of those differences.

Interpreting Cohen’s d is straightforward:

• A value around 0.2 indicates a small effect
• A value around 0.5 indicates a medium effect
• A value around 0.8 or higher suggests a large effect

These guidelines help you understand the practical significance of the effects observed in your data.

paired samples

In statistics, paired samples refer to data collected from the same subjects under different conditions. This is particularly useful when examining changes over time or the impact of a treatment. For this exercise, the peak responding times are measured before and after the NAC surgery for each rat, creating paired observations.

The benefit of using paired samples lies in reducing variability. Since measurements are taken from the same subject, extraneous variables are controlled, resulting in more reliable comparisons. By evaluating differences within each pair, we focus on the effect of the treatment itself—in this case, the surgery on the NAC.

A paired sample design is ideal when:

• Subjects are tested under different conditions, like before and after a treatment
• The same subjects are used to ensure more consistent results
• The goal is to minimize the influence of individual variability

For this study, the paired sample approach ensures that changes in peak responding times relate directly to the surgery rather than other external factors.

effect size

Effect size is a vital statistic that quantifies the magnitude of a change, difference, or relationship in data. Understanding the effect size is crucial, as it tells you how meaningful the difference is in a real-world context, beyond just identifying that one exists at a statistical level.

While a t-test indicates if a difference is statistically significant, effect size provides insights into how large and impactful that difference really is. In the context of this exercise, after finding out whether the removal of the NAC affects peak responding times through hypothesis testing, you can use effect size to understand the practical implications.

The effect size can be particularly powerful in:

• Comparing the magnitude of differences across different studies
• Assessing the real-world significance of study outcomes
• Understanding if an intervention or treatment leads to a substantial, noticeable change

Ultimately, calculating effect size helps in evaluating the practical application of research findings, which is invaluable in interpreting the results of experiments involving biological or psychological phenomena.