A study was conducted to investigate the effects of teaching method and class size on student performance in an educational setting. The researchers randomly assigned students to different teaching methods (Traditional Lecture, Interactive Lecture, Problem-based Learning) and different class sizes (Small, Medium, Large). The performance of the students was measured using a standardized test, and the data collected is provided in the dataset "Student Performance".

a) [5 marks] Describe the experimental design used in this study. Identify the factors, the number of levels for each factor, the number of treatments, the number of replicates for each treatment, and the total number of experimental units.

b) [5 marks] Plot the residuals against the fitted values corresponding to the two-way ANOVA model for student performance. What two key model assumptions can be examined with this plot, and do they appear to be warranted?

c) [8 marks] Test the following hypotheses at the 1% significance level; make sure you follow all the steps for hypothesis testing indicated in the instructions.

i. Is there a significant difference in the mean student performance across different teaching methods?

ii. Is there a significant difference in the mean student performance across different class sizes?

iii. Is there a significant interaction effect between teaching method and class size on student performance?

d) [2 marks] Create the interaction plot for teaching method and class size on student performance and explain if the interaction plot confirms the results in part (c).

e) [3 marks] Calculate the Bonferroni-adjusted margin of error for the confidence intervals based on all pairwise differences between the mean student performances (treatment means). Show your manual calculations and use an overall 99% confidence level.

f) [1 mark] Explain why the Bonferroni method for comparing treatment means is necessary.

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g) [4 marks] Based on the calculated margin of error and the pairwise comparisons approach, determine:

i. which combination of teaching methods have the highest and lowest mean student performances.

ii. For problem-based learning teaching method, does large class size have significantly higher student performance than medium class size?

iii. For medium class size, does interactive lecture teaching method have significantly lower student performance than problem-based learning?

Teaching Method	Class Size
	Small	Medium	Large
Traditional Lecture	70	78	82
	75	80	85
	68	76	80
	72	79	84
	73	77	83
	69	75	81
	71	79	84
	74	81	87
Interactive Lecture	75	83	89
	78	85	92
	71	81	87
	76	84	91
	77	82	90
	73	80	88
	76	84	91
	79	86	94
Problem-Based Learning	80	88	94
	82	90	97
	79	87	92
	84	92	96
	85	91	85
	83	89	91
	81	90	87
	86	88	89

Note: Same Data (StatCrunch Input)
Teaching Method	Class Size	Student Performance
Traditional Lecture	Small	70
Traditional Lecture	Small	75
Traditional Lecture	Small	68
Traditional Lecture	Small	72
Traditional Lecture	Small	73
Traditional Lecture	Small	69
Traditional Lecture	Small	71
Traditional Lecture	Small	74
Traditional Lecture	Medium	78
Traditional Lecture	Medium	80
Traditional Lecture	Medium	76
Traditional Lecture	Medium	79
Traditional Lecture	Medium	77
Traditional Lecture	Medium	75
Traditional Lecture	Medium	79
Traditional Lecture	Medium	81
Traditional Lecture	Large	82
Traditional Lecture	Large	85
Traditional Lecture	Large	80
Traditional Lecture	Large	84
Traditional Lecture	Large	83
Traditional Lecture	Large	81
Traditional Lecture	Large	84
Traditional Lecture	Large	87
Interactive Lecture	Small	75
Interactive Lecture	Small	78
Interactive Lecture	Small	71
Interactive Lecture	Small	76
Interactive Lecture	Small	77
Interactive Lecture	Small	73
Interactive Lecture	Small	76
Interactive Lecture	Small	79
Interactive Lecture	Medium	83
Interactive Lecture	Medium	85
Interactive Lecture	Medium	81
Interactive Lecture	Medium	84
Interactive Lecture	Medium	82
Interactive Lecture	Medium	80
Interactive Lecture	Medium	84
Interactive Lecture	Medium	86
Interactive Lecture	Large	89
Interactive Lecture	Large	92
Interactive Lecture	Large	87
Interactive Lecture	Large	91
Interactive Lecture	Large	90
Interactive Lecture	Large	88
Interactive Lecture	Large	91
Interactive Lecture	Large	94
Problem-Based Learning	Small	80
Problem-Based Learning	Small	82
Problem-Based Learning	Small	79
Problem-Based Learning	Small	84
Problem-Based Learning	Small	85
Problem-Based Learning	Small	83
Problem-Based Learning	Small	81
Problem-Based Learning	Small	86
Problem-Based Learning	Medium	88
Problem-Based Learning	Medium	90
Problem-Based Learning	Medium	87
Problem-Based Learning	Medium	92
Problem-Based Learning	Medium	91
Problem-Based Learning	Medium	89
Problem-Based Learning	Medium	90
Problem-Based Learning	Medium	88
Problem-Based Learning	Large	94
Problem-Based Learning	Large	97
Problem-Based Learning	Large	92
Problem-Based Learning	Large	96
Problem-Based Learning	Large	85
Problem-Based Learning	Large	91
Problem-Based Learning	Large	87
Problem-Based Learning	Large	89

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Here are my answers to the questions a The experimental design used was a 33 factorial design There ... View the full answer