Question

The article "An Alternative Vote: Applying Science to the Teaching of Science" (The Economist, May 12,2011 ) describes an experiment conducted at the University of British Columbia. A total of 850 engineering students enrolled in a physics course participated in the experiment. Students were randomly assigned to one of two experimental groups. Both groups attended the same lectures for the first 11 weeks of the semester. In the twelfth week, one of the groups was switched to a style of teaching where students were expected to do reading assignments prior to class, and then class time was used to focus on problem solving, discussion, and group work. The second group continued with the traditional lecture approach. At the end of the twelfth week, students were given a test over the course material from that week. The mean test score for students in the new teaching method group was \(74,\) and the mean test score for students in the traditional lecture group was \(41 .\) Suppose that the two groups each consisted of 425 students. Also suppose that the standard deviations of test scores for the new teaching method group and the traditional lecture method group were 20 and 24 , respectively. Estimate the difference in mean test score for the two teaching methods using a \(95 \%\) confidence interval. Be sure to give an interpretation of the interval.

Short answer

Estimated difference in mean test scores: 33 +/- 1.96(1.5616); Confidence Interval (95%): 30.93 - 35.07.

Step-by-step solution

Compute the difference in means

Firstly, calculate the difference in means between the two groups. Given that the mean test score for the new teaching method group is 74 and the mean test score for the traditional lecture group is 41, the difference in means \(\(\mu_1 - \mu_2\)\) is \(74 - 41 = 33\).

Calculate Standard Error

The standard error of the difference in means is computed using the formula: \(\(\sqrt{\frac{{s_1}^2}{n_1} + \frac{{s_2}^2}{n_2}}\)\), where \(s_1\) and \(s_2\) are the standard deviations of the two groups and \(n_1\) and \(n_2\) are the sizes of the two groups. Plugging in the given values: \(\sqrt{\frac{{20}^2}{425} + \frac{{24}^2}{425}}\) = 1.5616.

Determine Z-Score for 95% Confidence Interval

To find the Z-score for a 95% confidence interval, refer to a standard normal distribution table, or use a calculator or software that can find the Z-score. The Z-score that corresponds to a 95% confidence interval is approximately 1.96.

Calculate the Confidence Interval

The 95% confidence interval for the difference in means is calculated using the formula: \((\mu_1-\mu_2) \pm Z(SE)\), where \(SE\) is the standard error, \(Z\) is the Z-score and \((\mu_1-\mu_2)\) is the difference in means. Substituting the values, we find the confidence interval is \(33 \pm 1.96(1.5616)\). This gives an interval of \(30.93, 35.07\).

Interpret the Results

The interpretation of the confidence interval is that we are 95% confident that the true difference in mean test scores between the new teaching method and the traditional lecture method group lies between 30.93 and 35.07.

Key concepts

Statistics Education

When faced with the task of understanding an experiment's results, like the one conducted at the University of British Columbia, the role of statistics education cannot be overstated. It's essential in aiding students to make sense of data, draw accurate conclusions, and apply them to real-world situations.

In the context of the experiment with engineering students, statistics education empowers learners to interpret the mean test scores, standard deviations, and what the confidence intervals tell us about the two different teaching methods. Such educational groundwork is the foundation upon which solid scientific inquiry is built, enabling future professionals to incorporate data-driven decision-making deeply into their fields.

Standard Error

The standard error is a vital concept in statistics that measures the precision of a sample mean by indicating how much the sample mean is expected to fluctuate as a result of sampling variability. In essence, it informs us about the 'error' or deviation one can expect when using a sample mean to estimate the population mean.

For the university experiment, the standard error helps us assess how confidently we can speak about the mean difference in test scores between the two groups. A smaller standard error would indicate that the sample mean is a more precise estimate of the population mean. In this case, the relatively small value of 1.5616 suggests the sample means are likely to be close to the true means of the population.

Experimental Group Design

Experimental group design is a key aspect of research that involves structuring different groups to compare the effects of various conditions. In the described experiment, there were two groups: one experiencing a new teaching method and another continuing with traditional lectures.

This kind of comparative analysis is fundamental to understanding the efficacy of one method over another. Randomly assigning students to each group helps to eliminate biases and ensures that differences in outcomes can be attributed more confidently to the teaching methods rather than other variables.

Mean Difference Estimation

Understanding mean difference estimation is crucial when comparing two experimental groups. It involves calculating the difference between the mean outcomes of each group. In the UBC experiment, the mean difference estimation tells us how much more effective one teaching method is over the other, in terms of the students' average test scores.

The calculated mean difference of 33 points is significant, suggesting a substantial improvement with the new teaching method. However, this point estimate alone doesn't give a full picture; we also consider the confidence interval, which in this case tells us we can be 95% confident the true mean difference lies between 30.93 and 35.07, affirming the superiority of the new teaching approach with a quantifiable level of certainty.