A Rasch-model-based hierarchical framework for statistical literacy and learning

Figures & data

Figure 1. Item characteristic curve.

Table 1. Content of the Business Statistics unit.

1.	Data Exploration: Data collection, Exploratory data analysis. (1 Week)
2.	Probability: Basic ideas, Discrete Random variables, expectation and variance, binomial and Poisson distributions, Hypothesis test for binomial proportion and Poisson mean. Continuous random variables, probability density function, mean and variance for simple distribution, Uniform distribution. (3 weeks).
3.	Jointly distributed discrete random variables: independence, covariance, independence, sum of random variables. (1 week)
4.	Chi square tests for Contingency tables. (0.5 week)
5.	Normal distribution: probabilities, sum of normal random variables. (1 week)
6.	Distribution of estimators: sample mean, sample proportion. (0.5 week)
7.	Estimation: confindence intervals for population proportion and population mean. (1 week)
8.	One sample tests: population proportion and population mean. (1 week)
9.	Two-sample test for population mean: paired and independent samples. (1 week)
10.	ANOVA, Kramer-Tukey comparisons. (1 week)
11.	Simple linear regression. (0.5 week)
12.	Multiple linear regression, including categorical covariates. (0.5 week)

Figure 2. A question from the examination.

Table 2. Rescored items and new scoring structure.

Rescored	Item	Original	Post-Rescore	Concepts
	1a	1	1	1a.1: Null and Alternative Hypothese
	1b	2	2	1b.1: Expression for test statistic
				1b.2: Distribution of test statistic
Y	1c	2	1	1c.1: Calculate test statistic
Y	1d	2	1	1d.1: Expression for p-value
	1e	1	1	1e.1: Test p-value
	1f	2	2	1f.1: Inferences made from analysis
				1f.2: Justification of Inference
Y	1g	2	1	1g.1: Consequence of Type I error
	2a	1	1	2a.1: Null and Alternative Hypothese
Y	2b	3	2	2b.1: Expression for test statistic
				2b.2: Distribution for test statistic
Y	2c	2	1	2c.1: Calculate test statistic
	2d	1	1	2d.1: Expression for p-value
	2e	1	1	2e.1: Test p-value
Y	2f	2	1	2f.1: Inference made from analysis
Y	2g	4	3	2g.1: Assumptions of the model
				2g.2: Assumption of model or Discuss validity previous assumption
				2g.3 Discuss validity
Y	2h	2	1	2h.1: Which is worse Type I or Type II error
Y	3a	2	1	3a.1: Null and Alternative hypothese
	3b	2	2	3b.1: Calculate test statistic
				3b.2: Distribution for test statistic
	3c	1	1	3c.1: Expression for p-value
	3d	2	2	3d.1: Test p-value
				3d.2: Test p-value
Y	3e	2	1	3e.1: Inference made from analysis
Y	4a	2	1	4a.1: Null and alternative hypothese
	4b	1	1	4b.1: Distribution of test statistic
Y	4c	2	1	4c.1: Test p-value
Y	4d	2	1	4d.1: Inference made from analysis
	4e	2	2	4e.1: Assumption of analysis
Y	4f	4	3	4f.1: Confidence Interval: Pool variance
				4f.2: Calculation of Confidence Interval
				4f.3: Translate CI back into context
Y	5a	2	1	5a.1: Knowledge of data gathering
Y	5b	2	1	5b.1: Null and Alternate Hypothese
Y	5c	3	2	5c.1: Assumption of analysis
				5c.2: Validate assumption
	5d	2	2	5d.1: Test p-value
				5d.2: Inference made from analysis
Y	5e	4	1	5e.1: ANOVA TABLE: Complete Table
Y	5f	4	2	5f.1: Inference made from analysis
				5f.2: Validate inference
Y	5g	6	2	5g.1: Confidence Interval: Width components
				5g.2: Translate CI back into context
Y	5hi	2	1	5h.1: Confidence Interval: Width + Sample Size calculation
				5h.2: Confidence Interval: New degrees of freedom
Y	5i	2	1	5i.1: Inference made from analysis
Y	6a	6	2	6a.1: State linear regression equation
				6a.2: Inference made from model
Y	6b	4	2	6b.1: Linear regression calculations
				6b.2: Calculations from understanding context
Y	6c	3	2	6c.1: Assumptions of model
				6c.2: Validate assumptions of model

Figure 3. Person and item estimates distributions.

