Question

Twenty third graders were randomly separated into four equal groups, and each group was taught a math concept using a different teaching method. At the end of the teaching period, progress was measured by a unit test. The scores are shown below (one child in group 3 was absent on the day that the test was administered). $$ \begin{array}{rrrr} \hline & {\text { State }} \\ \hline 1 & 2 & 3 & 4 \\ \hline \$ 261 & \$ 236 & \$ 250 & \$ 265 \\ 255 & 240 & 245 & 270 \\ 258 & 225 & 255 & 258 \\ 267 & 233 & 248 & 275 \\ 270 & 240 & 260 & 275 \\ \hline \end{array} $$ a. What type of design has been used in this experiment? b. Construct an ANOVA table for the experiment. c. Do the data present sufficient evidence to indicate a difference in the average scores for the four teaching methods? Test using \(\alpha=.05 .\)

Short answer

#Tag_title#Sample Short Answer #Tag_content# Based on the one-way ANOVA performed, we rejected the null hypothesis, suggesting that there is a significant difference in the average scores for the four teaching methods at a 0.05 significance level. The experimental design used is a one-way ANOVA with four groups.

Step-by-step solution

a. Identifying the experimental design

The experiment consists of 23 students divided into four equal groups, and each group is subjected to a unique teaching technique. The students are randomly assigned, so the experimental design used is a one-way ANOVA (Analysis of Variance) with four groups. It is used to test if there are any significant differences between the means of the four groups.

b. Constructing an ANOVA table

To construct the ANOVA table, follow these steps: 1. Calculate the mean score for each group. Let's call them \(\bar{x}_1, \bar{x}_2, \bar{x}_3,\) and \(\bar{x}_f\). 2. Calculate the overall mean score. Let's call it \(\bar{x}_{total}\). 3. Calculate the Sum of Squares for Treatments (SST), Sum of Squares for Errors (SSE), and Total Sum of Squares (SSTotal). 4. Calculate mean squares for treatments (MST) and errors (MSE) by dividing SST and SSE by their respective degrees of freedom. 5. Calculate the F-value by dividing MST by MSE. 6. Calculate the critical F-value and compare it with the calculated F-value to test the hypothesis. Let's execute these steps now: 1. Group means:\ Group 1: \(\bar{x}_1 = \frac{261+255+258+267+270}{5}=262.2\)\ Group 2: \(\bar{x}_2 = \frac{236+240+225+233+240}{5}=234.8\)\ Group 3: \(\bar{x}_3 = \frac{250+245+255+248+260}{5}=251.6\)\ Group 4: \(\bar{x}_4 = \frac{265+270+258+275+275}{5}=268.6\) 2. Overall mean:\ \(\bar{x}_{total} = \frac{261+255+258+267+270+236+240+225+233+240+250+245+255+248+260+265+270+258+275+275}{20}=251.15\) 3. Calculate the Sum of Squares:\ SST = \(5[(\bar{x}_1 - \bar{x}_{total})^2 + (\bar{x}_2 - \bar{x}_{total})^2 + (\bar{x}_3 - \bar{x}_{total})^2 + (\bar{x}_4 - \bar{x}_{total})^2]=5374.2\)\ SSE = Sum of squares of the differences between individual scores in each group and their respective group means = 2923.00\ SSTotal = SST + SSE = 8297.2 4. Calculate mean squares:\ MST = \(\frac{SST}{k-1}\), where k is the number of groups\ MST = \(\frac{5374.2}{4-1}=1791.4\)\ MSE = \(\frac{SSE}{n-k}\), where n is the total number of students\ MSE = \(\frac{2923}{20-4}=162.39\) 5. Calculate F-value:\ F-value = \(\frac{MST}{MSE}=11.03\)

c. Hypothesis testing

We have to test if there is a significant difference in the average scores of the four teaching methods, using a significance level of 0.05. Null hypothesis \((H_0)\): There is no significant difference in the average scores. Alternative hypothesis \((H_1)\): There is a significant difference in the average scores. To test this, we compare the calculated F-value and the critical F-value. The critical F-value can be obtained from the F-distribution table, using df1 = k - 1 = 3 and df2 = n - k = 16 and \(\alpha=0.05\). Critical F-value = 3.240 Since our calculated F-value (11.03) is greater than the critical F-value (3.240), we reject the null hypothesis \((H_0)\) in favor of the alternative hypothesis \((H_1)\). The data provide sufficient evidence at a 0.05 significance level to indicate a difference in the average scores for the four teaching methods.

Key concepts

Experimental Design in Education

Experimental design plays a crucial role in educational research, particularly when investigating the effectiveness of various teaching methods. In the exercise provided, a one-way ANOVA (Analysis of Variance) design is employed to compare the math scores of third graders who are taught with different techniques.

Such an experimental design in education involves randomly assigning participants to different groups. Each group receives a unique instructional approach. This randomization attempts to ensure that the groups are comparable and that the results can be attributed to the teaching method rather than to other extraneous variables.

When designing an educational experiment, it's essential to consider the size of the groups, the homogeneity of the participants, and the measures used to assess outcomes. Larger and more homogeneous groups can lead to more reliable and valid conclusions. A well-designed education experiment, like the one described, provides a solid foundation for determining the best practices in teaching methods, ultimately enhancing the learning experience for students.

ANOVA Test for Teaching Methods

The ANOVA test is a statistical method used to compare the means of three or more groups, like in the exercise where four different teaching methods were compared. In education, the ANOVA test can be instrumental in assessing the effectiveness of various instructional strategies on student learning outcomes.

In the step-by-step solution, the ANOVA table is constructed by calculating means, sum of squares, mean squares, and the F-value. Each step is fundamental, with the F-value being a critical statistic that compares the variance between the group means to the variance within the groups.

The F-value obtained from this process helps in determining whether the teaching methods have statistically significant differences in their effectiveness. If significant, educators might consider the teaching method that leads to the highest mean scores for future implementation. The importance of this test in educational settings cannot be overstated as it aids in making data-driven decisions that could potentially improve the quality of education.

Statistical Hypothesis Testing in Education

Statistical hypothesis testing is a methodological procedure used by educators and researchers to make inferences about the population based on sample data. In the context of the given exercise, hypothesis testing is used to determine whether variations in third graders' math scores are due to the teaching method or are a product of random chance.

Hypothesis testing starts with the formulation of two hypotheses: the null hypothesis, which suggests no effect or difference, and the alternative hypothesis, which suggests that there is an effect or a difference. In our case, the null hypothesis states that there is no significant difference in average scores across the teaching methods, while the alternative posits that a significant difference does exist.

The \(p\)-value associated with the F-value is compared to the significance level (usually 0.05) to determine whether to accept or reject the null hypothesis. A \(p\)-value less than 0.05 would lead to a rejection of the null hypothesis, indicating that the teaching methods do significantly differ in their effectiveness. This process is invaluable in guiding educators as it promotes the use of evidence-based practices in teaching and learning.