Using the standard Wald confidence interval for a population proportion hypothesis test is a common mistake

Employing a Wald confidence interval to test hypotheses about population proportions could lead to an increase in Type I or Type II errors unless the hypothesized value, p0, is used in computing its standard error rather than the sample proportion.     

Coke or Pepsi?

A binomial experiment dealing with a volunteer's ability to distinguish Coca-Cola from Pepsi-Cola is used to introduce the concepts of hypothesis testing and Type I and Type II errors.     

Presentation of Type I and Type II Error Rates to Non‐statisticians

This article describes a simple computer program which graphically demonstrates both Type I and Type II statistical errors.     

Using Johnson's Transformation With Approximate Test Statistics for the Simple Regression Slope Homogeneity

The authors used Johnson's transformation with approximate test statistics to test the homogeneity of simple linear regression slopes when both x_ij and x_ij may have nonnormal distributions and there is Type I heteroscedasticity, Type II heteroscedasticity, or complete heteroscedasticity. The test statistic t was first transformed by Johnson's method for each group to correct the nonnormality and to correct the heteroscedasticity; also an approximate test, such as the Welch test or the DeShon-Alexander test, was applied to test the homogeneity of the regression slopes. Computer simulations showed that the proposed technique can control Type I error rate under various circumstances. Finally, the authors provide an example to demonstrate the calculation.     

Overcoming challenges in learning probability vocabulary

Students can struggle to understand and use terms that describe probabilities. Such struggles lead to difficulties comprehending classroom conversations. In this article, we describe some specific misunderstandings a group of students (ages 11–12) held in regard to vocabulary such as certain, likely and unlikely. We discuss our efforts to help the students use such terms appropriately. In particular, we show how engaging students in a game requiring the use of the terms helped them begin to develop their vocabulary more fully. We also explain how a probability ladder visual organizer helped students begin to organize their thinking about the meanings of terms relative to one another.     

Revisiting the Model Size Effect in Structural Equation Modeling

Fitting a large structural equation modeling (SEM) model with moderate to small sample sizes results in an inflated Type I error rate for the likelihood ratio test statistic under the chi-square reference distribution, known as the model size effect. In this article, we show that the number of observed variables (p) and the number of free parameters (q) have unique effects on the Type I error rate of the likelihood ratio test statistic. In addition, the effects of p and q cannot be fully explained using degrees of freedom (df). We also evaluated the performance of 4 correctional methods for the model size effect, including Bartlett’s (1950), Swain’s (1975), and Yuan’s (2005) corrected statistics, and Yuan, Tian, and Yanagihara’s (2015) empirically corrected statistic. We found that Yuan et al.’s (2015) empirically corrected statistic generally yields the best performance in controlling the Type I error rate when fitting large SEM models.     

Teaching principles of inference with ANOVA

Analysis of variance (ANOVA) is a test of mean differences, but the reference to variances in the name is often overlooked. Classroom activities are presented to illustrate how ANOVA works with emphasis on how to think critically about inferential reasoning.     

Having a Ball with Confidence Intervals

Confidence intervals are often misinterpreted. This article describes an activity which helps students understand what it means to have confidence in a Process.     

The Place of the Fable in the Character Training of Children

Type I error rate and power for the t test, Wilcoxon-Mann-Whitney (U) test, van der Waerden Normal Scores (NS) test, and Welch-Aspin-Satterthwaite (W) test were compared for two independent random samples drawn from nonnormal distributions. Data with varying degrees of skewness (S) and kurtosis (K) were generated using Fleishman's (1978) power function. Five sample size combinations were used with both equal and unequal variances. For nonnormal data with equal variances, the power of the U test exceeded the power of the t test regardless of sample size. When the sample sizes were equal but the variances were unequal, the t test proved to be the most powerful test. When variances and sample sizes were unequal, the W test became the test of choice because it was the only test that maintained its nominal Type I error rate.     

