Assessing Knowledge Structures in a Constructive Statistical Learning Environment

Figures & data

Figure 1. A histogram of the total recall scores for the concept of confidence intervals.

Table 1. Contribution of each category to the total recall score for the concept of confidence intervals.

	Mean score	Standard error	Proportional contribution (%) in the concept map	Proportional contribution (%) in the student responses
Statistical term	0.58	0.06	5.27	79
Formula	0.07	0.02	0.64	9
Procedure	0.00	0.00	0.00	0
Condition	0.00	0.00	0.00	0
Interpretation	0.09	0.03	0.81	12
Theoretical background	0.00	0.00	0.00	0
Rest category	0.00	0.00	0.00	0
Total	0.74		6.72	100

Table 2. Prototypical examples of incompletely and incorrectly recalled elements for the concept of confidence intervals.

Incomplete	Incorrect
If the outcome lies within the acceptance area, the null hypothesis is not rejected.	… A confidence interval is chosen: α.
A confidence interval can be used to determine the plausibility of the null hypothesis.	The level of significance depends on the confidence interval.
If you have determined the confidence interval it is possible to see whether or not the null hypothesis should be rejected.	½ α = confidence interval

Figure 2. A histogram of the total recall scores for the concepts of one-sample z- and t-tests.

Table 3. Contribution of each category to the total recall score for the concept of one- sample z- and t-tests.

	Mean score	Standard error	Proportional contribution (%) in the concept map	Proportional contribution (%) in the student responses
Statistical term	5.16	0.23	17.20	54
Formula	2.43	0.20	8.10	26
Procedure	0.33	0.06	1.10	3
Condition	0.63	0.06	2.10	7
Interpretation	0.88	0.11	2.93	9
Theoretical background	0.00	0.00	0.00	0
Rest category	0.05	0.02	0.17	1
Total	9.48		31.60	100

Table 4. Prototypical examples of incompletely and incorrectly recalled elements for the concepts of one-sample z- and t-tests.

Incomplete	Incorrect
The t-test is used if the standard deviation of the population is unknown.	Reject the null hypothesis if the p-value = α.
If the calculated value falls within the critical area, the null hypothesis should be rejected.	The alternative hypothesis is the assumption that is tested.
T-tests and z-test are used to determine the reliability of the null hypothesis.	α = the probability that the null hypothesis is rejected when the null hypothesis is wrong.

Figure 3. A histogram for the total recall scores for the concept errors in statistical inference.

Table 5. Contribution of each category to the total recall score for the concept errors in statistical inference.

	Mean score	Standard error	Proportional contribution (%) in the concept map	Proportional contribution (%) in the student responses
Statistical term	1.37	0.12	9.79	63
Formula	0.46	0.07	3.29	21
Procedure	0.03	0.03	0.21	1
Condition	0.00	0.00	0.00	0
Interpretation	0.33	0.07	2.36	15
Theoretical background	0.00	0.00	0.00	0
Rest category	0.00	0.00	0.00	0
Total	2.19		15.65	100

Table 6. Prototypical examples of incompletely and incorrectly recalled informational elements for the concept of errors in statistical inference.

Incomplete	Incorrect
α : the null hypothesis is rejected, although the null hypothesis is correct.	Type 1 error: reject the null hypothesis when the null hypothesis is incorrect.
β: the null hypothesis is not rejected although the null hypothesis is incorrect.	Type 2 error: reject the null hypothesis when the null hypothesis is correct.
Power, 1 – β, is somehow related to the alternative hypothesis.	One kind of error occurs in case of a disproportional large α. This will result in a faulty decision.

Table

Terms	Formula(s)	Procedure	Conditions of application	Interpretation	Theoretical Background
Confidence interval (95%) Significance level (α) Population mean Sample mean z-score Standard error		Arithmetic procedure for calculating the confidence interval formula.	A confidence interval can be used if one would like to determine, with an a-priori stated reliability, the range of the population mean on the basis of a sample mean.	There is a 95% chance that the population parameter falls within the given confidence interval. or Of 100 samples with sample-size n, 95 sample-means will fall within the confidence interval.	The systematic behavior of sample size means obtained from samples with a large enough sample size.

Table

Terms	Formula(s)	Procedure	Conditions of application	Interpretation	Theoretical Background
Null hypothesis Alternative Hypothesis z-test t-test One-tailed testing Two-tailed testing Population mean Sample mean Standard error p-value Significance level Acceptance area Critical area Degrees of freedom	H0 H1 α n −1	Arithmetic procedures for calculating the z- and t-test.	If one would like to see whether or not a sample mean differs significantly from a known or a pre-supposed population mean. If both the population mean and the standard deviation are known, then use the z-test. If the population mean and the population standard deviation are unknown then a t-test should be used. If n > 30 then the t-test can be approached as a z-test.	Null hypothesis: The assumption that there is no difference between the population and the sample mean. Alternative hypothesis: This hypothesis states that there is a difference between the population and the sample mean. The p-value is the probability of finding the given or a more extreme outcome, provided that the null hypothesis is correct. If the p-value = α, then reject the null hypothesis. If the p-value = α, then do not reject the null hypothesis.	The systematic behavior of sample size means obtained from samples with a large enough sample size.

Table

Terms	Formula(s)	Procedure	Conditions of application	Interpretation	Theoretical Background
Alpha (significance level) Beta Power Type 1 error Type 2 error	α β 1 – β	Arithmetic procedure for calculating the power.		Type 1 error: Reject the null hypothesis when the null hypothesis was in fact correct. Type 2 error: Failing to reject the null hypothesis when the null hypothesis was incorrect. α: The probability of making a Type 1 error. β: The probability of making a Type 2 error. Power: The power of a test, that is, the probability of indicating a significant difference when there is a difference.