In Thinking and Reasoning (Apr. 2007), researchers at...

In Thinking and Reasoning (Apr. 2007), researchers at Columbia University conducted a series of studies to assess the cognitive skills required for successful arguments. One study focused on whether one would choose to argue by weakening the opposing position or by strengthening the favored position. A sample of 52 graduate students in psychology was equally divided into two groups. Group 1 was presented with 10 items such that the argument always attempts to strengthens the favored position. Group 2 was presented with the same 10 items, but in this case the argument always attempts to weaken the nonfavored position. Each student then rated the 10 arguments on a five-point scale from very weak (1) to very strong (5). The variable of interest was the sum of the 10 item scores, called the total rating. Summary statistics for the data are shown in the accompanying table. Use the methodology of this chapter to compare the mean total ratings for the two groups at $\alpha=.05 .$ Give a practical interpretation of the results in the words of the problem. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Group 1 (support } \\ \text { favored position) } \end{array} & \begin{array}{c} \text { Group } 2 \text { (weaken } \\ \text { opposing position) } \end{array} \\ \hline \text { Sample size } & 26 & 26 \\ \text { Mean } & 28.6 & 24.9 \\ \text { Standard } & 12.5 & 12.2 \\ \text { deviation } & & \\ \hline \end{array} $$

In Thinking and Reasoning (Apr. 2007), researchers at Columbia University conducted a series of studies to assess the cognitive skills required for successful arguments. One study focused on whether one would choose to argue by weakening the opposing position or by strengthening the favored position. A sample of 52 graduate students in psychology was equally divided into two groups. Group 1 was presented with 10 items such that the argument always attempts to strengthens the favored position. Group 2 was presented with the same 10 items, but in this case the argument always attempts to weaken the nonfavored position. Each student then rated the 10 arguments on a five-point scale from very weak (1) to very strong (5). The variable of interest was the sum of the 10 item scores, called the total rating. Summary statistics for the data are shown in the accompanying table. Use the methodology of this chapter to compare the mean total ratings for the two groups at $\alpha=.05 .$ Give a practical interpretation of the results in the words of the problem.
$$
\begin{array}{lcc}
\hline & \begin{array}{c}
\text { Group 1 (support } \\
\text { favored position) }
\end{array} & \begin{array}{c}
\text { Group } 2 \text { (weaken } \\
\text { opposing position) }
\end{array} \\
\hline \text { Sample size } & 26 & 26 \\
\text { Mean } & 28.6 & 24.9 \\
\text { Standard } & 12.5 & 12.2 \\
\text { deviation } & & \\
\hline
\end{array}
$$

Key Concepts

Independent Samples t-Test

The independent samples t-test is a statistical technique used to compare the means of two unrelated groups. It determines whether any observed difference between the group means is statistically significant, considering the variability in the sample data and the sample sizes, making it a fundamental tool in comparing two conditions or treatments.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) typically states that there is no difference between the group means, while the alternative hypothesis (H?) asserts that there is a difference. This framework is crucial to determine whether any observed effect in the data is due to chance or reflects a true underlying effect.

Significance Level (?)

The significance level, denoted as ? (commonly set at 0.05), is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the error made when a true null hypothesis is incorrectly rejected. Setting ? helps control the risk of false positives in statistical testing.

Test Statistic and p-Value

The test statistic is a standardized value derived from the sample data that measures the degree of difference between the groups relative to the variability within the groups. The p-value indicates the probability of observing a test statistic as extreme as, or more extreme than, the one computed if the null hypothesis were true. Together, they help determine whether the observed differences are statistically significant.

Assumptions of the t-Test

The t-test relies on key assumptions including independence of observations, normality of the population distributions, and homogeneity of variances (equal variances) between groups. Verifying these assumptions is important to ensure that the results of the t-test are valid and that any conclusions drawn from the analysis are reliable.

Degrees of Freedom

Degrees of freedom, which reflect the sample size used to estimate variability, are a parameter in the t-distribution that influences the critical values and shape of the curve. In a two-sample t-test, degrees of freedom are typically calculated based on the sizes of both samples, and they account for the estimation of group means from the data.

Transcript

00:01 Here's a solution for 925.

00:03 So we have two samples, and this is about winning an argument or strengthening an argument.

00:09 And in group one, it says that each student got a sample 52 graduate students in a psychology, it was equally divided in two groups.

00:20 Group one was presented with 10 items, such that the argument always attempts to strengthen the favorite position.

00:25 And then group two was presented with the same 10 items, but this case, in this case, the argument always attempts to weaken the non -favor position.

00:35 So each student then rated the 10 arguments on a 5 -point scale from 1 to 5, and we wanted to look at the sum of the 10 item scores called the total rating.

00:45 So here are the summary stats.

00:47 So we have equal sample sizes, and here are the mean ratings, and then the standard deviations.

00:53 So first, let's write down what our hypotheses would be in this case.

00:57 So we have h -0 being that mu -1 minus mu -2 equals 0, meaning there's no difference between the two meaning ratings.

01:05 And then we don't know which one is necessarily bigger.

01:08 So we just want to see if there's a difference between having just a positive or a negative argument.

01:13 Okay, so now let's find the test statistic.

01:16 And we're going to use the t -value for that, the t -test, and we'll also find the p -value.

01:21 And i'm going to use a calculator for this.

01:23 So if you go to stat and then test, and it's the fourth one down, the test.

01:27 Two sample t test...

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