Efficiency in Minitab

This simple Minitab simulation exercise illustrates the concept of efficiency of estimators. Students compare the sampling distributions of the sample mean, median and mid-range from a normal distribution and from a uniform distribution. The striking differences observed give rise to much discussion.     

Why is S a Biased Estimate of σ?

This article outlines an experiment that leads one to discover and verify that S is a biased estimator of σ.     

Lexical ambiguity: making a case against spread

We argue for decreasing the use of the word spread when describing the statistical idea of dispersion or variability in introductory statistics courses. In addition, we argue for increasing the use of the word variability as a replacement for spread.     

Random Permutation Testing Applied to Measurement Invariance Testing with Ordered-Categorical Indicators

We describe and evaluate a random permutation test of measurement invariance with ordered-categorical data. To calculate a p-value for the observed (?)χ², an empirical reference distribution is built by repeatedly shuffling the grouping variable, then saving the χ² from a configural model, or the ?χ² between configural and scalar-invariance models, fitted to each permuted dataset. The current gold standard in this context is a robust mean- and variance-adjusted ?χ² test proposed by Satorra (2000), which yields inflated Type I errors, particularly when thresholds are asymmetric, unless samples sizes are quite large (Bandalos, 2014; Sass et al., 2014). In a Monte Carlo simulation, we compare permutation to three implementations of Satorra’s robust χ² across a variety of conditions evaluating configural and scalar invariance. Results suggest permutation can better control Type I error rates while providing comparable power under conditions that the standard robust test yields inflated errors.     

Pattern analysis on the PIAT

Reynolds and Gutkin's approach to pattern analysis on the PIAT is modified to allow for the fact that multiple comparisons are involved. The modified strategy uses Bonferroni is ts to control the familywise Type I error rate.     

Web‐Based Project and Key Skills Work

Pupils in England and Wales are increasingly being asked to undertake investigative‐type work, be it the new compulsory projects in data handling for GCSE Mathematics (age 14–16) (see Browne 2002 ) or the Key Skills topic application of number. This article shows how teachers can generate realistic project scenarios using real data and produce indicative model solutions from the same data. The projects range from simple presentational problems for data, through hypothesis testing to complex modelling scenarios.     

What is The International Statistical Institute?

This is the second of a short series of articles briefly describing the organisations that sponsor Teaching Statistics.     

Teacher education and the choice to enter the teaching profession: A prospective study

This two-wave survey study aimed at testing a hypothetical model of teacher education graduates’ decisions about whether or not to take a teaching position upon graduation. The model focuses on the relationship between teacher education and graduates’ choice on job entrance. Using path analysis and logistic regression, this model was tested in a sample of student teachers (n = 436), subsequently graduates (n = 251) of teacher training for secondary education. The results validate the relationship between teacher education variables and nearly graduates’ intention to enter the teaching profession. Furthermore, this intention proves an imperative predictor of graduates’ actual entrance.     

The Most Compact Subdomain of a Continuous Probability Distribution

This article describes the most compact 100c% interval for a probability density for 0 < c < 1. An unbiased, distribution‐free estimator when c = 0.5 is presented.     

A Visual Model for the Variance and Standard Deviation

This paper shows how the variance and standard deviation can be represented graphically by looking at each squared deviation as a graphical object – in particular, as a square. A series of displays show how the standard deviation is the size of the average square.     

Missing Data and Multiple Imputation in the Context of Multivariate Analysis of Variance

Multivariate analysis of variance (MANOVA) is widely used in educational research to compare means on multiple dependent variables across groups. Researchers faced with the problem of missing data often use multiple imputation of values in place of the missing observations. This study compares the performance of 2 methods for combining p values in the context of a MANOVA, with the typical default for dealing with missing data: listwise deletion. When data are missing at random, the new methods maintained the nominal Type I error rate and had power comparable to the complete data condition. When 40% of the data were missing completely at random, the Type I error rates for the new methods were inflated, but not for lower percents.