Table 3. Concepts and corresponding estimated difficulty for subset 1.

	Subset 1
Item	Sample mean Hypothesis testing	Difficulty
1e.1	Test p-value	−3.081191
1b.1	Expression for test statistic	−1.755735
1a.1	Null and Alternative hypotheses	−1.508292
1f.1	Inferences made from analysis	−1.428720
1d.1	Expression for p-value	−1.256588
1c.1	Calculate test statistic	−1.133645
1b.2	Distribution of test statistic	−.062591
1g.1	Consequence of Type I error	.666767
1f.2	Justification of Inference	.892913

Table 4. Concepts and corresponding estimated difficulty for subset 2.

	Subset 2
Item	Population proportion Hypothesis testing	Difficulty
2a.1	Null and Alternative hypotheses	−1.480074
2d.1	Expression for p-value	−1.422848
2e.1	Test p-value	−1.399515
2f.1	Inference made from analysis	−.443215
2c.1	Calculate test statistic	−.300311
2b.1	Expression for test statistic	−.072824
2g.1	Assumptions of the model	.200807
2b.2	Distribution for test statistic	.427942
2g.2	Assumption of model or Discuss validity previous assumption	1.078447
2h.1	Which is worse Type I or Type II error	1.384398
2g.3	Discuss validity of assumption	1.906115

Table 5. Concepts and corresponding estimated difficulty for subset 3.

	Subset 3
Item	Difference in mean hypothesis test	Difficulty
3d.1	Test p-value	−3.252977
3d.2	Test p-value	−1.022625
3a.1	Null and Alternative hypotheses	−.737017
3b.1	Calculate test statistic	−.082433
3c.1	Expression for p-value	.001383
3b.2	Distribution for test statistic	.0971366
3e.1	Inference made from analysis	.132357

Table 6. Concepts and corresponding estimated difficulty for subset 4.

	Subset 4
Item	Two sample t-test	Difficulty
4a.1	Null and alternative hypotheses	−1.663054
4d.1	Inference made from analysis	−1.421445
4c.1	Test p-value	−.998989
4f.1	Confidence Interval: Pool variance	−.187210
4e.1	Assumption of analysis	.142804
4f.2	Calculation of Confidence Interval	.240334
4e.2	Verify one key assumption for the analysis.	.257029
4b.1	Distribution of test statistic	.628882
4f.3	Translate CI back into context	.699284

Table 7. Concepts and corresponding estimated difficulty for subset 5.

Item	ANOVA	Difficulty
5b.1	Null and Alternate hypotheses	−1.868861
5g.1	Confidence Interval: Width components	−.521850
5f.1	Inference made from analysis	−.512741
5c.1	Assumption of analysis	−.540048
5d.1	Test p-value	−.281008
5g.2	Translate CI back into context	−.017847
5e.1	ANOVA TABLE: SUM sq total	.103894
5e.2	ANOVA TABLE: Complete Table	.186920
5a.1	Knowledge of data gathering	1.069715
5f.2	Validate inference	1.570322
5h.1	Confidence Interval: Width + Sample Size calculation	1.886909
5d.2	Inference made from analysis	2.062039
5h.2	Confidence Interval: New degrees of freedom	2.321119
5c.2	Validate assumption	2.479875
5i.1	Inference made from analysis	3.722651

Table 8. Concepts and corresponding estimated difficulty for subset 6.

	Subset 6
Item	Subset 6:Linear regression	Difficulty
6ab.1	State linear regression equation	.704380
6ab.2	Inference made from model	1.339925
6c.1	Assumptions of model	1.361929
6ab.3	Linear regression calculations	1.864438
6ab.4	Calculations from understanding context	2.939116
6c.2	Validate assumptions of model	3.423044

Figure 4. Proposed Hierarchy for Statistical Literacy.