Question

The paper "Fudging the Numbers: Distributing Chocolate Influences Student Evaluations of an Undergraduate Course" (Teaching of Psychology [2007]: \(245-247\) ) describes an experiment in which 98 students at the University of Illinois were assigned at random to one of two groups. All students took a class from the same instructor in the same semester. Students were required to report to an assigned room at a set time to fill out a course evaluation. One group of students reported to a room where they were offered a small bar of chocolate as they entered. The other group reported to a different room where they were not offered chocolate. Summary statistics for the overall course evaluation score are given in the accompanying table. \begin{tabular}{lccc} Group & \(n\) & \(\bar{x}\) & \(s\) \\ Chocolate & 49 & 4.07 & 0.88 \\ No Chocolate & 49 & 3.85 & 0.89 \\ \hline \end{tabular} a. Use the given information to construct and interpret a \(95 \%\) confidence interval for the mean difference in overall course evaluation score. b. Does the confidence interval from Part (a) support the statement made in the title of the paper? Explain.

Short answer

The 95% confidence interval for the difference in mean course evaluation scores between the 'Chocolate' group and the 'No Chocolate' group is approximately \(-0.14, 0.58\). As the interval contains zero, it cannot definitively confirm the statement made in the paper's title. However, the presence of positive values demonstrates that there is a possibility of the chocolate group having higher scores.

Step-by-step solution

Calculate the mean difference

First, subtract the mean of the 'No Chocolate' group (\(3.85\)) from the mean of the 'Chocolate' group (\(4.07\)). This gives the observed mean difference between the two groups. \( \bar{x}_{d} = 4.07 - 3.85 = 0.22 \).

Calculate the standard error

Next, calculate the standard error (SE) of the mean difference. The standard error is a measure of the statistical accuracy of an estimate, which in this case is our mean difference. It can be calculated by using the formula \[SE = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}\] where \(s_{1}\) and \(s_{2}\) are the standard deviations of the first and the second groups, and \(n_{1}\) and \(n_{2}\) are the sizes of the first and the second groups. Substituting the gleaned values into the formula, we get \[SE = \sqrt{\frac{(0.88)^{2}}{49} + \frac{(0.89)^{2}}{49}} = 0.18118\].

Construct the 95% confidence interval

The 95% confidence interval is calculated using the formula \[(\bar{x}_{d} -t_{\alpha/2,df}*SE, \bar{x}_{d} + t_{\alpha/2,df}*SE )\] where \(\bar{x}_{d}\) is our calculated mean difference, \(t_{\alpha/2,df}\) is the t-score value that captures 95% of values (read from t-distribution table or calculated using statistical software) with 'df' being the degrees of freedom which is \(n_{1}+n_{2}-2\), and \( SE\) is our calculated standard error. Here, with 96 degrees of freedom, the t value for a two-tailed test at 95% confidence level is approximately 1.984. Substituting these values into the formula gives \[(0.22 - (1.984* 0.18118), 0.22 + (1.984* 0.18118)) = (-0.139906, 0.579906)\]

Interpret the Confidence Interval

Since the confidence interval includes zero, we can interpret this to mean we are 95% confident that the true population mean difference in course evaluation scores between the 'Chocolate' group and the 'No Chocolate' group could potentially be zero. This implies that there is a possibility that there is no significant difference in overall course evaluation scores between the two groups. We cannot therefore state definitively whether offering chocolate influences course evaluations.

Validate Title Statement

Since the confidence interval includes zero, this suggests that there may be no real difference between the groups, contradicting the paper's title, 'Fudging the Numbers: Distributing Chocolate Influences Student Evaluations of an Undergraduate Course'. However, it's also important to note that the interval also includes positive values, meaning there could be a difference (higher scores for the chocolate group), so it's not completely clear cut.